Math 104: Calculus I – Homework
Section 004 - Spring 2014
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1.E: Functions and Graphs (Exercises)
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1.1: Review of Functions
For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.
Solution: a. Domain = {\(−3,−2,−1,0,1,2,3\)}, range = {\(0,1,4,9\)}
b. Yes, a function
Solution:a. Domain = {\(0,1,2,3\)}, range = {\(−3,−2,−1,0,1,2,3\)}
b. No, not a function
Solution: a. Domain = {\(3,5,8,10,15,21,33\)}, range = {\(0,1,2,3\)}
For the following exercises, find the values for each function, if they exist, then simplify.
a. \(f(0)\) b. \(f(1)\) c. \(f(3)\) d. \(f(−x)\) e. \(f(a)\) f. \(f(a+h)\)
1) \(f(x)=5x−2\)
Solution: a. \(−2\) b. \(3\) c. \(13\) d. \(−5x−2\) e. \(5a−2\) f. \(5a+5h−2\)
2) \(f(x)=4x^2−3x+1\)
3) \(f(x)=\frac{2}{x}\)
Solution: a. Undefined b. \(2\) c. \(23\) d. \(−\frac{2}{x}\) e \(\frac{2}{a}\) f. \(\frac{2}{a+h}\)
4) \(f(x)=|x−7|+8\)
5) \(f(x)=\sqrt{6x+5}\)
Solution: a. \(\sqrt{5}\) b. \(\sqrt{11}\) c. \(\sqrt{23}\) d. \(\sqrt{−6x+5}\) e. \(\sqrt{6a+5}\) f. \(\sqrt{6a+6h+5}\)
6) \(f(x)=\frac{x−2}{3x+7}\)
7) \(f(x)=9\)
Solution: a. 9 b. 9 c. 9 d. 9 e. 9 f. 9
For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
1) \(f(x)=\frac{x}{x^2−16}\)
2) \(g(x)=\sqrt{8x−1}\)
Solution: \(x≥\frac{1}{8};y≥0;x=\frac{1}{8}\); no y-intercept
3) \(h(x)=\frac{3}{x^2+4}\)
4) \(f(x)=−1+\sqrt{x+2}\)
Solution: \(x≥−2;y≥−1;x=−1;y=−1+\sqrt{2}\)
5) \(f(x)=1x−\sqrt{9}\)
6) \(g(x)=\frac{3}{x−4}\)
Solution: \(x≠4;y≠0\); no x-intercept; \(y=−\frac{3}{4}\)
7) \(f(x)=4|x+5|\)
8) \(g(x)=\sqrt{\frac{7}{x−5}}\)\
Solution: \(x>5;y>0\); no intercepts
For the following exercises, set up a table to sketch the graph of each function using the following values: \(x=−3,−2,−1,0,1,2,3.\)
1) \(f(x)=x^2+1\)
2) \(f(x)=3x−6\)
3) \(f(x)=\frac{1}{2}x+1\)
4) \(f(x)=2|x|\)
5) \(f(x)=-x^2\)
6) \(f(x)=x^3\)
For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph:
Domain and range
\(x\) -intercept, if any (estimate where necessary)
\(y\)-Intercept, if any (estimate where necessary)
The intervals for which the function is increasing
The intervals for which the function is decreasing
The intervals for which the function is constant
Symmetry about any axis and/or the origin
Whether the function is even, odd, or neither
Solution: Function; a. Domain: all real numbers, range: \(y≥0\) b. \(x=±1\) c. \(y=1\) d. \(−1<x<0\) and \(1<x<∞ e\). \(−∞<x<−1\) and \(0<x<1\) f. Not constant g. \(y\)-axis h. Even
Solution: Function; a. Domain: all real numbers, range: \(−1.5≤y≤1.5\) b. \(x=0\) c. \(y=0\) d. all real numbers e. None f. Not constant g. Origin h. Odd
Solution: Function; a. Domain: \(−∞<x<∞\), range: \(−2≤y≤2\) b. \(x=0\) c. \(y=0\) d. \(−2<x<2\) e. Not decreasing f. \(−∞<x<−2\) and \(2<x<∞\) g. Origin h. Odd
Solution: Function; a. Domain: \(−4≤x≤4\), range: \(−4≤y≤4\) b. \(x=1\).2 c. \(y=4\) d. Not increasing e. \(0<x<4\) f. \(−4<x<0\) g. No Symmetry h. Neither
For the following exercises, for each pair of functions, find a. \(f+g\) b. \(f−g\) c. \(f⋅g\) d. \(f/g\). Determine the domain of each of these new functions.
1) \(f(x)=3x+4,g(x)=x−2\)
2) \(f(x)=x−8,g(x)=5x^2\)
Solution: a. \(5x^2+x−8\); all real numbers b. \(−5x^2+x−8\); all real numbers c. \(5x^3−40x^2\); all real numbers d. \(\frac{x−8}{5x^2}\);\(x≠0\)
3) \(f(x)=3x^2+4x+1,g(x)=x+1\)
4) \(f(x)=9−x^2,g(x)=x^2−2x−3\)
Solution: a. \(−2x+6\); all real numbers b. \(−2x^2+2x+12\); all real numbers c. \(−x^4+2x^3+12x^2−18x−27\); all real numbers d. \(−\frac{x+3}{x+1};x≠−1,3\)
5) \(f(x)=\sqrt{x},g(x)=x−2\)
6) \(f(x)=6+\frac{1}{x},g(x)=\frac{1}{x}\)
Solution: \(a. 6+\frac{2}{x};x≠0 b. 6; x≠0 c. 6x+\frac{1}{x^2};x≠0 d. 6x+1;x≠0\)
For the following exercises, for each pair of functions, find a. \((f∘g)(x)\) and b. \((g∘f)(x)\) Simplify the results. Find the domain of each of the results.
1) \(f(x)=3x,g(x)=x+5\)
2) \(f(x)=x+4,g(x)=4x−1\)
Solution: a. \(4x+3\); all real numbers b. \(4x+15\); all real numbers
3) \(f(x)=2x+4,g(x)=x^2−2\)
4) \(f(x)=x^2+7,g(x)=x^2−3\)
Solution:a. \(x^4−6x^2+16\); all real numbers b. \(x^4+14x^2+46\); all real numbers
5) \(f(x)=\sqrt{x}, g(x)=x+9\)
6) \(f(x)=\frac{3}{2x+1},g(x)=\frac{2}{x}\)
Solution: a. \(\frac{3x}{4+x};x≠0,−4\) b. \(\frac{4x+2}{3};x≠−12\)
7) \(f(x)=|x+1|,g(x)=x^2+x−4\)
8) The table below lists the NBA championship winners for the years 2001 to 2012.
- Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.
- Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.
Solution: a. Yes, because there is only one winner for each year.
b. No, because there are three teams that won more than once during the years 2001 to 2012.
9) [T] The area \(A\) of a square depends on the length of the side s.
1.Write a function \(A(s)\) for the area of a square.
2.Find and interpret \(A(6.5)\).
3.Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units.
10) [T] The volume of a cube depends on the length of the sides s.
Write a function \(V(s)\) for the area of a square.
Find and interpret \(V(11.8)\).
Solution: a. \(V(s)=s^3\) b. \(V(11.8)≈1643\); a cube of side length 11.8 each has a volume of approximately 1643 cubic units.
11) [T] A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost C to rent a car is a function of the hours \( t\) the car is rented plus the flat fee.
- Write the formula for the function that models this situation.
- Find the total cost to rent a car for 2 days and 7 hours.
- Determine how long the car was rented if the bill is $432.73.
12) [T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles N that can be driven depends on the amount of gas x in the tank.
1.Write a formula that models this situation.
2.Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas.
3.Determine the domain and range of the function.
4.Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.
a. \(N(x)=15x\) b. i. \(N(20)=15(20)=300\); therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. \(N(15)=225\); therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: \(0≤x≤20\); range: [\(0,300\)] d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.
13) [T] The volume V of a sphere depends on the length of its radius as \(V=(4/3)πr3\). Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius \(6.371×106\) m.
14) [T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by \(r(t)=6−\)[\(5/(t2+1)\)], where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.
1.Express the area of the bacteria as a function of time.
2.Find the exact and approximate area of the bacterial culture in 3 hours.
3.Express the circumference of the bacteria as a function of time.
4.Find the exact and approximate circumference of the bacteria in 3 hours.
Solution: a. \(A(t)=A(r(t))=π⋅(6−\frac{5}{t^2+1})^2\) b. Exact: \(\frac{121π}{4}\); approximately 95 cm2 c. \(C(t)=C(r(t))=2π(6−\frac{5}{t^2+1})\) d. Exact: \(11π\); approximately 35 cm
15) [T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function \(E(x)=0.79x\), where x is the number of U.S. dollars and \(E(x)\) is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is \(D(x)=1.245x\), where x is the number of Euros and \(D(x)\) is the equivalent number of U.S. dollars.
1.Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process?
2.Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $200 when she arrived in Paris.
16) [T] The manager at a skateboard shop pays his workers a monthly salary S of $750 plus a commission of $8.50 for each skateboard they sell.
1.Write a function \(y=S(x)\) that models a worker’s monthly salary based on the number of skateboards x he or she sells.
2.Find the approximate monthly salary when a worker sells 25, 40, or 55 skateboards.
3.Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line \(y=1400\).)
Solution: a. \(S(x)=8.5x+750\) b. $962.50, $1090, $1217.50 c. 77 skateboards
17) [T] Use a graphing calculator to graph the half-circle \(y=\sqrt{25−(x−4)^2}\). Then, use the INTERCEPT feature to find the value of both the \(x\)- and \(y\)-intercepts.
1.2: Basic Classes of Functions
For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.
1) \((-2,4)\) and \((1,1)\)
Solution: a. −1 b. Decreasing
2) \((-1,4)\) and \((3,-1)\)
3 \((3,5)\) and \((-1,2)\)
Solution: a. 3/4 b. Increasing
4) \((6,4)\) and \((4,-3)\)
5) \((2,3)\) and \((5,7)\)
Solution: a. 4/3 b. Decreasing
6) \((1,9)\) and \((-8,5)\)
7) \((2,4)\) and \((1,4)\)
Solution: a. 0 b. Horizontal
8) \((1,4)\) and \((1,0)\)
For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.
1) Slope =\(−6\), passes through \((1,3)\)
Solution: \(y=−6x+9\)
2) Slope =\(3\), passes through \((-3,2)\)
3) Slope =\(\frac{1}{3}\), passes through \((0,4)\)
Solution: \(y=\frac{1}{3}x+4\)
4) Slope =\(\frac{2}{5}\), \(x\)-intercept =\(8\)
5) Passing through \((2,1\) and \((−2,−1)\)
Solution: \(y=\frac{1}{2}x\)
6) Passing through \((−3,7)\) and \((1,2)\)
7) \(x\)-intercept =\(5\) and \(y\)-intercept =\(−3\)
Solution:\(y=\frac{3}{5}x−3\)
8) \(x\)-Intercept =−\(6\) and \(y\)-intercept =\(9\)
For the following exercises, for each linear equation, a. give the slope \(m\) and \(y\)-intercept b, if any, and b. graph the line.
1) \(y=2x−3\)
Solution: a. \((m=2,b=−3)\)
2) \(y=−\frac{1}{7}x+1\)
3) \(f(x)=-6x\)
a. \((m=−6,b=0)\)
4) \(f(x)=−5x+4\)
5) \(4y+24=0\)
Solution: a. \((m=0,b=−6)\)
6) \(8x-4=0\)
7) \(2x+3y=6\)
Solution: a. \((m=−\frac{2}{3},b=2)\)
8) \(6x−5y+15=0\)
For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the \(y\)-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither.
1) \(f(x)=2x^2−3x−5\)
Solution: a. 2 b. \(\frac{5}{2}\),−1; c. −5 d. Both ends rise e. Neither
2) \(f(x)=−3x^2+6x\)
3) \(f(x)=\frac{1}{2}x^2−1\)
Solution: a. 2 b. ±\(\sqrt{2}\) c. −1 d. Both ends rise e. Even
4) \(f(x)=x^3+3x^2−x−3\)
5) \(f(x)=3x−x^3\)
Solution: a. 3 b. 0, ±\(\sqrt{3}\) c. 0 d. Left end rises, right end falls e. Odd
For the following exercises, use the graph of \(f(x)=x^2\) to graph each transformed function \(g\).
1) \(g(x)=x^2−1\)
2) \(g(x)=(x+3)^2+1\)
For the following exercises, use the graph of \(f(x)=\sqrt{x}\) to graph each transformed function \(g\).
1) \(g(x)=\sqrt{x+2}\)
2) \(g(x)=−\sqrt{x}−1\)
For the following exercises, use the graph of \(y=f(x)\) to graph each transformed function \(g\).
1) \(g(x)=f(x)+1\)
2) \(g(x)=f(x−1)+2\)
For the following exercises, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph.
1) \(f(x)=\begin{cases}4x+3, & x≤0\\ -x+1, & x>0\end{cases} ;f(−3);f(0);f(2)\)
2) \(f(x)=\begin{cases}x^2-3, & x≤0\\ 4x+3, & x>0\end{cases} ;f(−4);f(0);f(2)\)
Solution: a. \(13,−3,5\)
3) \(h(x)=\begin{cases}x+1, &x≤5\\4, &x>5\end{cases} ;h(0);h(π);h(5)\)
4) \(g(x)=\begin{cases}\frac{3}{x−2}, &x≠2\\4, &x=2\end{cases} ;g(0);g(−4);g(2)\)
Solution: a. \(\frac{−3}{2},\frac{−1}{2},4\)
For the following exercises, determine whether the statement is true or false . Explain why.
1) \(f(x)=(4x+1)/(7x−2)\) is a transcendental function.
2) \(g(x)=\sqrt[3]{x}\) is an odd root function
Solution: True; \(n=3\)
3) A logarithmic function is an algebraic function.
4) A function of the form \(f(x)=x^b\), where \(b\) is a real valued constant, is an exponential function.
Solution: False; \(f(x)=x^b\), where \(b\) is a real-valued constant, is a power function
5) The domain of an even root function is all real numbers.
6) [T] A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300.
1.Find a function \(y=V(t)\) that determines the value V of the equipment at the end of t years.
2.Find and interpret the meaning of the \(x\)- and \(y\)-intercepts for this situation.
3.What is the value of the equipment at the end of 5 years?
