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Calculus Volume 1

Gilbert Strang

Chapter 1

Functions and Graphs

Educators


Problem 1

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Linh V.
Numerade Educator

Problem 2

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Mihir N.
Numerade Educator

Problem 3

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Linh V.
Numerade Educator

Problem 4

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Mihir N.
Numerade Educator

Problem 5

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Linh V.
Numerade Educator

Problem 6

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Mihir N.
Numerade Educator

Problem 7

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. $f(a+h)$
$$f(x)=5 x-2$$

Linh V.
Numerade Educator

Problem 8

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. $f(a+h)$
$$f(x)=4 x^{2}-3 x+1$$

Mihir N.
Numerade Educator

Problem 9

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. $f(a+h)$
$$f(x)=\frac{2}{x}$$

Linh V.
Numerade Educator

Problem 10

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. $f(a+h)$
$$f(x)=|x-7|+8$$

Mihir N.
Numerade Educator

Problem 11

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. $f(a+h)$
$$f(x)=\sqrt{6 x+5}$$

Linh V.
Numerade Educator

Problem 12

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. $f(a+h)$
$$f(x)=\frac{x-2}{3 x+7}$$

Mihir N.
Numerade Educator

Problem 13

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. $f(a+h)$
$$f(x)=9$$

Linh V.
Numerade Educator

Problem 14

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
$$
f(x)=\frac{x}{x^{2}-16}
$$

Mihir N.
Numerade Educator

Problem 15

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
$$
g(x)=\sqrt{8 x-1}
$$

Linh V.
Numerade Educator

Problem 16

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
$$
h(x)=\frac{3}{x^{2}+4}
$$

Mihir N.
Numerade Educator

Problem 17

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
$$
f(x)=-1+\sqrt{x+2}
$$

Linh V.
Numerade Educator

Problem 18

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
$$
f(x)=\frac{1}{\sqrt{x-9}}
$$

Mihir N.
Numerade Educator

Problem 19

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
$$
g(x)=\frac{3}{x-4}
$$

Linh V.
Numerade Educator

Problem 20

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

Mihir N.
Numerade Educator

Problem 21

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.
$$
g(x)=\sqrt{\frac{7}{x-5}}
$$

Linh V.
Numerade Educator

Problem 22

For the following exercises, set up a table to sketch the graph of each function using the following values: $x=-3,-2,-1,0,1,2,3$
$$f(x)=x^{2}+1$$

Mihir N.
Numerade Educator

Problem 23

For the following exercises, set up a table to sketch the graph of each function using the following values: $x=-3,-2,-1,0,1,2,3$
$$
f(x)=3 x-6
$$

Linh V.
Numerade Educator

Problem 24

For the following exercises, set up a table to sketch the graph of each function using the following values: $x=-3,-2,-1,0,1,2,3$
$$
f(x)=\frac{1}{2} x+1
$$

Mihir N.
Numerade Educator

Problem 25

For the following exercises, set up a table to sketch the graph of each function using the following values: $x=-3,-2,-1,0,1,2,3$
$$
f(x)=2|x|
$$

Linh V.
Numerade Educator

Problem 26

For the following exercises, set up a table to sketch the graph of each function using the following values: $x=-3,-2,-1,0,1,2,3$
$$
f(x)=-x^{2}
$$

Mihir N.
Numerade Educator

Problem 27

For the following exercises, set up a table to sketch the graph of each function using the following values: $x=-3,-2,-1,0,1,2,3$
$$
f(x)=x^{3}
$$

Linh V.
Numerade Educator

Problem 28

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Mihir N.
Numerade Educator

Problem 29

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Linh V.
Numerade Educator

Problem 30

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Linh V.
Numerade Educator

Problem 31

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Linh V.
Numerade Educator

Problem 32

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Mihir N.
Numerade Educator

Problem 33

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Linh V.
Numerade Educator

Problem 34

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Linh V.
Numerade Educator

Problem 35

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:
a. Domain and range
b. $x$ -intercept, if any (estimate where necessary)
c. $y$ -Intercept, if any (estimate where necessary)
d. The intervals for which the function is increasing
e. The intervals for which the function is decreasing
f. The intervals for which the function is decreasing
f. The intervals for which the function is conigin
g. Symmetry about any axis and/or the origin
h. Whether the function is even, odd, or neither

Linh V.
Numerade Educator

Problem 36

For the following exercises, for each pair of functions, find a. $f+g$ b. $f-g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.
$$
f(x)=3 x+4, g(x)=x-2
$$

Mihir N.
Numerade Educator

Problem 37

For the following exercises, for each pair of functions, find a. $f+g$ b. $f-g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.
$$
f(x)=x-8, g(x)=5 x^{2}
$$

