🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus Volume 1 ## Gilbert Strang ## Chapter 1 ## Functions and Graphs ## Educators  ### Problem 1 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. Linh V. Numerade Educator ### Problem 2 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. Mihir N. Numerade Educator ### Problem 3 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. Linh V. Numerade Educator ### Problem 4 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. Mihir N. Numerade Educator ### Problem 5 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. Linh V. Numerade Educator ### Problem 6 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. Mihir N. Numerade Educator ### Problem 7 For the following exercises, find the values for each function, if they exist, then simplify. a.$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.

### 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.

### 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.

### 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.

### 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.
[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. 