4.When will the value of the equipment be $3000?
Solution: a. \(V(t)=−2733t+20500\) b. \((0,20,500)\) means that the initial purchase price of the equipment is $20,500; \((7.5,0)\) means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years
7) [T] Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 \((t=0)\),total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion.
1. Find a linear function S that estimates the total online holiday sales in the year t.
2. Interpret the slope of the graph of S.
3. Use part a. to predict the year when online shopping during Christmas will reach $60 billion.
8) [T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost \(C\) as a function of number of cupcakes made.
1.Find a linear function that relates cost C to x, the number of cupcakes made.
2.Find the cost to bake 160 cupcakes.
3.If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)
Solution: a. \(C=0.75x+125\) b. $245 c. 167 cupcakes
9) [T] A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years.
1. Find a linear function that models the price P of the house versus the number of years t since the original purchase.
2. Interpret the slope of the graph of P.
3. Find the price of the house 15 years from when it was originally purchased.
10) [T] A car was purchased for $26,000. The value of the car depreciates by $1500 per year.
1. Find a linear function that models the value V of the car after t years.
2. Find and interpret \(V(4)\).
Solution: a. \(V(t)=−1500t+26,000\) b. In 4 years, the value of the car is $20,000.
11) [T] A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation.
12) [T] The total cost C (in thousands of dollars) to produce a certain item is modeled by the function \(C(x)=10.50x+28,500\), where x is the number of items produced. Determine the cost to produce 175 items.
Solution: $30,337.50
13) [T] A professor asks her class to report the amount of time t they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four-page assignment and about 1.5 hours to type a nine-page assignment.
1. Find the linear function \(y=N(t)\) that models this situation, where \(N\) is the number of pages typed and t is the time in minutes.
2. Use part a. to determine how many pages can be typed in 2 hours.
3. Use part a. to determine how long it takes to type a 20-page assignment.
14) [T] The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function \(P(t)=1.8576t+68.052\), where t is time in years and \(t=0\) corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.
Solution: 96% of the total capacity
15) [T] The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll.
1. Find the linear function \(y=N(x)\), where \(N\) is the number of students that actually enroll and \(x\) is the number of all students offered admission to the class of 2019.
2. If the university wants the 2019 freshman class size to be 1350, determine how many students should be admitted.
1.3: Trigonometric Functions
For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of \(π\).
1) \(240°\)
Solution: \(\frac{4π}{3} rad\)
2) \(15°\)
3) \(60°\)
Solution: \(\frac{-π}{3} rad\)
4) \(-225°\)
5) \(330°\)
Solution: \(\frac{11π}{6} rad\)
For the following exercises, convert each angle in radians to degrees.
1) \(\frac{π}{2} rad\)
2) \(\frac{7π}{6} rad\)
Solution: \(210°\)
3) \(\frac{11π}{2} rad\)
4) \(-3π rad\)
Solution:\(-540°\)
5) \(\frac{5π}{12} rad\)
Evaluate the following functional values.
1) \(cos(\frac{4π}{3}\))
Solution: -0.5
2) \(tan(\frac{19π}{4}\))
3) \(sin(-\frac{3π}{4}\))
Solution: \(-\frac{sqrt{2}}{2}\)
4) \(sec(-\frac{π}{6}\))
5) \(sin(-\frac{π}{12}\))
Solution: \(\frac{\sqrt{3}-1}{2\sqrt{2}}\)
6) \(cos(-\frac{5π}{12}\))
For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place.
1) \(a=4, c=7)\)
Solution: \(a. b=5.7 b. sinA=\frac{4}{7},cosA=\frac{5.7}{7},tanA=\frac{4}{5.7} ,cscA=\frac{7}{4} ,secA=\frac{7}{5.7} ,cotA=\frac{5.7}{4}\)
2) \(a=21, c=29)\)
3) \(a=85.3, b=125.5)\)
Solution: \(a. c=151.7 b. sinA=0.5623,cosA=0.8273,tanA=0.6797,cscA=1.778,secA=1.209,cotA=1.471\)
4) \(b=40, c=41)\)
5) \(a=84, b=13)\)
Solution: \(a. c=85 b. sinA=\frac{84}{85},cosA=\frac{13}{85}, tanA=\frac{84}{13}, cscA=\frac{85}{84} ,secA=\frac{85}{13} ,cotA=\frac{13}{84}\)
6) \(b=28, c=35)\)
For the following exercises, \(P\) is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle \(θ\) with a terminal side that passes through point \(P\). Rationalize denominators.
1) \(P(\frac{7}{25},y), y>0\)
Solution:\(a.y=\frac{24}{25}b.sinθ=\frac{24}{25},cosθ=\frac{7}{25},tanθ=\frac{24}{7},cscθ=\frac{25}{24} , secθ=\frac{25}{7},cotθ=\frac{7}{24}\)
2) \(P(\frac{-15}{17},y), y>0\)
3) \(P(\frac{x}{\frac{\sqrt{7}}{3}}), y>0\)
Solution: a. \(x=−\frac{\sqrt{2}}{3} b. sinθ=\frac{\sqrt{7}}{3} ,cosθ=\frac{−\sqrt{2}}{3} ,tanθ=\frac{\sqrt{−14}}{2},cscθ=\frac{3\sqrt{7}}{7},secθ=\frac{−3\sqrt{2}}{2} ,cotθ=\frac{−\sqrt{14}}{7}\)
4) \(P(\frac{x}{\frac{-\sqrt{15}}{4}}), y>0\)
For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.
1) \(tan^2x+sinxcscx\)
Solution: \(sec^2x\)
2) \(secxsinxcotx\)
3)\(\frac{tan^2x}{sec^2x}\)
Solution: \(sin^2x\)
4) \(secx-cosx\)
5) \((1+tanθ)^2-2tanθ\)
Solution: sex^2θ
6) \(sinx(cscx-sinx)\)
7) \(\frac{cos t}{sin t}+\frac{sin t}{1+cos t}\)
Solution: \(1/sin t) = csc t\)
8) \(\frac{1+tan^2α}{1+cot^2α}\)
For the following exercises, verify that each equation is an identity.
1) \(\frac{tanθcotθ}{cscθ}=sinθ\)
2) \(\frac{sec^2θ}{tanθ}=secθcscθ\)
3) \(\frac{sin t}{csc t} + \frac{cos t}{sec t} = 1\)
4) \(\frac{sinx}{cosx+1}+\frac{cosx−1}{sinx}=0\)
5) \(cotγ+tanγ=secγcscγ\)
6) \(sin^2β+tan^2β+cos^2β=sec^2β\)
7) \(\frac{1}{1−sinα}+\frac{1}{1+sinα}=2sec^2α\)
8)\(\frac{tanθ−cotθ}{sinθcosθ}=sec^2θ−csc^2θ\)
For the following exercises, solve the trigonometric equations on the interval \(0≤θ<2π.\)
1) \(2sinθ−1=0\)
Solution: {\(\frac{π}{6},\frac{5π}{6}\)}
2) \(1+cosθ=\frac{1}{2}\)
3) \(2tan^2θ=2\)
Solution: {\(\frac{π}{4},\frac{3π}{4},\frac{5π}{4},\frac{7π}{4}\)}
4) \(4sin^2θ−2=0\)
5) \(\sqrt{3}cotθ+1=0\)
Solution: {\(\frac{2π}{3},\frac{5π}{3}\)}
6) \(3secθ−2\sqrt{3}=0\)
7) \(2cosθsinθ=sinθ\)
Solution: {\(0,π,\frac{π}{3},\frac{5π}{3}\)}
8) \(csc^2θ+2cscθ+1=0\)
For the following exercises, each graph is of the form \(y=AsinBx\) or \(y=AcosBx\), where \(B>0\). Write the equation of the graph.
Solution: \(y=4sin(\frac{π}{4}x)\)
Solution: \(y=cos(2πx)\)
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
1) \(y=sin(x−\frac{π}{4})\)
Solution: \(a. 1 b. 2 π c. \frac{π}{ 4}\) units to the right
2) \(y=3cos(2x+3)\)
3) \(y=−\frac{1}{2}sin(\frac{1}{4}x)\)
Solution: \(a. \frac{1}{2} b. 8π c. No phase shift\)
4) \(y=2cos(x−\frac{π}{3})\)
5) \(y=−3sin(πx+2)\)
Solution: \( a. 3 b. 2 c. \frac{2}{π}\) units to the left
6) \(y=4cos(2x−\frac{π}{2})\)
1) [T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of \(120\)°, how many inches does it move? Approximate to the nearest whole inch.
Solution: Approximately 42 in.
2) [T] Find the length of the arc intercepted by central angle \(θ\) in a circle of radius \(r\). Round to the nearest hundredth.
a. \(r=12.8\) cm, \(θ=5π6\) rad b. \(r=4.378\) cm, \(θ=7π6\) rad c. \(r=0.964\) cm, \(θ=50\)° d. \(r=8.55\) cm, \(θ=325\)°
3) [T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, \(ω\), and is given by \(ω=θ/t\), where \(θ\) is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.
a. \(θ=\frac{7π}{4}\) rad, \(t=10\) sec b. \(θ=\frac{3π}{5}\) rad, \(t=8\) sec c. \(θ=\frac{2π}{9}\) rad, \(t=1\) min d. \(θ=23.76\) rad, \(t=14\) min
Solution: \(a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min\)
4) [T] A total of 250,000 m2 of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.
a)Find the radius of the circular land area.
b)If the land area is to form a \(45\)° sector of a circle instead of a whole circle, find the length of the curved side.
5) [T] The area of an isosceles triangle with equal sides of length x is \(\frac{1}{2}x^2sinθ\),
where \(θ\) is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle \(θ=5π/12\) rad.
Solution: \(≈30.9in^2\)
6) [T] A particle travels in a circular path at a constant angular speed \(ω\). The angular speed is modeled by the function \(ω=9|cos(πt−π/12)|\). Determine the angular speed at \(t=9\) sec.
7) [T] An alternating current for outlets in a home has voltage given by the function
\(V(t)=150cos368t\),
where V is the voltage in volts at time t in seconds.
a) Find the period of the function and interpret its meaning.
b) Determine the number of periods that occur when 1 sec has passed.
Solution: a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods
8) [T] The number of hours of daylight in a northeast city is modeled by the function
\(N(t)=12+3sin[\frac{2π}{365}(t−79)]\),
where t is the number of days after January 1.
a) Find the amplitude and period.
b) Determine the number of hours of daylight on the longest day of the year.
c) Determine the number of hours of daylight on the shortest day of the year.
d) Determine the number of hours of daylight 90 days after January 1.
e) Sketch the graph of the function for one period starting on January 1.
9) [T] Suppose that \(T=50+10sin[\frac{π}{12}(t−8)]\) is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week.
a) Determine the amplitude and period.
b) Find the temperature 7 hours after midnight.
c) At what time does \(T=60\)°?
d) Sketch the graph of \(T\) over \(0≤t≤24\).
Solution: a. Amplitude = \(10;period=24\) b. \(47.4°F\) c. 14 hours later, or 2 p.m. d.
10) [T] The function \(H(t)=8sin(\frac{π}{6}t)\) models the height H (in feet) of the tide t hours after midnight. Assume that \(t=0\) is midnight.
b) Graph the function over one period.
c) What is the height of the tide at 4:30 a.m.?
1.4: Inverse Functions
For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.
Solution: Not one-to-one
Solution: One-to-one
For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.
1) \(f(x)=x^2−4 ,x≥0\)
Solution: a. \(f^{−1}(x)=\sqrt{x+4}\) b. Domain : \(x≥−4\),range:\(y≥0\)
2) \(f(x)=\sqrt[3]{x−4}\)
3) \(f(x)=^3+1\)
Solution: a. \(f^{−1}(x)=\frac{3}{x−1}\) b. Domain: all real numbers, range: all real numbers
4) \(f(x)=(x−1)^2, x≤1\)
5) \(f(x)=\sqrt{x−1}\)
Solution: a. \(f^{−1}(x)=x^2+1\), b. Domain: \(x≥0\), range: \(y≥1\)
6) \(f(x)=\frac{1}{x+2}\)
For the following exercises, use the graph of f to sketch the graph of its inverse function.
For the following exercises, use composition to determine which pairs of functions are inverses.
1) \(f(x)=8x, g(x)=\frac{x}{8}\)
Solution: These are inverses.
2) \(f(x)=8x+3, g(x)=\frac{x-3}{8}\)
3) \(f(x)=5x−7,g(x)=\frac{x+5}{7}\)
Solution: These are not inverses.
4) \(f(x)=\frac{2}{3}x+2, g(x)=\frac{3}{2}x+3\)
5) \(f(x)=\frac{1}{x−1}, x≠1, g(x)=\frac{1}{x}+1,x≠0\)
6) \(f(x)=x^3+1,g(x)=(x−1)^{1/3}\)
7) \(f(x)=x^2+2x+1,x≥−1, g(x)=−1+\sqrt{x},x≥0\)
8) \(f(x)=\sqrt{4−x^2},0≤x≤2, g(x)=\sqrt{4−x^2},0≤x≤2\)
For the following exercises, evaluate the functions. Give the exact value.
1) \(tan^{−1}(\frac{\sqrt{3}}{3})\)
Solution: \(\frac{π}{6}\)
2) \(cos^{−1}(−\frac{\sqrt{2}}{2})\)
3) \(cot^{−1}(1)\)
Solution: \(\frac{π}{4}\)
4) \(sin^{−1}(−1)\)
5) \(cos^{−1}(\frac{\sqrt{3}}{2})\)
6) \(cos(tan^{−1}(\sqrt{3}))\)
7) \(sin(cos^{−1}(\frac{\sqrt{2}}{2}))\)
Solution: \(\frac{\sqrt{2}}{2}\)
8) \(sin^{−1}(sin(\frac{π}{3}))\)
9) \(tan^{−1}(tan(−\frac{π}{6}))\)
Solution: \(-\frac{π}{6}\)
1) The function \(C=T(F)=(5/9)(F−32)\) converts degrees Fahrenheit to degrees Celsius.
a) Find the inverse function \(F=T^{−1}(C)\)
b) What is the inverse function used for?
2) [T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function \(V=f(x)=500(0.04−x^2)\) for \(0≤x≤0.2.\)
a) Find \(x=f^{−1}(V).\)
b) Interpret what the inverse function is used for.
c) Find the distance from the center of an artery with a velocity of 15 cm/sec, 10 cm/sec, and 5 cm/sec.
