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Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 4

Integration

Educators


Problem 1

In Exercises $1-4,$ verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$\int\left(-\frac{6}{x^{4}}\right) d x=\frac{2}{x^{3}}+C$$

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Problem 2

In Exercises $1-4,$ verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$\int\left(8 x^{3}+\frac{1}{2 x^{2}}\right) d x=2 x^{4}-\frac{1}{2 x}+C$$

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Problem 3

In Exercises $1-4,$ verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$\int(x-4)(x+4) d x=\frac{1}{3} x^{3}-16 x+C$$

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Problem 4

In Exercises $1-4,$ verify the statement by showing that the derivative of the right side equals the integrand of the left side.
$$\int \frac{x^{2}-1}{x^{3 / 2}} d x=\frac{2\left(x^{2}+3\right)}{3 \sqrt{x}}+C$$

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Problem 5

In Exercises $5-8$ , find the general solution of the differential equation and check the result by differentiation.
$$\frac{d y}{d t}=9 t^{2}$$

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Problem 6

In Exercises $5-8$ , find the general solution of the differential equation and check the result by differentiation.
$$\frac{d r}{d \theta}=\pi$$

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Problem 7

In Exercises $5-8$ , find the general solution of the differential equation and check the result by differentiation.
$$\frac{d y}{d x}=x^{3 / 2}$$

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Problem 8

In Exercises $5-8$ , find the general solution of the differential equation and check the result by differentiation.
$$\frac{d y}{d x}=2 x^{-3}$$

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Problem 9

In Exercises $9-14,$ complete the table.
$$\begin{array}{l}{\text { Original Integral }} & \text { Rewrite } & \text { Integrate }& \text { Simplify }\\ {\int \sqrt[3]{x} d x}\end{array}$$

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Problem 10

In Exercises $9-14,$ complete the table.
$$\begin{array}{l}{\text { Original Integral }} & \text { Rewrite } & \text { Integrate }& \text { Simplify }\\ {\int \frac{1}{4 x^{2}} d x}\end{array}$$

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Problem 11

In Exercises $9-14,$ complete the table.
$$\begin{array}{l}{\text { Original Integral }} & \text { Rewrite } & \text { Integrate }& \text { Simplify }\\ {\int \frac{1}{x \sqrt{x}} d x}\end{array}$$

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Problem 12

In Exercises $9-14,$ complete the table.
$$\begin{array}{l}{\text { Original Integral }} & \text { Rewrite } & \text { Integrate }& \text { Simplify }\\ {\int x\left(x^{3}+1\right) d x}\end{array}$$

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Problem 13

In Exercises $9-14,$ complete the table.
$$\begin{array}{l}{\text { Original Integral }} & \text { Rewrite } & \text { Integrate }& \text { Simplify }\\ {\int \frac{1}{2 x^{3}} d x}\end{array}$$

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Problem 14

In Exercises $9-14,$ complete the table.
$$\begin{array}{l}{\text { Original Integral }} & \text { Rewrite } & \text { Integrate }& \text { Simplify }\\ {\int \frac{1}{(3 x)^{2}} d x}\end{array}$$

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Problem 15

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int(x+7) d x$$

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Problem 16

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int(13-x) d x$$

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Problem 17

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(2 x-3 x^{2}\right) d x$$

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Problem 18

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(8 x^{3}-9 x^{2}+4\right) d x$$

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Problem 19

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(x^{5}+1\right) d x$$

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Problem 20

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(x^{3}-10 x-3\right) d x$$

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Problem 21

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(x^{3 / 2}+2 x+1\right) d x$$

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Problem 22

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x$$

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Problem 23

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int \sqrt[3]{x^{2}} d x$$

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Problem 24

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(\sqrt[4]{x^{3}}+1\right) d x$$

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Problem 25

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int \frac{1}{x^{5}} d x$$

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Problem 26

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int \frac{1}{x^{6}} d x$$

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Problem 27

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int \frac{x+6}{\sqrt{x}} d x$$