Linh V.
Numerade Educator

Problem 38

For the following exercises, for each pair of functions, find a. $f+g$ b. $f-g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.
$$
f(x)=3 x^{2}+4 x+1, g(x)=x+1
$$

Mihir N.
Numerade Educator

Problem 39

For the following exercises, for each pair of functions, find a. $f+g$ b. $f-g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.
$$
f(x)=9-x^{2}, g(x)=x^{2}-2 x-3
$$

Linh V.
Numerade Educator

Problem 40

For the following exercises, for each pair of functions, find a. $f+g$ b. $f-g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.
$$
f(x)=\sqrt{x}, g(x)=x-2
$$

Mihir N.
Numerade Educator

Problem 41

For the following exercises, for each pair of functions, find a. $f+g$ b. $f-g$ c. $f \cdot g$ d. $f / g .$ Determine the domain of each of these new functions.
$$
f(x)=6+\frac{1}{x}, g(x)=\frac{1}{x}
$$

Linh V.
Numerade Educator

Problem 42

For the following exercises, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results.
$$
f(x)=3 x, g(x)=x+5
$$

Mihir N.
Numerade Educator

Problem 43

For the following exercises, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results.
$$
f(x)=x+4, g(x)=4 x-1
$$

Linh V.
Numerade Educator

Problem 44

For the following exercises, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results.
$$
f(x)=2 x+4, g(x)=x^{2}-2
$$

Mihir N.
Numerade Educator

Problem 45

For the following exercises, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results.
$$
f(x)=x^{2}+7, g(x)=x^{2}-3
$$

Linh V.
Numerade Educator

Problem 46

For the following exercises, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results.
$$
f(x)=\sqrt{x}, g(x)=x+9
$$

Mihir N.
Numerade Educator

Problem 47

For the following exercises, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results.
$$
f(x)=\frac{3}{2 x+1}, g(x)=\frac{2}{x}
$$

Linh V.
Numerade Educator

Problem 48

For the following exercises, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results.
$$
f(x)=|x+1|, g(x)=x^{2}+x-4
$$

Mihir N.
Numerade Educator

Problem 49

The table below lists the NBA championship winners for the years 2001 to 2012 .
a. 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.
b. 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.

Linh V.
Numerade Educator

Problem 50

The area A of a square depends on the length of the side s.
a. Write a function A(s) for the area of a square.
b. Find and interpret A(6.5).
c. Find the exact and the two-significant-digit approximation to the length of the sides of a square
with area 56 square units.

Mihir N.
Numerade Educator

Problem 51

[T] The volume of a cube depends on the length of the sides s.
a. Write a function V(s) for the area of a square.
b. Find and interpret V(11.8).

Linh V.
Numerade Educator

Problem 52

[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.
a. Write the formula for the function that models this situation.
b. Find the total cost to rent a car for 2 days and 7 hours.
c. Determine how long the car was rented if the bill is $432.73.

Mihir N.
Numerade Educator

Problem 53

[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.
a. Write a formula that models this situation.
b. 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.
c. Determine the domain and range of the function.
d. Determine how many times the driver had to stop for gas if she has driven a total of 578 $\mathrm{mi}$ .

Linh V.
Numerade Educator

Problem 54

[T] The volume $V$ of a sphere depends on the length of its radius as $V=(4 / 3) \pi r^{3}$ . 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 $6.371 \times 10^{6} \mathrm{m} .$

Mihir N.
Numerade Educator

Problem 55

[T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r(t)=6-\left[5 /\left(t^{2}+1\right)\right], \quad$ where $t$ is time
measured in hours since a circle of a 1 -cm radius of the bacterium was put into the culture.
a. Express the area of the bacteria as a function of time.
b. Find the exact and approximate area of the bacterial culture in 3 hours.
c. Express the circumference of the bacteria as a function of time.
d. Find the exact and approximate circumference of the bacteria in 3 hours.

Linh V.
Numerade Educator

Problem 56

[T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function $E(x)=0.79 x,$ 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 $D(x)=1.245 x,$ where $x$ is the number of Euros and
$D(x)$ is the equivalent number of U.S. dollars.
a. 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?
b. 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.

Mihir N.
Numerade Educator

Problem 57

[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.
a. Write a function $y=S(x)$ that models a worker's monthly salary based on the number of skateboards $x$ he or she sells.
b. Find the approximate monthly salary when a worker sells $25,40,$ or 55 skateboards.
c. 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 $y=1400 .$ )

Linh V.
Numerade Educator

Problem 58

[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.

Mihir N.
Numerade Educator