Solution: a. \(x=f^{−1}(V)\)=\sqrt{0.04−\frac{V}{500}}\) b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V. c. 0.1 cm; 0.14 cm; 0.17 cm
3) A function that converts dress sizes in the United States to those in Europe is given by \(D(x)=2x+24.\)
a) Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States.
b) Find the function that converts European dress sizes to U.S. dress sizes.
c) Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70.
4) [T] The cost to remove a toxin from a lake is modeled by the function \(C(p)=75p/(85−p),\) where \(C\) is the cost (in thousands of dollars) and \(p\) is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.
a) Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake.
b) Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.
Solution: a. $31,250, $66,667, $107,143 b. (\(p=\frac{85C}{C+75}\)) c. 34 ppb
5) [T] A race car is accelerating at a velocity given by \(v(t)=\frac{25}{4}t+54,\)
where v is the velocity (in feet per second) at time t.
a) Find the velocity of the car at 10 sec.
b) Find the inverse function.
c) Use part b. to determine how long it takes for the car to reach a speed of 150 ft/sec.
6) [T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by \(μ=2sin^{−1}(\frac{1}{M}).\)
Find the Mach angle (to the nearest degree) for the following Mach numbers.
a. μ=1.4
b. μ=2.8
c. μ=4.3
Solution: a. \(~92°\) b. \(~42°\) c. \(~27°\)
7) [T] Using \(μ=2sin^{−1}(\frac{1}{M})\), find the Mach number M for the following angles.
a. μ=\(\frac{π}{6}\)
b. μ=\(\frac{2π}{7}\)
c. μ=\(\frac{3π}{8}\)
8) [T] The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function
\(T(x)=5+18sin[\frac{π}{}6(x−4.6)],\)
where \(x\) is time in months and \(x=1.00\) corresponds to January 1. Determine the month and day when the temperature is \(21°C.\)
Solution: \(x≈6.69,8.51\); so, the temperature occurs on June 21 and August 15
9) [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function \(D(t)=5sin(\frac{π}{6}t−\frac{7π}{6})+8,\) where \(t\) is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft.
10) [T] An object moving in simple harmonic motion is modeled by the function \(s(t)=−6cos(\frac{πt}{2}),\) where \(s\) is measured in inches and t is measured in seconds. Determine the first time when the distance moved is 4.5 ft.
Solution: \(~1.5\) sec
11) [T] A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle \(θ\) can be modeled by the function \(θ=tan^{−1}\frac{5.5}{x}−tan^{−1}\frac{2.5}{x}\), where \(x\) is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.
12) [T] Use a calculator to evaluate \(tan^{−1}(tan(2.1))\) and \(cos^{−1}(cos(2.1))\). Explain the results of each.
Solution: \(tan^{−1}(tan(2.1))≈−1.0416\); the expression does not equal \(2.1\) since \(2.1>1.57=\frac{π}{2}\)—in other words, it is not in the restricted domain of \(tanx\). \(\cos^{−1}(cos(2.1))=2.1\), since \(2.1\) is in the restricted domain of \(cosx\).
13) [T] Use a calculator to evaluate \(sin(sin^{−1}(−2))\) and \(tan(tan^{−1}(−2))\). Explain the results of each.
1.5: Exponential and Logarithmic Functions
For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.
1) \(f(x)=5^x\) a. \(x=3\) b. \(x=\frac{1}{2}\) c. \(x=\sqrt{2}\)
Solution: \(a. 125 b. 2.24 c. 9.74\)
2) \(f(x)=(0.3)^x\) a. \(x=−1\) b. \(x=4\) c. \(x=−1.5\)
3) \(f(x)=10^x\) a. \(x=−2\) b. \(x=4\) c. \(x=\frac{5}{3}\)
Solution: \(a. 0.01 b. 10,000 c. 46.42\)
4) \(f(x)=e^x\) a. \(x=2\) b. \(x=−3.2\) c. \(x=π\)
For the following exercises, match the exponential equation to the correct graph.
a. \(y=4^{−x}\)
b. \(y=3^{x−1}\)
c. \(y=2^{x+1}\)
d. \(y=(\frac{1}{2})^x+2\)
e. \(y=−3^{−x}\)
f. \(y=1−5^x\)
Solution: d
Solution: b
Solution: e
For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.
1) \(f(x)=e^x+2\)
Solution: Domain: all real numbers, range: \((2,∞),y=2\)
2) \(f(x)= −2^x\)
3) \(f(x)=3^{x+1}\)
Solution: Domain: all real numbers, range: \((0,∞),y=0\)
4) \(f(x)=4^x−1\)
5) \(f(x)=1−2^{−x}\)
Solution: Domain: all real numbers, range: \((−∞,1),y=1\)
6) \(f(x)=5^{x+1}+2\)
7) \(f(x)=e^{−x}−1\)
Solution: Domain: all real numbers, range: \((−1,∞),y=−1\)
For the following exercises, write the equation in equivalent exponential form.
1) \(log_381=4\)
2) \(log_82=\frac{1}{3}\)
Solution: \(8^{1/3}=2\)
3) \(log_51=0\)
4) \(log_525=2\)
Solution: \(5^2=25\)
5) \(log0.1=−1\)
6) \(ln(\frac{1}{e^3})=−3\)
Solution: \(e^{−3}=\frac{1}{e^3}\)
7) \(log_93=0.5\)
8) \(ln1=0\)
Solution: \(e^0=1\)
For the following exercises, write the equation in equivalent logarithmic form.
1) \(2^3=8\)
2) \(4^{−2}=\frac{1}{16}\)
Solution: \(log_4(\frac{1}{16})=−2\)
3) \(10^2=100\)
4) \(9^0=1\)
Solution: \(log_91=0\)
5) \((\frac{1}{3})^3=\frac{1}{27}\)
6) \(\sqrt[3]{64}=4\)
Solution: \(log_{64}4=\frac{1}{3}\)
7) \(e^x=y\)
8) \(9^y=150\)
Solution: \(log_9150=y\)
9) \(b^3=45\)
10) \(4^{-3/2}=0.125\)
Solution: \(log_40.125=−\frac{3}{2}\)
For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.
1) \(f(x)=3+lnx\)
2) \(f(x)=ln(x−1)\)
Solution: Domain: \((1,∞)\), range: \((−∞,∞),x=1\)
3) \(f(x)=ln(−x)\)
4) \(f(x)=1−lnx\)
Solution: Domain: \((0,∞)\), range: \((−∞,∞),x=0\)
5) \(f(x)=\log x−1\)
6) \(f(x)=ln(x+1)\)
Solution: Domain: \((−1,∞)\), range: \((−∞,∞)\), \(x=−1\)
For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.
1) \(logx^4y\)
2) \(log_3\frac{9a^3}{b}\)
Solution: \(2+3log_3a−log_3b\)
3) \(lna\sqrt[3]{b}\)
4) \(log_5\sqrt{125xy^3}\)
Solution: \(\frac{3}{2}+\frac{1}{2}log_5x+\frac{3}{2}log_5y\)
5) \(log_\frac{\sqrt[3]{xy}}{64}\)
6) \(ln(\frac{6}{\sqrt{e^3}})\)
Solution: \(−\frac{3}{2}+ln6\)
For the following exercises, solve the exponential equation exactly.
1) \(5^x=125\)
2) \(e^{3x}−15=0\)
Solution: \(\frac{ln15}{3}\)
3) \(8^x=4\)
4) \(4^{x+1}−32=0\)
Solution: \(\frac{3}{2}\)
5) \(3^{x/14}=\frac{1}{10}\)
6) \(10^x=7.21\)
Solution: \(log7.21\)
7) \(4⋅2^{3x}−20=0\)
8) \(7^{3x−2}=11\)
Solution: \(\frac{2}{3}+\frac{log11}{3log7}\)
For the following exercises, solve the logarithmic equation exactly, if possible.
1) \(log_3x=0\)
2) \(log_5x=−2\)
Solution: \(x=\frac{1}{25}\)
3) \(log_4(x+5)=0\)
4) \(log(2x−7)=0\)
Solution: \(x=4\)
5) \(ln\sqrt{x+3}=2\)
6) \(log_6(x+9)+log_6x=2\)
Solution: \(x=3\)
7) \(log_4(x+2)−log_4(x−1)=0\)
8) \(lnx+ln(x−2)=ln4\)
Solution: \(1+\sqrt{5}\)
For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.
1) \(log_547\)
2) \(log_782\)
Solution: \((\frac{log82}{log7}≈2.2646)\)
3) \(log_6103\)
4) \(log_{0.5}211\)
Solution: \((\frac{log211}{log0.5}≈−7.7211)\)
5) \(log_2π\)
6) \(log_{0.2}0.452\)
Solution: \((\frac{log0.452}{log0.2}≈0.4934)\)
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1) Rewrite the following expressions in terms of exponentials and simplify.
a. \(2cosh(lnx)\) b. \(cosh4x+sinh4x\) c. \(cosh2x−sinh2x\) d. \(ln(coshx+sinhx)+ln(coshx−sinhx)\)
2) [T] The number of bacteria N in a culture after t days can be modeled by the function \(N(t)=1300⋅(2)^{t/4}\). Find the number of bacteria present after 15 days.
Solution: \(~17,491\)
3) [T] The demand D (in millions of barrels) for oil in an oil-rich country is given by the function \(D(p)=150⋅(2.7)^{−0.25p}\), where p is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.
4) [T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by \(A(t)=100,000⋅e^{0.055t}\). Find the amount A accumulated in 5 years.
Solution: Approximately $131,653 is accumulated in 5 years.
5) [T] An investment is compounded monthly, quarterly, or yearly and is given by the function \(A=P(1+\frac{j}{n})^{nt}\), where \(A\) is the value of the investment at time \(t\), \(P\) is the initial principle that was invested, \(j\) is the annual interest rate, and n is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount \(A\) accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.
6) [T] The concentration of hydrogen ions in a substance is denoted by \([H+]\), measured in moles per liter. The pH of a substance is defined by the logarithmic function \(pH=−log[H+]\). This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.
a. Find the pH of the following substances. Round answers to one digit.
b. Determine whether the substance is an acid or a base.
i. Eggs: \([H+]=1.6×10^{−8}\) mol/L
ii. Beer: \([H+]=3.16×10^{−3}\) mol/L
iii. Tomato Juice: \([H+]=7.94×10^{−5}\) mol/L
Solution: i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid
7) [T] Iodine-131 is a radioactive substance that decays according to the function \(Q(t)=Q_0⋅e^{−0.08664t}\), where \(Q_0\) is the initial quantity of a sample of the substance and t is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.
8) [T] According to the World Bank, at the end of 2013 \((t=0)\) the U.S. population was 316 million and was increasing according to the following model:
\(P(t)=316e^{0.0074t}\),
where P is measured in millions of people and t is measured in years after 2013.
a. Based on this model, what will be the population of the United States in 2020?
b. Determine when the U.S. population will be twice what it is in 2013.
Solution: a. \(~333\) million b. 94 years from 2013, or in 2107
9) [T] The amount A accumulated after 1000 dollars is invested for t years at an interest rate of 4% is modeled by the function \(A(t)=1000(1.04)^t\).
a. Find the amount accumulated after 5 years and 10 years.
b. Determine how long it takes for the original investment to triple.
10) [T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.
a. Use the exponential function \(Q=Q_0e^{kt}\)to determine the value \(k\), which is the growth rate of the bacteria. Round to four decimal places.
b. Determine approximately how long it takes for 200,000 bacteria to grow.
Solution: a. \(k≈0.0578\) b. ≈\(92\) hours
11) [T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.
a. Use the exponential function \(P=P_0a^t\) to determine the growth rate constant \(a\). Round to four decimal places.
b. Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.
12) [T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?
Solution: The San Francisco earthquake had \(10^{3.4} or ~2512\) times more energy than the Japan earthquake.
Chapter Review Exercises
True or False? Justify your answer with a proof or a counterexample.
1) A function is always one-to-one.
2) \(f∘g=g∘f\), assuming f and g are functions.
Solution: False
3) A relation that passes the horizontal and vertical line tests is a one-to-one function.
4) A relation passing the horizontal line test is a function.
For the following problems, state the domain and range of the given functions:
\(f=x^2+2x−3\), \(g=ln(x−5)\), \(h=\frac{1}{x+4}\)
Solution: Domain: \(x>5\), range: all real numbers
3) \(h∘f\)
4) \(g∘f\)
Solution: Domain: \(x>2\) and \(x<−4\), range: all real numbers
Find the degree, y-intercept, and zeros for the following polynomial functions.
1) \(f(x)=2x^2+9x−5\)
2) \(f(x)=x^3+2x^2−2x\)
Solution: Degree of 3, \(y\)-intercept: 0, zeros: \(0, \sqrt{3}−1,−1−\sqrt{3}\)
Simplify the following trigonometric expressions.
1) \(\frac{tan^2x}{sec^2x}+{cos^2x}\)
2) \(cos(2x)=sin^2x\)
Solution: \(cos(2x)\) or \(\frac{1}{2}(cos(2x)+1)\)
Solve the following trigonometric equations on the interval \(θ=[−2π,2π]\) exactly.
1) \(6cos2x−3=0\)
2) \(sec^2x−2secx+1=0\)
Solution: \(0,±2π\)
Solve the following logarithmic equations.
1) \(5^x=16\)
2) \log_2(x+4)=3\)
Solution: 4
Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse \(f^{−1}(x)\) of the function. Justify your answer.
1) \(f(x)=x^2+2x+1\)
2) \(f(x)=\frac{1}{x}\)
Solution: One-to-one; yes, the function has an inverse; inverse: \(f^{−1}(x)=\frac{1}{y}\)
For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.
1) \(f(x)=\sqrt{9−x}\)
2) \(f(x)=x^2+3x+4\)
Solution: \(x≥−\frac{3}{2},f^{−1}(x)=−\frac{3}{2}+\frac{1}{2}\sqrt{4y−7}\)
3) A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?
For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.
1) a. Find the equation \(C=f(x)\) that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.
Solution: a. \(C(x)=300+7x\) b. 100 shirts
2) a. Find the inverse function \(x=f^{−1}(C)\) and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has $8000 to spend.
For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.
1) The population can be modeled by \(P(t)=82.5−67.5cos[(π/6)t]\), where \(t\) is time in months (\(t=0\) represents January 1) and \(P\) is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?