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Problem 28

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int \frac{x^{2}+2 x-3}{x^{4}} d x$$

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Problem 29

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int(x+1)(3 x-2) d x$$

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Problem 30

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int\left(2 t^{2}-1\right)^{2} d t$$

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Problem 31

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int y^{2} \sqrt{y} d y$$

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Problem 32

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int(1+3 t) t^{2} d t$$

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Problem 33

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int d x$$

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Problem 34

In Exercises $15-34$ , find the indefinite integral and check the result by differentiation.
$$\int 14 d t$$

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Problem 35

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int(5 \cos x+4 \sin x) d x$$

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Problem 36

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int\left(t^{2}-\cos t\right) d t$$

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Problem 37

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int(1-\csc t \cot t) d t$$

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Problem 38

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int\left(\theta^{2}+\sec ^{2} \theta\right) d \theta$$

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Problem 39

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int\left(\sec ^{2} \theta-\sin \theta\right) d \theta$$

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Problem 40

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int \sec y(\tan y-\sec y) d y$$

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Problem 41

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int\left(\tan ^{2} y+1\right) d y$$

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Problem 42

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int\left(4 x-\csc ^{2} x\right) d x$$

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Problem 43

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int \frac{\cos x}{1-\cos ^{2} x} d x$$

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Problem 44

In Exercises $35-44,$ find the indefinite integral and check the result by differentiation.
$$\int \frac{\sin x}{1-\sin ^{2} x} d x$$

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Problem 45

In Exercises $45-48$ , the graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, go to the website www. mathgraphs.com.
GRAPH CAN'T COPY

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Problem 46

In Exercises $45-48$ , the graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, go to the website www. mathgraphs.com.
GRAPH CAN'T COPY

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Problem 47

In Exercises $45-48$ , the graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, go to the website www. mathgraphs.com.
GRAPH CAN'T COPY

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Problem 48

In Exercises $45-48$ , the graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, go to the website www. mathgraphs.com.
GRAPH CAN'T COPY

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Problem 49

In Exercises 49 and 50, find the equation of $y,$ given the derivative and the indicated point on the curve.
$$\frac{d y}{d x}=2 x-1$$

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Problem 50

In Exercises 49 and 50, find the equation of $y,$ given the derivative and the indicated point on the curve.
$$\frac{d y}{d x}=2(x-1)$$

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Problem 51

Slope Fields In Exercises $51-54,$ a differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, go to the website www. mathgraphs.com.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$\frac{d y}{d x}=\frac{1}{2} x-1,(4,2)$$

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Problem 52

Slope Fields In Exercises $51-54,$ a differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, go to the website www. mathgraphs.com.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$\frac{d y}{d x}=x^{2}-1,(-1,3)$$

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Problem 53

Slope Fields In Exercises $51-54,$ a differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, go to the website www. mathgraphs.com.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$\frac{d y}{d x}=\cos x,(0,4)$$

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Problem 54

Slope Fields In Exercises $51-54,$ a differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, go to the website www. mathgraphs.com.) (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a).
$$\frac{d y}{d x}=-\frac{1}{x^{2}}, x>0,(1,3)$$

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Problem 55

Slope Fields In Exercises 55 and 56, (a) use a graphing utility to graph a slope field for the differential equation, (b) use integration and the given point to find the particular solution of the differential equation, and (c) graph the solution and the slope field in the same viewing window.
$$\frac{d y}{d x}=2 x,(-2,-2)$$

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Problem 56

Slope Fields In Exercises 55 and 56, (a) use a graphing utility to graph a slope field for the differential equation, (b) use integration and the given point to find the particular solution of the differential equation, and (c) graph the solution and the slope field in the same viewing window.
$$\frac{d y}{d x}=2 \sqrt{x},(4,12)$$

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Problem 57

In Exercises $57-64,$ solve the differential equation.
$$f^{\prime}(x)=6 x, f(0)=8$$

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Problem 58

In Exercises $57-64,$ solve the differential equation.
$$g^{\prime}(x)=6 x^{2}, g(0)=-1$$