Solution: The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2
2) In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as \(P(t)=82.5−67.5cos[(π/6)t]+t\), where t is time in months (\(t=0\) represents January 1) and \(P\) is population (in thousands). When is the first time the population reaches 200,000?
For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation \(y=e^{rt}\), where \(y\) is the percentage of radiocarbon still present in the material, t is the number of years passed, and \(r=−0.0001210\) is the d78.51%ecay rate of radiocarbon.
1) If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?
Solution: 78.51%
2) Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?
f ( 1 ) = 3 f ( 1 ) = 3 and f ( a + h ) = a 2 + 2 a h + h 2 − 3 a − 3 h + 5 f ( a + h ) = a 2 + 2 a h + h 2 − 3 a − 3 h + 5
Domain = { x | x ≤ 2 } , { x | x ≤ 2 } , range = { y | y ≥ 5 } { y | y ≥ 5 }
x = 0 , 2 , 3 x = 0 , 2 , 3
( f g ) ( x ) = x 2 + 3 2 x − 5 . ( f g ) ( x ) = x 2 + 3 2 x − 5 . The domain is { x | x ≠ 5 2 } . { x | x ≠ 5 2 } .
( f ∘ g ) ( x ) = 2 − 5 x . ( f ∘ g ) ( x ) = 2 − 5 x .
( g ∘ f ) ( x ) = 0.63 x ( g ∘ f ) ( x ) = 0.63 x
f ( x ) f ( x ) is odd.
Domain = ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , range = { y | y ≥ −4 } . { y | y ≥ −4 } .
m = 1 / 2 . m = 1 / 2 . The point-slope form is
y − 4 = 1 2 ( x − 1 ) . y − 4 = 1 2 ( x − 1 ) .
The slope-intercept form is
y = 1 2 x + 7 2 . y = 1 2 x + 7 2 .
The zeros are x = 1 ± 3 / 3 . x = 1 ± 3 / 3 . The parabola opens upward.
The domain is the set of real numbers x x such that x ≠ 1 / 2 . x ≠ 1 / 2 . The range is the set { y | y ≠ 5 / 2 } . { y | y ≠ 5 / 2 } .
The domain of f f is (−∞, ∞). (−∞, ∞). The domain of g g is { x | x ≥ 1 / 5 } . { x | x ≥ 1 / 5 } .
C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3 C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3
Shift the graph y = x 2 y = x 2 to the left 1 unit, reflect about the x x -axis, then shift down 4 units.
7 π / 6 ; 7 π / 6 ; 330°
cos ( 3 π / 4 ) = − 2 / 2 ; sin ( − π / 6 ) = −1 / 2 cos ( 3 π / 4 ) = − 2 / 2 ; sin ( − π / 6 ) = −1 / 2
θ = 3 π 2 + 2 n π , π 6 + 2 n π , 5 π 6 + 2 n π θ = 3 π 2 + 2 n π , π 6 + 2 n π , 5 π 6 + 2 n π for n = 0 , ± 1 , ± 2 ,… n = 0 , ± 1 , ± 2 ,…
To graph f ( x ) = 3 sin ( 4 x ) − 5 , f ( x ) = 3 sin ( 4 x ) − 5 , the graph of y = sin ( x ) y = sin ( x ) needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function f f will have a period of π / 2 π / 2 and an amplitude of 3.
f −1 ( x ) = 2 x x − 3 . f −1 ( x ) = 2 x x − 3 . The domain of f −1 f −1 is { x | x ≠ 3 } . { x | x ≠ 3 } . The range of f −1 f −1 is { y | y ≠ 2 } . { y | y ≠ 2 } .
The domain of f −1 f −1 is ( 0 , ∞ ) . ( 0 , ∞ ) . The range of f −1 f −1 is ( − ∞ , 0 ) . ( − ∞ , 0 ) . The inverse function is given by the formula f −1 ( x ) = −1 / x . f −1 ( x ) = −1 / x .
f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 . f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 .
x / ( 2 y 3 ) x / ( 2 y 3 )
A ( t ) = 750 e 0.04 t . A ( t ) = 750 e 0.04 t . After 30 30 years, there will be approximately $ 2 , 490.09 . $ 2 , 490.09 .
x = ln 3 2 x = ln 3 2
x = 1 e x = 1 e
1.29248 1.29248
The magnitude 8.4 8.4 earthquake is roughly 10 10 times as severe as the magnitude 7.4 7.4 earthquake.
( x 2 + x −2 ) / 2 ( x 2 + x −2 ) / 2
1 2 ln ( 3 ) ≈ 0.5493 . 1 2 ln ( 3 ) ≈ 0.5493 .
Section 1.1 Exercises
a. Domain = { −3 , −2 , −1 , 0 , 1 , 2 , 3 } , { −3 , −2 , −1 , 0 , 1 , 2 , 3 } , range = { 0 , 1 , 4 , 9 } { 0 , 1 , 4 , 9 } b. Yes, a function
a. Domain = { 0 , 1 , 2 , 3 } , { 0 , 1 , 2 , 3 } , range = { −3 , −2 , −1 , 0 , 1 , 2 , 3 } { −3 , −2 , −1 , 0 , 1 , 2 , 3 } b. No, not a function
a. Domain = { 3 , 5 , 8 , 10 , 15 , 21 , 33 } , { 3 , 5 , 8 , 10 , 15 , 21 , 33 } , range = { 0 , 1 , 2 , 3 } { 0 , 1 , 2 , 3 } b. Yes, a function
a. −2 −2 b. 3 c. 13 d. −5 x − 2 −5 x − 2 e. 5 a − 2 5 a − 2 f. 5 a + 5 h − 2 5 a + 5 h − 2
a. Undefined b. 2 c. 2 3 2 3 d. − 2 x − 2 x e 2 a 2 a f. 2 a + h 2 a + h
a. 5 5 b. 11 11 c. 23 23 d. −6 x + 5 −6 x + 5 e. 6 a + 5 6 a + 5 f. 6 a + 6 h + 5 6 a + 6 h + 5
a. 9 b. 9 c. 9 d. 9 e. 9 f. 9
x ≥ 1 8 ; y ≥ 0 ; x = 1 8 ; x ≥ 1 8 ; y ≥ 0 ; x = 1 8 ; no y -intercept
x ≥ −2 ; y ≥ −1 ; x = −1 ; y = −1 + 2 x ≥ −2 ; y ≥ −1 ; x = −1 ; y = −1 + 2
x ≠ 4 ; y ≠ 0 ; x ≠ 4 ; y ≠ 0 ; no x -intercept; y = − 3 4 y = − 3 4
x > 5 ; y > 0 ; x > 5 ; y > 0 ; no intercepts
Function; a. Domain: all real numbers, range: y ≥ 0 y ≥ 0 b. x = ± 1 x = ± 1 c. y = 1 y = 1 d. −1 < x < 0 −1 < x < 0 and 1 < x < ∞ 1 < x < ∞ e. − ∞ < x < − 1 − ∞ < x < − 1 and 0 < x < 1 0 < x < 1 f. Not constant g. y -axis h. Even
Function; a. Domain: all real numbers, range: −1.5 ≤ y ≤ 1.5 −1.5 ≤ y ≤ 1.5 b. x = 0 x = 0 c. y = 0 y = 0 d. all real numbers all real numbers e. None f. Not constant g. Origin h. Odd
Function; a. Domain: − ∞ < x < ∞ , − ∞ < x < ∞ , range: −2 ≤ y ≤ 2 −2 ≤ y ≤ 2 b. x = 0 x = 0 c. y = 0 y = 0 d. −2 < x < 2 −2 < x < 2 e. Not decreasing f. − ∞ < x < − 2 − ∞ < x < − 2 and 2 < x < ∞ 2 < x < ∞ g. Origin h. Odd
Function; a. Domain: −4 ≤ x ≤ 4 , −4 ≤ x ≤ 4 , range: −4 ≤ y ≤ 4 −4 ≤ y ≤ 4 b. x = 1.2 x = 1.2 c. y = 4 y = 4 d. Not increasing e. 0 < x < 4 0 < x < 4 f. −4 < x < 0 −4 < x < 0 g. No Symmetry h. Neither
a. 5 x 2 + x − 8 ; 5 x 2 + x − 8 ; all real numbers b. −5 x 2 + x − 8 ; −5 x 2 + x − 8 ; all real numbers c. 5 x 3 − 40 x 2 ; 5 x 3 − 40 x 2 ; all real numbers d. x − 8 5 x 2 ; x ≠ 0 x − 8 5 x 2 ; x ≠ 0
a. −2 x + 6 ; −2 x + 6 ; all real numbers b. −2 x 2 + 2 x + 12 ; −2 x 2 + 2 x + 12 ; all real numbers c. − x 4 + 2 x 3 + 12 x 2 − 18 x − 27 ; − x 4 + 2 x 3 + 12 x 2 − 18 x − 27 ; all real numbers d. − x + 3 x + 1 ; x ≠ − 1 , 3 − x + 3 x + 1 ; x ≠ − 1 , 3
a. 6 + 2 x ; x ≠ 0 6 + 2 x ; x ≠ 0 b. 6; x ≠ 0 x ≠ 0 c. 6 x + 1 x 2 ; x ≠ 0 6 x + 1 x 2 ; x ≠ 0 d. 6 x + 1 ; x ≠ 0 6 x + 1 ; x ≠ 0
a. 4 x + 3 ; 4 x + 3 ; all real numbers b. 4 x + 15 ; 4 x + 15 ; all real numbers
a. x 4 − 6 x 2 + 16 ; x 4 − 6 x 2 + 16 ; all real numbers b. x 4 + 14 x 2 + 46 ; x 4 + 14 x 2 + 46 ; all real numbers
a. 3 x 4 + x ; x ≠ 0 , −4 3 x 4 + x ; x ≠ 0 , −4 b. 4 x + 2 3 ; x ≠ − 1 2 4 x + 2 3 ; x ≠ − 1 2
a. Yes, because there is only one winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.
a. V ( s ) = s 3 V ( s ) = s 3 b. V ( 11.8 ) ≈ 1643 ; V ( 11.8 ) ≈ 1643 ; a cube of side length 11.8 each has a volume of approximately 1643 cubic units.
a. N ( x ) = 15 x N ( x ) = 15 x b. i. N ( 20 ) = 15 ( 20 ) = 300 ; N ( 20 ) = 15 ( 20 ) = 300 ; therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. N ( 15 ) = 225 ; N ( 15 ) = 225 ; therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: 0 ≤ x ≤ 20 ; 0 ≤ x ≤ 20 ; range: [ 0 , 300 ] [ 0 , 300 ] d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.
a. A ( t ) = A ( r ( t ) ) = π · ( 6 − 5 t 2 + 1 ) 2 A ( t ) = A ( r ( t ) ) = π · ( 6 − 5 t 2 + 1 ) 2 b. Exact: 121 π 4 ; 121 π 4 ; approximately 95 cm 2 c. C ( t ) = C ( r ( t ) ) = 2 π ( 6 − 5 t 2 + 1 ) C ( t ) = C ( r ( t ) ) = 2 π ( 6 − 5 t 2 + 1 ) d. Exact: 11 π ; 11 π ; approximately 35 cm
a. S ( x ) = 8.5 x + 750 S ( x ) = 8.5 x + 750 b. $962.50, $1090, $1217.50 c. 77 skateboards
Section 1.2 Exercises
a. −1 b. Decreasing
a. 3/4 b. Increasing
a. 4/3 b. Increasing
a. 0 b. Horizontal
y = −6 x + 9 y = −6 x + 9
y = 1 3 x + 4 y = 1 3 x + 4
y = 1 2 x y = 1 2 x
y = 3 5 x − 3 y = 3 5 x − 3
a. ( m = 2 , b = −3 ) ( m = 2 , b = −3 ) b.
a. ( m = −6 , b = 0 ) ( m = −6 , b = 0 ) b.
a. ( m = 0 , b = −6 ) ( m = 0 , b = −6 ) b.
a. ( m = − 2 3 , b = 2 ) ( m = − 2 3 , b = 2 ) b.
a. 2 b. 5 2 , −1 ; 5 2 , −1 ; c. −5 d. Both ends rise e. Neither
a. 2 b. ± 2 ± 2 c. −1 d. Both ends rise e. Even
a. 3 b. 0, ± 3 ± 3 c. 0 d. Left end rises, right end falls e. Odd
a. 13 , −3 , 5 13 , −3 , 5 b.
a. −3 2 , −1 2 , 4 −3 2 , −1 2 , 4 b.
True; n = 3 n = 3
False; f ( x ) = x b , f ( x ) = x b , where b b is a real-valued constant, is a power function
a. V ( t ) = −2733 t + 20500 V ( t ) = −2733 t + 20500 b. ( 0 , 20 , 500 ) ( 0 , 20 , 500 ) means that the initial purchase price of the equipment is $20,500; ( 7.5 , 0 ) ( 7.5 , 0 ) means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years
a. C = 0.75 x + 125 C = 0.75 x + 125 b. $245 c. 167 cupcakes
a. V ( t ) = −1500 t + 26,000 V ( t ) = −1500 t + 26,000 b. In 4 years, the value of the car is $20,000.