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Problem 59

In Exercises $57-64,$ solve the differential equation.
$$h^{\prime}(t)=8 t^{3}+5, h(1)=-4$$

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Problem 60

In Exercises $57-64,$ solve the differential equation.
$$f^{\prime}(s)=10 s-12 s^{3}, f(3)=2$$

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Problem 61

In Exercises $57-64,$ solve the differential equation.
$$f^{\prime \prime}(x)=2, f^{\prime}(2)=5, f(2)=10$$

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Problem 62

In Exercises $57-64,$ solve the differential equation.
$$f^{\prime \prime}(x)=x^{2}, f^{\prime}(0)=8, f(0)=4$$

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Problem 63

In Exercises $57-64,$ solve the differential equation.
$$f^{\prime \prime}(x)=x^{-3 / 2}, f^{\prime}(4)=2, f(0)=0$$

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Problem 64

In Exercises $57-64,$ solve the differential equation.
$$f^{\prime \prime}(x)=\sin x, f^{\prime}(0)=1, f(0)=6$$

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Problem 65

Tree Growth An evergreen nursery usually sells a certain type of shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by $d h / d t=1.5 t+5$ , where $t$ is the time in years and $h$ is the height in centimeters. The seedlings are 12 centimeters tall when planted $(t=0) .$
(a) Find the height after $t$ years.
(b) How tall are the shrubs when they are sold?

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Problem 66

Population Growth The rate of growth $d P / d t$ of a population of bacteria is proportional to the square root of $t,$ where $P$ is the population size and $t$ is the time in days $(0 \leq t \leq 10)$ . That is, $d P / d t=k \sqrt{t}$ . The initial size of the population is $500 .$ After 1 day the population has grown to $600 .$ Estimate the population after 7 days.

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Problem 67

What is the difference, if any, between finding the antiderivative of $f(x)$ and evaluating the integral $\int f(x) d x ?$

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Problem 68

Consider $f(x)=\tan ^{2} x$ and $g(x)=\sec ^{2} x .$ What do you notice about the derivatives of $f(x)$ and $g(x) ?$ What can you conclude about the relationship between $f(x)$ and $g(x) ?$

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Problem 69

The graphs of $f$ and $f^{\prime}$ each pass through the origin. Use the graph of $f^{\prime \prime}$ shown in the figure to sketch the graphs of $f$ and $f^{\prime} .$ To print an enlarged copy of the graph, go to the website www. mathgraphs.com.

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Problem 70

Use the graph of $f^{\prime}$ shown in the figure to answer the following, given that $f(0)=-4 .$
(a) Approximate the slope of $f$ at $x=4 .$ Explain.
(b) Is it possible that $f(2)=-1 ?$ Explain.
(c) $\operatorname{ls} f(5)-f(4)>0 ?$ Explain.
(d) Approximate the value of $x$ where $f$ is maximum. Explain.
(e) Approximate any intervals in which the graph of $f$ is concave upward and any intervals in which it is concave downward. Approximate the $x$ -coordinates of any points of inflection.
(f) Approximate the $x$ -coordinate of the minimum of $f^{\prime \prime}(x)$ .
(g) Sketch an approximate graph of $f .$ To print an enlarged copy of the graph, go to the website www. mathgraphs,com.

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Problem 71

Vertical Motion In Exercises $71-74,$ use $a(t)=-32$ feet per second per second as the acceleration due to gravity. (Neglect air resistance.)
A ball is thrown vertically upward from a height of 6 feet with an initial velocity of 60 feet per second. How high will the ball go?

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Problem 72

Vertical Motion In Exercises $71-74,$ use $a(t)=-32$ feet per second per second as the acceleration due to gravity. (Neglect air resistance.)
Show that the height above the ground of an object thrown upward from a point $s_{0}$ feet above the ground with an initial velocity of $v_{0}$ feet per second is given by the function $f(t)=-16 t^{2}+v_{0} t+s_{0}$

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Problem 73

Vertical Motion In Exercises $71-74,$ use $a(t)=-32$ feet per second per second as the acceleration due to gravity. (Neglect air resistance.)
With what initial velocity must an object be thrown upward (from ground level) to reach the top of the Washington Monument (approximately 550 feet)?