96% of the total capacity
Section 1.3 Exercises
4 π 3 rad 4 π 3 rad
− π 3 − π 3
11 π 6 rad 11 π 6 rad
210 ° 210 °
−540 ° −540 °
− 2 2 − 2 2
3 − 1 2 2 3 − 1 2 2
a. b = 5.7 b = 5.7 b. sin A = 4 7 , cos A = 5.7 7 , tan A = 4 5.7 , csc A = 7 4 , sec A = 7 5.7 , cot A = 5.7 4 sin A = 4 7 , cos A = 5.7 7 , tan A = 4 5.7 , csc A = 7 4 , sec A = 7 5.7 , cot A = 5.7 4
a. c = 151.7 c = 151.7 b. sin A = 0.5623 , cos A = 0.8273 , tan A = 0.6797 , csc A = 1.778 , sec A = 1.209 , cot A = 1.471 sin A = 0.5623 , cos A = 0.8273 , tan A = 0.6797 , csc A = 1.778 , sec A = 1.209 , cot A = 1.471
a. c = 85 c = 85 b. sin A = 84 85 , cos A = 13 85 , tan A = 84 13 , csc A = 85 84 , sec A = 85 13 , cot A = 13 84 sin A = 84 85 , cos A = 13 85 , tan A = 84 13 , csc A = 85 84 , sec A = 85 13 , cot A = 13 84
a. y = 24 25 y = 24 25 b. sin θ = 24 25 , cos θ = 7 25 , tan θ = 24 7 , csc θ = 25 24 , sec θ = 25 7 , cot θ = 7 24 sin θ = 24 25 , cos θ = 7 25 , tan θ = 24 7 , csc θ = 25 24 , sec θ = 25 7 , cot θ = 7 24
a. x = − 2 3 x = − 2 3 b. sin θ = 7 3 , cos θ = − 2 3 , tan θ = − 14 2 , csc θ = 3 7 7 , sec θ = −3 2 2 , cot θ = − 14 7 sin θ = 7 3 , cos θ = − 2 3 , tan θ = − 14 2 , csc θ = 3 7 7 , sec θ = −3 2 2 , cot θ = − 14 7
sec 2 x sec 2 x
sin 2 x sin 2 x
sec 2 θ sec 2 θ
1 sin t ( = csc t ) 1 sin t ( = csc t )
{ π 6 , 5 π 6 } { π 6 , 5 π 6 }
{ π 4 , 3 π 4 , 5 π 4 , 7 π 4 } { π 4 , 3 π 4 , 5 π 4 , 7 π 4 }
{ 2 π 3 , 5 π 3 } { 2 π 3 , 5 π 3 }
{ 0 , π , π 3 , 5 π 3 } { 0 , π , π 3 , 5 π 3 }
y = 4 sin ( π 4 x ) y = 4 sin ( π 4 x )
y = cos ( 2 π x ) y = cos ( 2 π x )
a. 1 b. 2 π 2 π c. π 4 π 4 units to the right
a. 1 2 1 2 b. 8 π 8 π c. No phase shift
a. 3 b. 2 2 c. 2 π 2 π units to the left
Approximately 42 in.
a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min
≈ 30.9 in 2 ≈ 30.9 in 2
a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods
a. Amplitude = 10 ; period = 24 10 ; period = 24 b. 47.4 ° F 47.4 ° F c. 14 hours later, or 2 p.m. d.
Section 1.4 Exercises
Not one-to-one
a. f −1 ( x ) = x + 4 f −1 ( x ) = x + 4 b. Domain : x ≥ −4 , range : y ≥ 0 : x ≥ −4 , range : y ≥ 0
a. f −1 ( x ) = x − 1 3 f −1 ( x ) = x − 1 3 b. Domain: all real numbers, range: all real numbers
a. f −1 ( x ) = x 2 + 1 , f −1 ( x ) = x 2 + 1 , b. Domain: x ≥ 0 , x ≥ 0 , range: y ≥ 1 y ≥ 1
These are inverses.
These are not inverses.
− π 6 − π 6
a. x = f −1 ( V ) = 0.04 − V 500 x = f −1 ( V ) = 0.04 − V 500 b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V . c. 0.1 cm; 0.14 cm; 0.17 cm
a. $31,250, $66,667, $107,143 b. ( p = 85 C C + 75 ) ( p = 85 C C + 75 ) c. 34 ppb
a. ~ 92 ° ~ 92 ° b. ~ 42 ° ~ 42 ° c. ~ 27 ° ~ 27 °
x ≈ 6.69 , 8.51 ; x ≈ 6.69 , 8.51 ; so, the temperature occurs on June 21 and August 15
~ 1.5 sec ~ 1.5 sec
tan −1 ( tan ( 2.1 ) ) ≈ − 1.0416 ; tan −1 ( tan ( 2.1 ) ) ≈ − 1.0416 ; the expression does not equal 2.1 since 2.1 > 1.57 = π 2 2.1 > 1.57 = π 2 —in other words, it is not in the restricted domain of tan x . cos −1 ( cos ( 2.1 ) ) = 2.1 , tan x . cos −1 ( cos ( 2.1 ) ) = 2.1 , since 2.1 is in the restricted domain of cos x . cos x .
Section 1.5 Exercises
a. 125 b. 2.24 c. 9.74
a. 0.01 b. 10,000 c. 46.42
Domain: all real numbers, range: ( 2 , ∞ ) , y = 2 ( 2 , ∞ ) , y = 2
Domain: all real numbers, range: ( 0 , ∞ ) , y = 0 ( 0 , ∞ ) , y = 0
Domain: all real numbers, range: ( − ∞ , 1 ) , y = 1 ( − ∞ , 1 ) , y = 1
Domain: all real numbers, range: ( −1 , ∞ ) , y = −1 ( −1 , ∞ ) , y = −1
8 1 / 3 = 2 8 1 / 3 = 2
5 2 = 25 5 2 = 25
e −3 = 1 e 3 e −3 = 1 e 3
e 0 = 1 e 0 = 1
log 4 ( 1 16 ) = −2 log 4 ( 1 16 ) = −2
log 9 1 = 0 log 9 1 = 0
log 64 4 = 1 3 log 64 4 = 1 3
log 9 150 = y log 9 150 = y
log 4 0.125 = − 3 2 log 4 0.125 = − 3 2
Domain: ( 1 , ∞ ) , ( 1 , ∞ ) , range: ( − ∞ , ∞ ) , x = 1 ( − ∞ , ∞ ) , x = 1
Domain: ( 0 , ∞ ) , ( 0 , ∞ ) , range: ( − ∞ , ∞ ) , x = 0 ( − ∞ , ∞ ) , x = 0
Domain: ( −1 , ∞ ) , ( −1 , ∞ ) , range: ( − ∞ , ∞ ) , x = −1 ( − ∞ , ∞ ) , x = −1
2 + 3 log 3 a − log 3 b 2 + 3 log 3 a − log 3 b
3 2 + 1 2 log 5 x + 3 2 log 5 y 3 2 + 1 2 log 5 x + 3 2 log 5 y
− 3 2 + ln 6 − 3 2 + ln 6
ln 15 3 ln 15 3
log 7.21 log 7.21
2 3 + log 11 3 log 7 2 3 + log 11 3 log 7
x = 1 25 x = 1 25
x = 4 x = 4
x = 3 x = 3
1 + 5 1 + 5
( log 82 log 7 ≈ 2.2646 ) ( log 82 log 7 ≈ 2.2646 )
( log 211 log 0.5 ≈ − 7.7211 ) ( log 211 log 0.5 ≈ − 7.7211 )
( log 0.452 log 0.2 ≈ 0.4934 ) ( log 0.452 log 0.2 ≈ 0.4934 )
~ 17 , 491 ~ 17 , 491
Approximately $131,653 is accumulated in 5 years.
i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid
a. ~ 333 ~ 333 million b. 94 years from 2013, or in 2107
a. k ≈ 0.0578 k ≈ 0.0578 b. ≈ 92 ≈ 92 hours
The San Francisco earthquake was 10 3.4 or ≈ 2512 10 3.4 or ≈ 2512 times more intense than the Japanese earthquake.
Review Exercises
Domain: x > 5 , x > 5 , range: all real numbers
Domain: x > 2 x > 2 and x < − 4 , x < − 4 , range: all real numbers
Degree of 3, y y -intercept: 0, zeros: 0, 3 − 1 , −1 − 3 3 − 1 , −1 − 3
cos 2 x - sin 2 x = cos 2 x = 1 - 2 sin 2 x = 2 cos 2 x - 1 cos 2 x - sin 2 x = cos 2 x = 1 - 2 sin 2 x = 2 cos 2 x - 1
0 , ± 2 π 0 , ± 2 π
One-to-one; yes, the function has an inverse; inverse: f −1 ( x ) = 1 y f −1 ( x ) = 1 y
x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7 x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7
a. C ( x ) = 300 + 7 x C ( x ) = 300 + 7 x b. 100 shirts
The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2
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Course Resources
Student guide.
Calculus I includes many interactive opportunities where you can strengthen your knowledge and practice using the concepts taught in the course. Research has shown that this type of learn-by-doing approach has a significant positive impact on learning. We encourage you to utilize as many resources in this course as possible to deepen your understanding over time. Since this course can feel daunting with all of its components, we’ve compiled some helpful tips and bits of information about the structure of the course to assist you as you navigate through.
- Every module begins and ends with a “Why It Matters” page and a “Putting It Together” page. The Why It Matters page presents a real-world example of a calculus concept to be taught in the module. Then the problem is restated, solved, and explained on the Putting It Together page once the module is completed.
- Each module is proceeded by a “Prerequisite Material” module and followed by a “Supplemental Content” module. The Prerequisite Material pages present skills needed for the upcoming module that you should be review from earlier math classes. The Supplemental Content pages contain the problems sets, discussions, etc. that specifically focus on the concepts presented in the prior module.
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Colored Emphasis Boxes
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The objective is for you to master the concepts and skills stated below. Mastery takes time, effort, practice, and application. Reading the content on the page is only one step in the learning process.
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The complete Learning Outcome spreadsheet can be used if a specific “LO” is referenced anywhere in the course, i.e. “LO 4.1.3”
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- A Brief Review of Algebra (for Reference) Lecture Notes
- A Brief Review of Trigonometry (for Reference) Lecture Notes
- 1. The Concept of a Limit Lecture Notes Homework
- 2. Limits and One Sided Limits Lecture Notes Homework
- 3. Calculating Limits Lecture Notes Homework
- 4. Infinite Limits Lecture Notes Homework
- 5. Limits at Infinity Lecture Notes Homework
- 6. Continuity Lecture Notes Homework
- 7. The Formal Definition of a Limit Lecture Notes Homework
- 8. The Derivative of a Function Lecture Notes Homework
- 9. The Graph of f'(x) Lecture Notes Homework
- 10. Some Differentiation Rules Lecture Notes Homework
- 11. The Product and Quotient Rules Lecture Notes Homework
- 12. Derivatives of Trigonometric Functions Lecture Notes Homework
- 13. Derivatives and Rates of Change Lecture Notes Homework
- 14. The Chain Rule Lecture Notes Homework
- 15. Implicit Differentiation Lecture Notes Homework
- 16. Related Rate Problems Lecture Notes Homework
- 17. Maximum and Minimum Values Lecture Notes Homework
- 18. Derivatives and the Shapes of Graphs Lecture Notes Homework
- 19. Graphing Functions Lecture Notes Homework
- 20. Optimization Problems Lecture Notes Homework
- 21. Linear Approximations and Differentials Lecture Notes Homework
- 22. The Mean Value Theorem Lecture Notes Homework
- 23. Antiderivatives Lecture Notes Homework
- 24. Approximating the Area under a Curve Lecture Notes Homework
- 25. Definite Integrals Lecture Notes Homework
- 26. The Fundamental Theorem of Calculus Lecture Notes Homework
- 27. More Properties of Integrals Lecture Notes Homework
- 28. The Substitution Rule Lecture Notes Homework
- 29. Net Change- Integrating the Derivative Lecture Notes Homework
- 30. Area Between Curves Lecture Notes Homework
Calculus Volume 1
(19 reviews)
Gilbert Strang, MIT
Edwin Herman, University of Wisconsin-Stevens Point
Copyright Year: 2016
ISBN 13: 9781938168024
Publisher: OpenStax
Language: English
Formats Available
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Learn more about reviews.
Reviewed by Pat Miceli, Associate Professor, City Colleges of Chicago on 4/10/23
The text covers all topics I was expecting except for a discussion about normal lines. This isn't required but is nice to get exposure to in Calculus 1. I would also say that about the midpoint rule for finding area under a curve, not necessary... read more
Comprehensiveness rating: 5 see less
The text covers all topics I was expecting except for a discussion about normal lines. This isn't required but is nice to get exposure to in Calculus 1. I would also say that about the midpoint rule for finding area under a curve, not necessary but gives the alternative to an upper sum and lower sum. It provides greater accuracy but uncertainty about if it's an upper or lower sum.
Content Accuracy rating: 5
I did not find any errors. The writing was clear. I appreciate the informal proofs about limits at infinity along with the formal proofs.
Relevance/Longevity rating: 5
I agree that the content is relevant. Many chapter openers and student projects involve situations that would interest students.
Clarity rating: 5
I thought the writing was very accessible. It was formal when necessary but otherwise friendly or not stuffy.
Consistency rating: 4
Generally the text is consistent. There were some bold terms that I would have like to see get a boxed definition. This would include many of the functions mentioned in chapter 1 (they could be included in the Precalculus review), the existence of a limit, the compound formula and continuous compound formula. The limit rule for sinx/x as x approaches 0 is explained in 2.3, but I would like this to have it's Theorem or Rule box. It could include the (1-cos x) /x rule.
Modularity rating: 5
Yes, the text is nicely broken into manageable sections that could be rearranged if necessary.
Organization/Structure/Flow rating: 5
Topics are in a logical order. I was curious why delta x(sub i) was not included in the definition of a Riemann sum but this was later explained in 5.2.
Interface rating: 5
No distortion issues. I did have trouble with some media links but I'm attributing that to the work computers and blocking access to some sites.
Grammatical Errors rating: 5
I found no grammatical errors.
Cultural Relevance rating: 5
I didn't find any instances of culturally insensitive or offensive text. Most references to individuals seems neutral.
The media links I was able to open I thought were very good. Images for some of the properties of definite integrals could be helpful. The explanation of the average value function was very good. A visual or applet for the Fundamental Theorem of Calculus would be great.
Reviewed by Mikheil Elashvili, Part-time Faculty, Bridgewater State University on 12/9/22
Textbook represents a well-thought compromise of comprehensive mathematical material, providing the full amount of knowledge on the given subject, but sometimes omitting the details which might be overburdening for the students, especially for... read more
Comprehensiveness rating: 4 see less
Textbook represents a well-thought compromise of comprehensive mathematical material, providing the full amount of knowledge on the given subject, but sometimes omitting the details which might be overburdening for the students, especially for those of non-math major. So I would say the Textbook is comprehensive but not ideal, which does not reduce its value. I found it much more comfortable for students to work with. The index is accurate and easy to navigate, though slight changes in the sequence of chapters might be introduced.
The content is accurate, I never found any factual errors in the Textbook, and provided information was accurate. The text is clear and well formulated, It does not cause any bias or misinterpretation.
Relevance/Longevity rating: 4
Math as a subject is quite pre-defined in terms of factual content and there is really nothing to update. But, the way the math subjects are explained in the Textbook is modern and well up-to-date. Though a bit more connection to modern/practical applications would be beneficial.