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Problem 74

Vertical Motion In Exercises $71-74,$ use $a(t)=-32$ feet per second per second as the acceleration due to gravity. (Neglect air resistance.)
A balloon, rising vertically with a velocity of 16 feet per second, releases a sandbag at the instant it is 64 feet above the ground.
(a) How many seconds after its release will the bag strike the ground?
(b) At what velocity will it hit the ground?

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Problem 75

Vertical Motion In Exercises $75-78$ , use $a(t)=-9.8$ meters per second per second as the acceleration due to gravity. (Neglect air resistance.)
Show that the height above the ground of an object thrown upward from a point $s_{0}$ meters above the ground with an initial velocity of $v_{0}$ meters per second is given by the function
$f(t)=-4.9 t^{2}+v_{0} t+s_{0}$.

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Problem 76

Vertical Motion In Exercises $75-78$ , use $a(t)=-9.8$ meters per second per second as the acceleration due to gravity. (Neglect air resistance.)
The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of the time $t$ in seconds. How long will it take the rock to hit the canyon floor?

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Problem 77

Vertical Motion In Exercises $75-78$ , use $a(t)=-9.8$ meters per second per second as the acceleration due to gravity. (Neglect air resistance.)
A baseball is thrown upward from a height of 2 meters with an initial velocity of 10 meters per second. Determine its maximum height.

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Problem 78

Vertical Motion In Exercises $75-78$ , use $a(t)=-9.8$ meters per second per second as the acceleration due to gravity. (Neglect air resistance.)
With what initial velocity must an object be thrown upward (from a height of 2 meters) to reach a maximum height of 200 meters?

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Problem 79

Lunar Gravity On the moon, the acceleration due to gravity is $-1.6$ meters per second per second. A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?

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Problem 80

Escape Velocity The minimum velocity required for an object to escape Earth's gravitational pull is obtained from the solution of the equation
$$\int v d v=-G M \int \frac{1}{y^{2}} d y$$
where $v$ is the velocity of the object projected from Earth, $y$ is the distance from the center of Earth, $G$ is the gravitational constant, and $M$ is the mass of Earth. Show that $v$ and $y$ are related by the equation
$$v^{2}=v_{0}^{2}+2 G M\left(\frac{1}{y}-\frac{1}{R}\right)$$
where $v_{0}$ is the initial velocity of the object and $R$ is the radius of Earth.

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Problem 81

Rectilinear Motion In Exercises $81-84,$ consider a particle moving along the $x$ -axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration.
$$x(t)=t^{3}-6 t^{2}+9 t-2, \quad 0 \leq t \leq 5$$
(a) Find the velocity and acceleration of the particle.
(b) Find the open $t$ -intervals on which the particle is moving to the right.
(c) Find the velocity of the particle when the acceleration is $0 .$

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Problem 82

Rectilinear Motion In Exercises $81-84,$ consider a particle moving along the $x$ -axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration.
Repeat Exercise 81 for the position function
$$x(t)=(t-1)(t-3)^{2}, 0 \leq t \leq 5$$

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Problem 83

Rectilinear Motion In Exercises $81-84,$ consider a particle moving along the $x$ -axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration.
A particle moves along the $x$ -axis at a velocity of $v(t)=1 / \sqrt{t}$ , $t>0$ . At time $t=1,$ its position is $x=4 .$ Find the acceleration and position functions for the particle.

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Problem 84

Rectilinear Motion In Exercises $81-84,$ consider a particle moving along the $x$ -axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration.
A particle, initially at rest, moves along the $x$ -axis such that its acceleration at time $t>0$ is given by $a(t)=\cos t .$ At the time $t=0,$ its position is $x=3$
(a) Find the velocity and position functions for the particle.
(b) Find the values of $t$ for which the particle is at rest.