Text is written with simple and understandable language, which is not an easy task for Math textbooks. Adequate terminology is used.
Consistency rating: 5
The textbook is consistent with its goal. The framework and the structure of the textbook is well architected.
Modularity rating: 4
Modularity is not a strong side of Math textbooks, since the sequence of sections matters, topics being built one on another. Nevertheless textbook provides some flexibility, to "jump" between the chapters in accordance with the instructor's requirement (syllabus content).
The topics in the text are presented in a logical, clear fashion. I have nothing to comment on it.
The text is free of any interface issues, no navigation problems, drawings are clear and high quality.
So far, no grammatical errors were noticed.
Because of the Math content, the text involves basically no culturally sensitive material.
I would recommend with no hesitation this textbook as a Calculus Teaching material for any Science and Engineering majors, though it might not be entirely suitable for Math or Physics majors.
Reviewed by Nicholas Wong, Director of Introductory Mathematics, Virginia Commonwealth University on 8/18/22
The textbook covers a full first semester of college calculus and with enough material to also be useful in an AP Calculus AB course for high school calculus teachers. read more
The textbook covers a full first semester of college calculus and with enough material to also be useful in an AP Calculus AB course for high school calculus teachers.
The content is accurate and presented in a relatively student friendly manner. Making more connections between concepts would make it relevant and valued for both students and instructors.
The calculus content doesn't really change, so this textbook, along with any other, will be relevant for quite a time. The order may not be be how all instructors teach, but instructors should be able to make the professional judgement to utilize the book "out of order" if it suits their teaching pedagogy.
Clarity rating: 4
The text is written at an appropriate level. Some more student-friendly language would be appreciated as many students avoid the textbook in this day and age. Providing a "layman's speak" for the mathematical text and language would make this more approachable for students.
The text is very consistent in terms of its use of terminology and the framework which they develop the key concepts for students.
The text is broken up into reasonable sections for instructors to break up if they feel the order of material should be different to suit their students. A "guide" for students would be beneficial as they don't necessarily understand why an instructor would teach material "out of order".
Organization/Structure/Flow rating: 4
The text is organized in a way for instructors to teach this course with early transcendentals or not although it will take some work on their part. Similarly to the note above, a "guide" for students would be beneficial as they are the ones who should be interacting, reading, and synthesizing the materials in the text. I believe many calculus textbooks, this one included, are written to favor faculty to choose the textbook rather than written in a manner that is approachable for students.
Interface rating: 4
The interface is pretty good and relatively easy to navigate without major issues.
Grammatical Errors rating: 4
No major grammatical errors that I've noticed through the book.
Cultural Relevance rating: 4
I wouldn't say it's inclusive, but rather it's more "neutral" in its presentation of material.
With the current issue of D/F/W rates in calculus courses across the country, having a small note at the beginning of each section to remind students of required prerequisite skills and concepts would benefit students reading the text to obtain some just in time remediation if needed to make sure they can be on-level or as close to that as possible. The notes could reference other OER course books as well. It may also be beneficial pedagogically for instructors and a great marketing tool to get more faculty on board utilizing OER.
Reviewed by Josh Hallam, Assistant Professor, Loyola Marymount University on 3/20/22
This textbook contains all the material that is typically covered in a first semester calculus course. It is written in such a way that one can do either early or late transcendentals. It also contains a review of pre-calculus material and... read more
This textbook contains all the material that is typically covered in a first semester calculus course. It is written in such a way that one can do either early or late transcendentals. It also contains a review of pre-calculus material and answers to odd numbered problems.
I have not found any errors in the text.
The content is up-to-date and contains numerous examples that will still be relevant in the foreseeable future.
The text does an excellent job straddling the line between being conversational and being rigorous. Often times, motivation for the material is first given and then the mathematics is explained. The figures are well-suited to develop a deeper understanding of the concepts described in the text. Proofs given in the book are written in a way that a first year student may be able to follow.
I did not find any issues with inconsistencies.
Each section could be taught in one or two lessons. The division of topics in the text is essentially the same as those found in a commercial calculus textbook.
The topics are presented in the typical fashion for a calculus text and is logical and clear. Typically, new sections are built on previous sections.
This is a place where I think this text stands out compared to other texts. There is a pdf version (which is like a standard print textbook) and an online version. I particularly like the online version. Topics are hyperlinked and in-text examples have drop-down menus to show solutions. Both the pdf and the online version work great on the mobile devices that I have tried.
I haven’t found any grammatical issues.
I haven’t found anything that is culturally insensitive or offensive.
I would highly recommend this textbook. I think it would be hard to argue that this textbook is worse than any of the commercially available textbooks. Personally, I think that it is better than many of them. At the very least, I think it is a good resource to offer to students even if it is not the official textbook for the course.
Reviewed by Jay Daigle, Teaching Assistant Professor of Mathematics, The George Washington University on 1/30/22
The book covers all of the topics I would expect in a first-semester "early transcendentals"-style calculus course. It contains a detailed table of contents and a thorough-seeming index. It does set itself a difficult challenge of being usable... read more
The book covers all of the topics I would expect in a first-semester "early transcendentals"-style calculus course. It contains a detailed table of contents and a thorough-seeming index.
It does set itself a difficult challenge of being usable for both "early transcendentals" and "late transcendentals" courses. While it contains all of the appropriate content for either style of course, it is much less effectivel ydesigned for a "late transcendentals" course.
I have not noticed any errors or mistakes in the text.
In general the content is up-to-date. I would love to see more applications to economics, biology, and other non-physics fields, especially in the integrals section. I also wish there were a bit more leveraging of interactive computer tools that students can use to explore some of these ideas.
The exposition is reasonably clear and straightforward.
I have not noticed any inconsistencies in the text.
The text is mostly divided appropriately into sections containing about a day's worth of material. These sections are described as pursuing multiple learning objectives in the section headers, but the divisions within the sections often seem inadequate to me; if I want to assign part of a section I generally have to say something like "all the material up through Checkpoint 4.13", and I wish the subheadings were better. (However, the numbering of individual examples and checkpoints is quite useful for this.)
The text attemps to be usable for "early transcendentals" and "late transcendentals" classes both, but is clearly much more strongly geared to "early transcendentals". The core "early transcendentals" material is separated out into its own sections, but later sections will include examples and exercises that rely on early transcendentals material.
The organization of the book is overall good, though I wish there were more and clearer subsection divisions.
The book has a useful website with a well-designed table of contents, but the interface has some odd hiccups.
The book's text is clean and grammatical.
The content of the book is relevant to everyone, but it makes no particular attempt to be inclusive or culturally relevant.
This is a very usable and easily-accessible textbook for a first course in calculus, early transcendentals. If you want to use it for a late transcendentals course you will have to do some work reordering and skipping sections and individual problems and examples.
Reviewed by Sybil Prince Nelson, Assistant, Washington & Lee University on 10/15/21
This book covers all the topics necessary for a Calculus 1 curriculum. It even covers things that I think are optional for Calc 1 like epsilon delta definition of a limit. read more
This book covers all the topics necessary for a Calculus 1 curriculum. It even covers things that I think are optional for Calc 1 like epsilon delta definition of a limit.
I have not found any notable mistakes in the book and I have been using it for half a semester.
The examples are relevant for today with topics that students are aware of.
Sometimes the way the homework questions are asked is a little confusing. Just be prepared before the lesson so you can clarify for students.
I have no complaints in this area.
The modularity is especially beneficial with the Canvas resources that are given.
Sometimes I do not agree with the order that topics are presented but I can just rearrange for myself.
The layout is fine.
I have not noticed any glaring grammatical errors.
I see no examples of culture offense in my half a semester of usage.
This open textbook covers all the subjects needed for an entry level calculus 1 course. Though at times I don’t agree with the order of the topics, every thing is there and can be adjusted based on your teaching style. The graphs, displays, and general graphics are a bit no frills but they are suitable and free of any major errors or inaccuracies. I really enjoy this textbook because of its ease of use and all of the extra features available. Students can access it at any time, there is a Canvas course shell provided which makes course planning super simple, and there are even powerpoints and a solutions manual available. I highly recommend this textbook for college courses especially considering the escalating prices of textbooks.
Reviewed by Cristina Villalobos, Professor, University of Texas Rio Grande Valley on 11/13/20
The textbook covers topics covered by commercial textbooks. read more
The textbook covers topics covered by commercial textbooks.
Content Accuracy rating: 4
I mentioned to an OpenStax representative that the book still has (many) errors. I've been using the book for 2-3 years now and I still find errors. Given that the book is not a commercial textbook, I would think that it would be quicker to make edits. This is the only disappointing part. I am trying to get my colleagues on board to adopt the book but they keep pointed out the errors in the book.
Content is good; good examples and good student projects.
I like how the textbook is written --not terse--but simple wording. The mathematical notation is also good and not terse.
Yes, internally consistent
The sections are divided well and one can easily reference information through titles/subtitles and the table of contents.
In general, organization is good. We move Section 4.6 after Section 2.4 so that everything dealing with limits is covered together. Of course L'Hopital's section stays where it is since it deals with derivatives.
I have not seen any grammatical errors.
I don't recall seeing any examples that are inclusive of races, ethnicities, and backgrounds. Examples tend to be neutral and not dealing with names of individuals.
Again just the issue that the book still has errors and these errors need to be worked on rightaway. It should not take longer than a year to correct ALL errors in the book, just have someone go through each line of the book.
Reviewed by Igor Baryakhtar, Instructor, Massachusetts Bay Community College on 6/30/20
This book covers the standard Calculus 1 course: traditional topics of differential calculus and the basic concepts of integral calculus. The compact review of functions helps to make a good start with calculus. The text is vivid and lucid and not... read more
This book covers the standard Calculus 1 course: traditional topics of differential calculus and the basic concepts of integral calculus. The compact review of functions helps to make a good start with calculus. The text is vivid and lucid and not overcomplicated, exercises are reasonably difficult. Learning objectives, at the beginning of each section, key terms, key concepts, and key equations at the end of each sections are very helpful. Therefore the text is very comprehensive.
Notations and explanations are accurate.
The text is relevant to the subject as it is supposed to to be taught these days. Derivatives and the Shape of a Graph are explained very well among other topics.
The text very clearly explains the concepts and examples and relations between them. Illustrations, mostly graphs, are excellent, with great attention to the most important information.
The text is consistent in both terminology and style.
Modularity is logically reasonable, text is well organized (see the Table of Contents) and is a natural part of the Calculus 1,2,3 sequence.
Logically reasonable and convenient. Organization of the book can be understood very well from the Preface.
Easy to navigate. This textbook is available in online, downloadable pdf, and print version. Interactive examples with hints and/or answers make the online version really valuable.
The grammar is good. I have not found any grammatical errors.
The text is relevant to all people and all cultures.
I taught the Calculus 1 course at two community colleges for many years and found that Calculus 1 from OpenStax is the best choice for both students and teachers.
Reviewed by Ashley Fuller, Associate Professor, Richard Bland College on 10/9/19
The text is well laid out and has topics broken down into appropriate subtopics within each larger chapter. There are extensive examples to go with each learning objective followed by practice problems to allow the student significant practice. read more
The text is well laid out and has topics broken down into appropriate subtopics within each larger chapter. There are extensive examples to go with each learning objective followed by practice problems to allow the student significant practice.
The text has content that is accurate and does not have errors nor is it biased. There are a variety of topics used in examples.
The text has relevant problems that relate directly to the topic at hand, but they are classic enough that the book will be able to be used for many semesters and will not quickly be obsolete.
The text is written in an easy to read manner that a mathematics student can use to follow-up on material taught in class or as a stand alone in an online format. Terms are defined in easy to read highlighted sections followed by examples that can be easily targeted based on topic.
Each chapter is laid out in a similar fashion in order to facilitate ease of learning and comfort as more topics are introduced and more complex concepts examined.
Within each section topics are broken down in even greater detail and then examples are given to coincide with those topics. THis allows the student and or instructor alike to readily assign and or discover new topics as they approach.
Each subsequent chapter flows in a logical and concise way. Introduction of material is followed by practice, and then further developed by application. Topics build on each other based upon a clear pathway.
The book in its PDF format is easy to read and is laid out nicely. I saw no images or features that showed issues.
no grammar errors were noted.
Examples are fairly straight forward and are universal in their application. No areas were noted as insensitive and/or offensive in any way.
Reviewed by Tai Jen Liu, Math instructor, St Cloud Tech & Community College on 6/24/19
This text covers a standard list of topics for the 1st course of calculus. It begins with a chapter of functions review which is particularly useful for those non-STEM students taking calculus. It continues to differentiation and integration,... read more
This text covers a standard list of topics for the 1st course of calculus. It begins with a chapter of functions review which is particularly useful for those non-STEM students taking calculus. It continues to differentiation and integration, with both theorems and application accompanying the main concepts. Abundant exercises for practice, and external resource, such as applets or interactive graph, also are occasionally included to help the visualization of ideas. Index and glossary with easy access page number are presented in the end for reference.
The text is free of error, and the examples and exercises (the even ones I have worked on) are accurate.
Calculus content text is relatively timeless. The examples within the content may need to be updated from time to time, and this textbook has done so to a satisfactory degree. Those interactive features are good examples of this work. It certainly can be more of them, comparing to some commercial calculus text, if time and resources allow.
The text is clearly written, and its colloquial style is a strong point of the book. Notation and symbols are used in a must-only fashion. I find this feature also is a plus for students whose algebra experience is limited, or of long past.
The text is consistent in the content, the level of difficulty in exercise for the content, and notation use.
The text is well suited for 15 weeks course. The subchapters are in logic order and easily adopted for use. I would not omit any of those sections for a complete 1st course, but it certainly can be tailored and grouped to individual instructor need.
The organization/structure of this text is as most standard calculus text. Its flow is fluent and logical (as mentioned in previous category).
No apparent problems in navigation, or distortion of images found during my reading. It is relatively easy access for the text.
No grammatical errors are found.
There is not much cultural reference made within this context. While generally cultural relevance is not a major issue in calculus text, it probably will be great if the text includes some historical background as the topics move in logical order, since calculus is developed with rich history and important mathematicians along the way.
This is a good book for a 1st calculus course, especially for those non-STEM majors. It is more focus on introducing the concepts with examples of application well worked out. I have also read through another older Calculus text in this open library by Strang. The older text is a traditional calculus book can be used in 2-3 semesters calculus sequence course. The older one probably requires a more rigorous algebra background. However, that older text contains very good narratives that explore/explains those ideas presented in calculus, some of them are thoughtfully placed to connect reader to the background why/how certain theorems emerge or being developed. I think these two texts can be good supplement to each other for a calculus sequence course, depending on the skill level and goal of the course. I consider this is the ultimate advantage of using OER material, instructor can put together a good curriculum material suitable for the audience without worry too much about hiking up the cost.
Reviewed by Leanne Merrill, Assistant Professor , Western Oregon University on 3/5/19
This book contains all of the topics and material you would expect to see in a first calculus course. It starts with a review of functions, moves to limits, and then proceeds through differentiation and integration. There is a nice mix of theory... read more
This book contains all of the topics and material you would expect to see in a first calculus course. It starts with a review of functions, moves to limits, and then proceeds through differentiation and integration. There is a nice mix of theory and applications throughout. The index and glossary are both easy to use. There are also several interactive applets that have easy click-throughs from the pdf and the ebook.
This book is completely accurate. I have not found errors in formulas, examples, or homework problems. They also give a precise definition of limit in the main text, which I like. The Fundamental Theorem of Calculus, often a tricky section, is handled clearly and intuitively.
Most of the real-world examples are from the last 5-10 years, so will lose some relevance as time goes by, but the math would of course still be perfectly valid. This book could do a bit more integrating coding or interactivity in ways that many completely online textbooks have already done. However, for a book that must exist as a static pdf, this is perfectly fine. There are also several projects that are suggested in the book that are in line with current calculus pedagogy trends.
I particularly like that this book is written in a more conversational tone than most math textbooks, and includes more helpful pictures embedded in the text than most textbooks do. (Often, the pictures in commercial textbooks appear off to the side, and it's not always clear which figures go with which text.) It also has relevant examples that allow students to connect the definitions to concepts with which they already have some familiarity.
There were no issues with consistency. The exercise sets are of similar difficulty and length throughout. Notation is used clearly and consistently throughout the book.
The book is divided into sections that could easily be taught in one or two days each. It would be possible to use this book for any length of course between 10 and 15 weeks (or longer). The sections stand alone, but also fit into a coherent narrative within each chapter and the book as a whole.
This follows the usual progression of a calculus textbook: limits, derivatives, applications, integrals, and more applications. This means it's a very easy switch. The derivative rules are presented in a logical order, with motivating examples.
This book is great because it's available as a pdf, or in an easily navigable ebook form. From what I can tell, both work well on mobile devices. The images size and scale nicely across formats.
I found no grammar or syntax errors -- the level of quality matches any traditional, expensive book.
This is not usually a huge concern for a mathematics textbook, but I can see that the authors took particular care to find authentic application problems from a wide variety of contexts. The physics applications are explained well enough so that a student would not need a physics background to be successful. There are numerous examples from economics and biology throughout the text as well.
Reviewed by Nicole Kraft, Math Instructor, Portland Community College on 6/19/18
From the start, this book gives a comprehensive (yet straight forward) review of the necessary function knowledge. There is even a “Review of Pre-Calculus” at the end of the text, which contains all relevant formulas and identities. Before... read more
From the start, this book gives a comprehensive (yet straight forward) review of the necessary function knowledge. There is even a “Review of Pre-Calculus” at the end of the text, which contains all relevant formulas and identities.
Before diving into calculus, there is a section which shows the student the basic ideas covered in differential and integral calculus. After limits, differentiation is covered (with applications contained in a following section) and then integration is covered (with applications in the subsequent section). For my institution, all relevant topics are covered.
Plenty of exercises are placed throughout the text and include solutions for about half of them, similar to what is done for the odd numbered problems in a physical textbook.
The book is accurate, although I did not complete all exercises in order to check their solutions. In reading through the written portions, I did not find any glaring mistakes. Not only is this textbook accurate, it is concise as well.
Calculus is a classic subject, so the topics here will not need to change. Since the text is thorough and organized in a clear way, instructors can easily select the topics to cover for a given course. As course topics change, the fact that the topics are presented separately here will make course changes straight forward. There won’t be too much picking and choosing of things from within a section.
The book is clear in its written language, definitions, and figures. There isn’t an overuse of greek letters, which I like. For the most part, functions are in x or t. The figures used throughout are clear, support learning, and are free of extraneous visual information.
All sections have a similar flow and the depth feels even. Exercises are place consistently throughout the written text.
This text is broken up nicely by topic. Each topic has a little backstory or introduction that may reference a previous section, but it does not heavily reference past sections. It feels like the sections could easily be selected and ordered at the instructor’s discretion, without too much confusion.
The topics are structured clearly as: function background, limits, derivatives, derivative applications, integrals, integral applications. In each section has 7-10 subsections. My only complaint is that the limits at infinity is listed with applications. I wish it were listed in the limits section.
The fact that this text has an online version is a huge plus for me. I wouldn't use a text that didn't. Also, the website is modern-looking and organized. There are enough colors used so that the user doesn't feel bored immediately or over time with the aesthetic. The images are of good quality and seem to be used at pretty much every opportunity.
My one complaint here is that the exercises don’t have numbers. It would be hard to assign specific problems without them having a label. Maybe you just assign all of them?
The grammar in this text is good and it is well written. Commas are used in correct places, which adds to clarity.
Cultural Relevance rating: 3
This text is a bit of a cultural void. Although the majority of the real-world objects referenced inanimate, the humans referenced are male. It seems like the text was intentionally written to be gender and culture free. The hes should be eliminated, if that is the case. Otherwise, the message is that only men are of note.
I really like this text book and would use it in the future if my college didn’t already have their own open textbook available for this course. I do find this text superior, though, and will recommend it to my students for use as supplementary study material.
Reviewed by Kira Hamman, Lecturer in Mathematics, Pennsylvania State University on 2/1/18
The book is comprehensive. It covers the entirety of the usual Calculus I curriculum and includes sections with applications that are particularly helpful. read more
The book is comprehensive. It covers the entirety of the usual Calculus I curriculum and includes sections with applications that are particularly helpful.
Alas, there are many errors in the print version of the book, some of which are also in the PDF version. I'm not sure why these could not be fixed at least in the electronic version as they are discovered. Some of these errors are minor typos, but others significantly change the meaning of the text, e.g. f'' where f' is meant, cos instead of sin, and so on.
The bulk of the material is timeless, but the examples used are modern and applicable.
The book is easy to understand, and most material is presented in a sensible order. To a large extent this is the traditional Calc I curriculum, in the traditional order.
The book is quite consistent. The terminology used, names of concepts and theorems, and so on are all standard in the discipline.
I have been very happy with how the text is broken up. Generally speaking, with few exceptions, it is possible to cover one section in a 50-minute class period. Sections that can be skipped are fairly evident. Of course, this is a subject which requires that prerequisite material be learned before later material, but the later sections to rely overmuch on the previous ones in terms of examples or definitions.
Again, the book uses the traditional sequence of topics for calculus I, as follows: 1. A review of algebra concepts 2. Introduction to limits 3. Derivatives 4. Integrals
The online version of the book and the downloadable PDF are both very easy to load, navigate, and read on-screen. However, the problems at the end of each section are not numbered in the online version of the book, and this makes it difficult for students to find the assigned problems unless they have a download (which they do not if, for example, they're working on a phone) or a hard copy.
I haven't noticed any English grammar errors, although as I said earlier there are some issues with mathematical errors (typos).
The text is about calculus, which is relevant to all cultures.
I recommend the book. The biggest problem I have had is with the errors that change meaning, but these are easy enough to spot and do not seem to be a problem in the online version. I have not had students complain about them much. A smaller but still irritating issue is the lack of numbers on the problems in the online version, which makes it difficult for students using the book on a mobile device to locate the homework. However, beyond those two things I find the book to be of excellent quality, particularly given that it is free.
Reviewed by Caleb Moxley, Visiting Assistant Professor, Randolph College on 8/15/17
The test covered all necessary topics for an introductory calculus course with a particularly strong eye to understanding functions. Glossaries appeared at the end of each section, and the index was useful and contained all expected references. A... read more
The test covered all necessary topics for an introductory calculus course with a particularly strong eye to understanding functions. Glossaries appeared at the end of each section, and the index was useful and contained all expected references. A universal glossary would have been useful.
I did not encounter any errors.
The material is timeless, and the examples used aren't too topical.
The text is easy to read and is pleasantly presented.
There are no consistency errors which I found.
The material is broken into manageable chunks and foundational material is covered before advanced material.
The text is well organized.
There are no interface issues.
I found very few typos or grammatical errors.
The book doesn't make a particular effort to include examples that contain a breadth of cultural relevance.
This is overall a very good text for an introductory calculus course.
Reviewed by Michelle Perschbacher, Adjunct Instructor of Mathematics, Northern Virginia Community College on 6/20/17
This book covers all major topics in a typical first calculus course. Our curriculum also includes numerical integration, which is in the corresponding Calculus II text, but that single section could be easily incorporated into our Calculus I... read more
This book covers all major topics in a typical first calculus course. Our curriculum also includes numerical integration, which is in the corresponding Calculus II text, but that single section could be easily incorporated into our Calculus I course. Extensive further-reaching problems and Student Projects for each chapter make this text suitable for honors sections as well. A comprehensive Table of Contents and Index are easily located at the beginning and end of the text, respectively. A variety of application problems requiring the use of technology (denoted with [T]) accompany solid pure math exercises.
No obvious errors jumped out at me.
The theoretical content is fairly timeless. Broad applications in biology, engineering, business, statistics, chemistry, and computer science for calculus are included. The real-world data will eventually require updating – a regular necessity for all textbooks – but individual problems can be seamlessly modernized as needed.
Corresponding diagrams and figures are strong. The addition of colored definition boxes (light blue) and problem-solving strategy boxes (light orange) makes key concepts easy to find. I appreciate that the authors took the time and space in example problem solutions to include algebraic steps that other texts tend to omit. I noticed some minor spacing problems with mathematical symbols, but this was more prominent in the online version than on the pdf.
Formatting is clear and consistent. This text provides a wide variety of examples and problems for each section.
The topics in this course are easily divided into the 6 chapters offered here. Each section is divided into subsections by objective, which can be customized to any curriculum. The text is organized in such a way to accommodate both Early Transcendental and Late Transcendental approaches.
The explanations of concepts are very readable. Section 2.1 gives a nice overview of calculus, providing scaffolding for students to see where the course is heading. Each chapter begins with an exploration of a real-world problem, which is tackled in more detail later in the chapter as the mathematical concepts for its solution develop.
Visually, I found the pdf version more appealing and easier to follow. The examples and section exercises are not numbered online in the same way as in the pdf format, making referencing difficult. That said, I appreciate the continuous numbering of section problems in the pdf version. For instance, having only one problem #450 in the entire text eliminates confusion. Links to helpful interactive applets and demonstrations through the Wolfram Demonstrations Project, GeoGebra, Khan Academy, as well as OpenStax are embedded in the text, although two of the links I tried were broken.
None that I found.
While particular emphasis on Newton and Leibniz is appropriate, this text could benefit from a wider span of historical features from other early contributors to calculus, including non-Europeans and women.
Overall, this is a solid reference text. Out of the partner resources that I was able to access with a guest login, no particular online software stood out from the crowd. Although the surface-type questions presented are sufficient for skill-building, I was unable to find more comprehensive, multi-step problems that require students to synthesize concepts while providing immediate feedback. Using one of these resources in tandem with some sort of paper-and-pencil assignments from the text is likely the best alternative but still requires hand grading. Nevertheless, seeing several software companies embrace the OER initiative is an encouraging first step.
Reviewed by Steve Leonhardi, Professor of Mathematics and Statistics, Winona State University on 6/20/17
The table of contents and material covered is very similar to most standard, traditional Calculus textbooks intended for the first semester of study. In that regard, this textbook is extremely comprehensive. I like the learning objectives... read more
The table of contents and material covered is very similar to most standard, traditional Calculus textbooks intended for the first semester of study. In that regard, this textbook is extremely comprehensive. I like the learning objectives clearly stated at the beginning of each section, and the chapter summary and review problems. The text follows the usual format of offering many instructive, detailed examples for students to mimic, but tends to emphasize computational skills over conceptual understanding. While the text does include some examples and exercises using graphical and tabular approaches, I would like to see more examples and exercises that emphasize conceptual understanding and that encourage the development of modeling skills. Many of the exercises are straightforward and simply computational. There are a reasonable number of problems that involve applications, but in most of these, students are given the formula to use as if it were “pulled out of a hat” rather than derived by the student’s reasoning from general principles. I would like to see more interesting problems that emphasize deep conceptual understanding, or that require students to creatively bring together pieces of knowledge that come from different sections of the course. The “Student Projects” are of this type, but I would like to see more of these. I would need to supplement this textbook more than I would need to supplement other commercial options.
The content is accurate and unbiased. I did not find any errors in the text.
The text follows the usual format of a standard Calculus course, which tends to change little over the decades. The links to web resources and online data are in many cases helpful and enticing, but will require updates over time, not only to maintain functioning URL’s, but also to continue to refer to up-to-date data and examples. The applets at CalculusApplets.com did not open, probably because of updated browser security requirements, and other applets seemed outdated or only partly functional. I like the idea of linking to external resources, but most commercial textbooks (in e-book form) would be more likely to have stable, functioning internal links to illustrations and applets.
The exposition is very clear, direct, succinct, and at an appropriate level of mathematical sophistication for my Calculus I students. That is, it addresses all important issues, but broken down into comprehensible steps, without being pedantic or overly technical. In several key sections, the text succeeds in pointing out and warning against common mistakes, such as incorrectly that assuming the converse of a conditional also holds, or using a delta that depends upon x. The clarity is one of the strongest features of this text.
The text is internally consistent in terms of terminology, notation, and framework.
The sections seem well-partitioned and well-paced (again, not varying much from the standard Calculus textbook). I would want to reorder my presentation of some of them, but it appears that would not cause any major problems.
The overall organization, structure, and flow is good. Personally, I would make the following changes: present Section 4.6: Limits at Infinity as part of Chapter 2 on limits. Present exponential derivatives earlier in Chapter 3. Present L’Hopital’s Rule earlier, when discussing using derivatives in graphing. But this is a matter of personal preference, and the modularity of the text makes all of these changes appear to be pretty easy for instructors to adapt to their preferred order of presentation.