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Problem 85

Acceleration The maker of an automobile advertises that it takes 13 seconds to accelerate from 25 kilometers per hour to 80 kilometers per hour. Assuming constant acceleration, compute the following.
(a) The acceleration in meters per second per second
(b) The distance the car travels during the 13 seconds

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Problem 86

Deceleration $A$ car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied.
(a) How far has the car moved when its speed has been reduced to 30 miles per hour?
(b) How far has the car moved when its speed has been reduced to 15 miles per hour?
(c) Draw the real number line from 0 to $132,$ and plot the points found in parts (a) and (b). What can you conclude?

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Problem 87

Acceleration At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car.
(a) How far beyond its starting point will the car pass the truck?
(b) How fast will the car be traveling when it passes the truck?

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Problem 88

Acceleration Assume that a fully loaded plane starting from rest has a constant acceleration while moving down a runway. The plane requires 0.7 mile of runway and a speed of 160 miles per hour in order to lift off. What is the plane's acceleration?

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Problem 89

Airplane Separation Two airplanes are in a straight-line landing pattern and, according to FAA regulations, must keep at least a three-mile separation. Airplane A is 10 miles from touchdown and is gradually decreasing its speed from 150 miles per hour to a landing speed of 100 miles per hour. Airplane $\mathrm{B}$ is 17 miles from touchdown and is gradually decreasing its speed from 250 miles per hour to a landing speed of 115 miles per hour.
(a) Assuming the deceleration of each airplane is constant, find the position functions $s_{A}$ and $s_{B}$ for airplane $A$ and airplane B. Let $t=0$ represent the times when the airplanes are 10 and 17 miles from the airport.
(b) Use a graphing utility to graph the position functions.
(c) Find a formula for the magnitude of the distance $d$ between the two airplanes as a function of $t .$ Use a graphing utility to graph $d .$ Is $d<3$ for some time prior to the landing of airplane A? If so, find that time.

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Problem 90

True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Each antiderivative of an $n$ th-degree polynomial function is an $(n+1)$ th-degree polynomial function.

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Problem 91

True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $p(x)$ is a polynomial function, then $p$ has exactly one antiderivative whose graph contains the origin.

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Problem 92

True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $F(x)$ and $G(x)$ are antiderivatives of $f(x),$ then $F(x)=G(x)+C .$

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Problem 93

True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $f^{\prime}(x)=g(x),$ then $\int g(x) d x=f(x)+C$.

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Problem 94

True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$$\int f(x) g(x) d x=\int f(x) d x \int g(x) d x$$

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Problem 95

True or False? In Exercises $90-95$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
The antiderivative of $f(x)$ is unique.

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Problem 96

Find a function $f$ such that the graph of $f$ has a horizontal tangent at $(2,0)$ and $f^{\prime \prime}(x)=2 x .$

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Problem 97

The graph of $f^{\prime}$ is shown. Sketch the graph of $f$ given that $f$ is continuous and $f(0)=1 .$

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Problem 98

If $f^{\prime}(x)=\left\{\begin{array}{cc}{1,} & {0 \leq x<2} \\ {3 x,} & {2 \leq x \leq 5}\end{array}, f \text { is continuous, and } f(1)=3,\right.$ find $f .$ Is $f$ differentiable at $x=2 ?$

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Problem 99

Let $s(x)$ and $c(x)$ be two functions satisfying $s^{\prime}(x)=c(x)$ and $c^{\prime}(x)=-s(x)$ for all $x .$ If $s(0)=0$ and $c(0)=1,$ prove that $[s(x)]^{2}+[c(x)]^{2}=1$.

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Problem 100

Suppose $f$ and $g$ are nonconstant, differentiable, real-valued functions on $R$ . Furthermore, suppose that for each pair of real numbers $x$ and $y, f(x+y)=f(x) f(y)-g(x) g(y)$ and $g(x+y)=f(x) g(y)+g(x) f(y) .$ If $f^{\prime}(0)=0,$ prove that $(f(x))^{2}+(g(x))^{2}=1$ for all $x .$

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