Interface rating: 3
Navigation in the PDF version of the text could be improved. For one thing, I could not find a table of contents to navigate between different sections. Links to future examples and exercises are somewhat helpful, but it was not obvious how to return to the previous point in reading with the pdf file. The online HTML version includes the table of contents and is easier to navigate, but was somewhat slow to reload with my internet connection.
I did not find any grammatical or typographical errors. That said, I’m much more likely to notice errors (or have them brought to my attention by students) when actually using the text for a course.
The text is not culturally insensitive or offensive in any way. However, I would prefer a text that contains more historical observations or side-notes than this one.
The strong points of this text are clear, straightforward explanation and examples of the standard computational techniques of Calculus. Any instructor wanting to focus on computational skills would be completely happy with this text. There is some inclusion of the “rule of four” (graphical, tabular, and verbal approaches in addition to symbolic computations), but not as much as I personally would like to see. The text could be improved, in my opinion, by greater inclusion of conceptual examples and exercises, and more modeling.
Reviewed by Angela Simons, Mathematics Instructor, Century College on 6/20/17
The text covers the same material that is covered in Calculus 1 textbooks that I have used in the past and that other members of the department still use. There is an index at the end of the text and there is a glossary at the end of each section.... read more
The text covers the same material that is covered in Calculus 1 textbooks that I have used in the past and that other members of the department still use. There is an index at the end of the text and there is a glossary at the end of each section. It would be helpful if there was also a comprehensive glossary, especially in the pdf of the book for when it is printed.
I used this textbook in Calculus 1 during fall semester 2016. We did many of the problems both in class and as assigned problems and found no errors. For some of the worked examples in the text the students sometimes had a difficult time understanding what was being done but this is not uncommon regardless of the textbook. They do make an effort to update any errors that might be found and sent to them, as is stated in the preface to the book "Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on openstax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on openstax.org."
The text makes attempts to give examples and problems that are current and up-to-date. Given the subject matter the text will likely stay relevance for a long time.
The text is written in a way that is generally easy to read although as mentioned before some of the examples students had a difficult time following. Also if using the online text, it is important that one uses the full screen view of the text as some of the diagrams become clutter and difficult to decipher because labeling is placed very close together. There are some pages where even if looking at the print version the diagrams are hard to fully understand. For example in section 3.1, Defining the Derivative the diagrams cluttered and students who are being exposed to the idea of the derivative for the first time may not understand what label goes with what.
Terminology is used in a consistent manner.
Some of the sections cover quite a lot of material, sometime too much to be covered with in a 50 minute class period which is not terribly uncommon. It was an easy task to find a suitable stopping point to fit within the allotted time.
When using the book in class I changed the order of some of the sections. Most specifically, section 4.6 "Limits at Infinity and Asymptotes" was covered in chapter 1 and chapter 2 while talking about functions and limits. When we got to chapter 4 the class was reminded of our previous discussion and we moved on.
The online text is easy to navigate to the start of a particular section using the table of contents. Also in the online text the sample problems have the solutions hidden so that the problems can be done without being influenced by their presence. A positive change to the online version would be if it were possible to jump to the exercises that appear at the end of the section. Some of the tables and diagrams in the sections seemed larger than necessary and not as organized as they might be. It would have been a nice addition to the online text to have links to animations that might illustrate a particular concept, like the derivative.
In the pdf version of the book, the problems at the end of the section are numbered, it would be nice if the online version used the same numbering. It is difficult to use the online version in class and call students attention to a problem in their printed pdf copy. Also it is not possible in the pdf version of the text to jump to a section once you have navigated to a chapter. Each of the sections should be clickable so that by doing so you are taken to the start of the section. Further there should be a way to navigate to the end of the section to access the exercises without having to scroll all the way through.
I don't recall any grammatical errors.
Cultural content is slight. There's the obligatory picture of Newton and Leibniz and a nod to Archimedes but little else.
As mentioned this book was used to teach Calculus 1 in the fall of 2016. I used the book in conjunction with MyOpenMath. Used together the students found the resources helpful. This text made a suitable replacement for the text that I had used previously.
Reviewed by Elijah Bunnell, Mathematics Instructor, Rogue Community College on 4/11/17
This text was very comprehensive. It covered every section that our current book covers for 251 and 252. read more
This text was very comprehensive. It covered every section that our current book covers for 251 and 252.
I found no errors. Of course there are probably many considering it is a newer math book. No bias was present.
Examples given covered topics that should endure for a good amount of time. Relativity, rockets, swimmers and runners, windows... this book won't feel dated in 10 years.
All of the author's explanations were exceedingly clear. This was one of the features I most appreciated. Diagrams were not overly cluttered, each page was free of distracting margin comments and very to the point.
I found no inconsistencies.
This book follows the traditional layout of a calculus book. The sections lined up almost exactly with our current book. In fact, we currently cover 251 in 24 sections, and Open stax covers the material in 25 sections.
Very logical flow. Again, this book structured in a similar way to our current book. Switching to open stax would be nearly effortless.
The images and graphs appeared to be lower budget. I should also not that the images and graphs were also free of clutter and easily understood. Many of the tables were oversized and distracting.
If there are grammar errors in the book, they did not distract from the content. Again this book is written in a simple clear manner.
The text is not culturally insensitive or offensive in any way. Examples and problems do not make reference to individuals race or ethnicity.
I found this book to be clear and logically laid out. There were nice pieces of history interjected. The layout was intuitive. Each section was well motivated with examples. I also appreciated that volume 1 only covered only differential and integral calculus.
Reviewed by Joshua Fitzgerald, Instructor, Miami University on 8/21/16
This text covers the same material as other common Calculus I textbooks. I was unable to find any major topic that is covered in my classes currently that wasn't covered in this book. There are helpful glossaries at the end of each chapter, but no... read more
This text covers the same material as other common Calculus I textbooks. I was unable to find any major topic that is covered in my classes currently that wasn't covered in this book. There are helpful glossaries at the end of each chapter, but no universal glossary for the entire textbook. There is an index at the back.
I worked through a few examples and exercises and did not find any errors.
The nature of the subject makes it difficult to imagine a calculus book becoming out-of-date. The non-mathematical content of some textbooks (like historical notes) can become irrelevant or outdated, but this textbook has very little non-mathematical content and so it is not in danger of becoming out-of-date quickly.
Clarity rating: 3
The text is written in an accessible way and the prose is easy to read. Most figures were well-designed, but a few were cluttered. In particular, the critical diagrams showing the construction of the derivative were difficult to decipher due to the labels being nearly on top of one another.
The textbook is very consistent in its visual presentation. I did not notice any inconsistencies in terminology.
This textbook easily divides into small sections and subsections, as most math textbooks do. The sections are often too long for an hour-long lesson but the divisibility of the book allows the instructor to shorten or lengthen a lesson to fit the time allowed.
This book has a similar structure to that of Stewart or Briggs. The content is broken up into 6 chapters covering essentially the same topics as those popular textbooks. One major difference: Limits at Infinity are not covered until just before Optimization, after the students have already been graphing functions using the derivative. The section on Limits at Infinity does not appear to rely on derivatives at all, so it could easily be taught with the rest of the material on limits if the instructor chooses.
The online interface is nearly identical to the static PDF file available for download. The online version hides solutions for the example problems by default, allowing the reader to attempt the problem without being influenced by a visible solution. Some of the diagrams were larger and easier to read in the online version. It is simple to navigate to a particular section using the Table of Contents in the online interface. However I could not find a way to navigate to a particular page by the page number.
I did not find any grammatical errors.
Cultural content is very thin in this book, so there isn't much to critique here. I did notice that Newton, Leibniz, and other European mathematicians are mentioned, while there is no mention of the contributions and discoveries of non-European mathematicians.
This book would make a suitable replacement for other popular calculus textbooks such as Stewart or Briggs.
As a part of this review, I was not able to use the accompanying online homework system, WeBWorK. In my experience, students spend more time interacting with the online homework system than they do the textbook. An online homework system that is easy to use for both the instructor and the student is essential.
Table of Contents
- Chapter 1: Functions and Graphs
- Chapter 2: Limits
- Chapter 3: Derivatives
- Chapter 4: Applications of Derivatives
- Chapter 5: Integration
- Chapter 6: Applications of Integrations
Ancillary Material
About the book.
Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 1 covers functions, limits, derivatives, and integration.
OpenStax College has compiled many resources for faculty and students, from faculty-only content to interactive homework and study guides.
About the Contributors
Gilbert Strang was an undergraduate at MIT and a Rhodes Scholar at Balliol College, Oxford. His Ph.D. was from UCLA and since then he has taught at MIT. He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences. He is a Professor of Mathematics at MIT, an Honorary Fellow of Balliol College, and a member of the National Academy of Sciences. Professor Strang has published eleven books.
He was the President of SIAM during 1999 and 2000, and Chair of the Joint Policy Board for Mathematics. He received the von Neumann Medal of the US Association for Computational Mechanics, and the Henrici Prize for applied analysis. The first Su Buchin Prize from the International Congress of Industrial and Applied Mathematics, and the Haimo Prize from the Mathematical Association of America, were awarded for his contributions to teaching around the world. His home page is math.mit.edu/~gs/ and his video lectures on linear algebra and on computational science and engineering are on ocw.mit.edu
Edwin "Jed" Herman , Professor, Department of Mathematical Sciences, University of Wisconsin-Stevens Point. Ph.D., Mathematics, University of Oregon?.?
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Math 1a Spring 2020
1a introduction to calculus.
Handouts and Homework
Math 1a, Home | Oliver Knill, [email protected] ,SciCenter 432, (617) 495-5549 | Department of Mathematics | FAS | Canvas | Harvard University
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All homework will be due at midnight (actually 11:59) on the day listed. Online Hw # 1 - Due Wednesday 1/29 Volume by Slicing (cross-section, disk, and washer) and Volume by Shells. Online Hw # 2 - Due Wednesday 2/5 Arc Length, Surface Area of Revolution, and Center of Mass. Online Hw # 3 - Due Wednesday 2/12 Integration by Parts.
Unit 1: Limits and continuity. 0/3500 Mastery points. Limits intro Estimating limits from graphs Estimating limits from tables Formal definition of limits (epsilon-delta) Properties of limits Limits by direct substitution Limits using algebraic manipulation Strategy in finding limits. Squeeze theorem Types of discontinuities Continuity at a ...
This booklet contains worksheets for the Math 180 Calculus 1 course at the University of Illinois at Chicago. There are 27 worksheets, each covering a certain topic of the course curriculum. At the end of the booklet there are 2 review worksheets, covering parts of the course (based on a two-midterm model). In a 15-week semester, completing 2 ...
Our mission is to improve educational access and learning for everyone. OpenStax is part of Rice University, which is a 501 (c) (3) nonprofit. Give today and help us reach more students. This free textbook is an OpenStax resource written to increase student access to high-quality, peer-reviewed learning materials.
A walk-through of the solutions for Exam 2 of Calculus 1 administered in Spring 2022.For more information: https://www.calc1.org/Presenter: Steve Butler (htt...
These are homework exercises to accompany Chapter 17 of OpenStax's "Calculus" Textmap. Thumbnail: The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. (GNU Free Documentation License, Version 1.3 and CC- SA-BY 3.0; Wikipedia). These are homework exercises to accompany ...
Finding definite integrals using area formulas. (Opens a modal) Definite integral over a single point. (Opens a modal) Integrating scaled version of function. (Opens a modal) Switching bounds of definite integral. (Opens a modal) Integrating sums of functions.
If you are looking for calculus exercises to practice your skills and test your knowledge, check out this webpage from Mathematics LibreTexts. It contains a variety of problems and solutions on functions and graphs, based on Chapter 1 of OpenStax's "Calculus" Textmap. You can also explore other related topics in biology, microbiology, and more from the LibreTexts library.
Answer Key Chapter 1 - Calculus Volume 1 | OpenStax. Our mission is to improve educational access and learning for everyone. OpenStax is part of Rice University, which is a 501 (c) (3) nonprofit. Give today and help us reach more students.
For both AB and BC courses. This version follows CollegeBoard's Course and Exam Description. It was built for a 45-minute class period that meets every day, so the lessons are shorter than our Calculus Version #2. Version #2. . Covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day. This ...
The Why It Matters page presents a real-world example of a calculus concept to be taught in the module. Then the problem is restated, solved, and explained on the Putting It Together page once the module is completed. ... The purpose of these questions is different than the questions on homework or quizzes, which are designed to assess mastery ...
This website contains lecture notes and homework assignments for 14 undergraduate and master's level math courses that I have taught at Hunter College, City College of New York, Columbia University and Stanford University. The courses cover most requirements at U.S. colleges/universities for both an undergraduate math major and for non-math majors in other fields.
www.cims.nyu.edu
Step-by-step solution. Step 1 of 3. The objective is to prove algebraically that a vertical or horizontal shift or stretch of a cubic function is also a cubic function. Step 2 of 3. Assume that the function and the vertical shift of the function are as follows. Shift the function vertically by the quantity .
Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes ...
Now, with expert-verified solutions from Calculus, Volume 1 1st Edition, you'll learn how to solve your toughest homework problems. Our resource for Calculus, Volume 1 includes answers to chapter exercises, as well as detailed information to walk you through the process step by step. With Expert Solutions for thousands of practice problems ...
Free math problem solver answers your calculus homework questions with step-by-step explanations. Mathway. Visit Mathway on the web. Start 7-day free trial on the app ... Download free on Amazon. Download free in Windows Store. get Go. Calculus. Basic Math. Pre-Algebra. Algebra. Trigonometry. Precalculus. Calculus. Statistics. Finite Math ...
Introduction to Calculus Spring 2020. Math 1a Spring 2020 1a Introduction to Calculus. Home; Syllabus; Handouts; Q & A; Exam; Data; Exhibit; Quizz; ... We have a strict no-late homework submission policy. If you should get help from AI (computer algebra systems in particular), acknowledge it. If you use external resources (like literature or ...
The Calculus Worksheets are randomly created and will never repeat so you have an endless supply of quality Calculus Worksheets to use in the classroom or at home. These Calculus Worksheets consist of Integration, Differential Equation, Differentiation, and applications Worksheets for your use. Our Calculus Worksheets are free to download, easy ...
Ace problem sets with instant answers & step-by-step solutions. Your on-demand CS tutor — get instant homework help with step-by-step solutions for your Discrete Math, Linear Algebra, Statistics, and Calculus problem sets. Use CompSciLib's homework help feature to help with your Computer Science theory and math courses.undefined.