Applied Calculus for the MLSS A Brief Approach

Soo T. Tan

Chapter 6

Integration - all with Video Answers

Educators

Section 1

Antiderivatives and the Rules of Integration

Problem 1

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=\frac{1}{3} x^{3}+2 x^{2}-x+2 ; f(x)=x^{2}+4 x-1$$

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=x e^{x}+\pi ; f(x)=e^{x}(1+x)$$

Lucas Finney

Numerade Educator

Problem 3

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=\sqrt{2 x^{2}-1} ; f(x)=\frac{2 x}{\sqrt{2 x^{2}-1}}$$

Lucas Finney

Numerade Educator

Problem 4

Verify directly that $F$ is an antiderivative of $f$
$$F(x)=x \ln x-x ; f(x)=\ln x$$

Lucas Finney

Numerade Educator

Problem 5

(a) verify that $G$ is an antiderivative of $f$
(b) find all antiderivatives of $f,$ and
(c) sketch the graphs of a few members of the family of antiderivatives found in part (b).
$$G(x)=2 x ; f(x)=2$$

Gregory Higby

Numerade Educator

Problem 6

(a) verify that $G$ is an antiderivative of $f$
(b) find all antiderivatives of $f,$ and
(c) sketch the graphs of a few members of the family of antiderivatives found in part (b).
$$G(x)=2 x^{2} ; f(x)=4 x$$

Gregory Higby

Numerade Educator

Problem 7

(a) verify that $G$ is an antiderivative of $f$
(b) find all antiderivatives of $f,$ and
(c) sketch the graphs of a few members of the family of antiderivatives found in part (b).
$$G(x)=\frac{1}{3} x^{3} ; f(x)=x^{2}$$

Gregory Higby

Numerade Educator

Problem 8

(a) verify that $G$ is an antiderivative of $f$
(b) find all antiderivatives of $f,$ and
(c) sketch the graphs of a few members of the family of antiderivatives found in part (b).
$$G(x)=e^{x} ; f(x)=e^{x}$$

Gregory Higby

Numerade Educator

Problem 9

Find the indefinite integral.
$$\int 6 d x$$

Nishant Tyagi

Numerade Educator

Problem 10

Find the indefinite integral.
$$\int \sqrt{2} d x$$

Nishant Tyagi

Numerade Educator

Problem 11

Find the indefinite integral.
$$\int x^{3} d x$$

Nishant Tyagi

Numerade Educator

Problem 12

Find the indefinite integral.
$$\int 2 x^{5} d x$$

Nishant Tyagi

Numerade Educator

Problem 13

Find the indefinite integral.
$$\int x^{-4} d x$$

Nishant Tyagi

Numerade Educator

Problem 14

Find the indefinite integral.
$$\int 3 t^{-7} d t$$

Nishant Tyagi

Numerade Educator

Problem 15

Find the indefinite integral.
$$\int x^{2 / 3} d x$$

Joseph Liao

Numerade Educator

Problem 16

Find the indefinite integral.
$$\int 2 u^{3 / 4} d u$$

Nishant Tyagi

Numerade Educator

Problem 17

Find the indefinite integral.
$$\int x^{-5 / 4} d x$$

Nishant Tyagi

Numerade Educator

Problem 18

Find the indefinite integral.
$$\int 3 x^{-2 / 3} d x$$

Nishant Tyagi

Numerade Educator

Problem 19

Find the indefinite integral.
$$\int \frac{2}{x^{3}} d x$$

Joseph Liao

Numerade Educator

Problem 20

Find the indefinite integral.
$$\int \frac{1}{3 x^{5}} d x$$

Nishant Tyagi

Numerade Educator

Problem 21

Find the indefinite integral.
$$\int \pi \sqrt{t} d t$$

Nishant Tyagi

Numerade Educator

Problem 22

Find the indefinite integral.
$$\int \frac{3}{\sqrt{t}} d t$$

Nishant Tyagi

Numerade Educator

Problem 23

Find the indefinite integral.
$$\int(3-4 x) d x$$

Nishant Tyagi

Numerade Educator

Problem 24

Find the indefinite integral.
$$\int\left(1+u+u^{2}\right) d u$$

Nishant Tyagi

Numerade Educator

Problem 25

Find the indefinite integral.
$$\int\left(x^{2}+x+x^{-3}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 26

Find the indefinite integral.
$$\int\left(0.3 t^{2}+0.02 t+2\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 27

Find the indefinite integral.
$$\int 5 e^{x} d x$$

Eduard Sanchez

Numerade Educator

Problem 28

Find the indefinite integral.
$$\int\left(1+e^{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 29

Find the indefinite integral.
$$\int\left(1+x+e^{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 30

Find the indefinite integral.
$$\int\left(2+x+2 x^{2}+e^{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 31

Find the indefinite integral.
$$\int\left(4 x^{3}-\frac{2}{x^{2}}-1\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 32

Find the indefinite integral.
$$\int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 33

Find the indefinite integral.
$$\int\left(x^{5 / 2}+2 x^{3 / 2}-x\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 34

Find the indefinite integral.
$$\int\left(t^{3 / 2}+2 t^{1 / 2}-4 t^{-1 / 2}\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 35

Find the indefinite integral.
$$\int\left(\sqrt{x}+\frac{2}{\sqrt{x}}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 36

Find the indefinite integral.
$$\int\left(\sqrt[3]{x^{2}}-\frac{1}{x^{2}}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 37

Find the indefinite integral.
$$\int\left(\frac{u^{3}+2 u^{2}-u}{3 u}\right) d u$$

Nishant Tyagi

Numerade Educator

Problem 38

Find the indefinite integral.
$$\int \frac{x^{4}-1}{x^{2}} d x$$

Nishant Tyagi

Numerade Educator

Problem 39

Find the indefinite integral.
$$\int(2 t+1)(t-2) d t$$

Nishant Tyagi

Numerade Educator

Problem 40

Find the indefinite integral.
$$\int u^{-2}\left(1-u^{2}+u^{4}\right) d u$$

Nishant Tyagi

Numerade Educator

Problem 41

Find the indefinite integral.
$$\int \frac{1}{x^{2}}\left(x^{4}-2 x^{2}+1\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 42

Find the indefinite integral.
$$\int \sqrt{t}\left(t^{2}+t-1\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 43

Find the indefinite integral.
$$\int \frac{d s}{(s+1)^{-2}}$$

Nishant Tyagi

Numerade Educator

Problem 44

Find the indefinite integral.
$$\int\left(\sqrt{x}+\frac{3}{x}-2 e^{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 45

Find the indefinite integral.
$$\int\left(e^{t}+t^{e}\right) d t$$

Nishant Tyagi

Numerade Educator

Problem 46

Find the indefinite integral.
$$\int\left(\frac{1}{x^{2}}-\frac{1}{\sqrt[3]{x^{2}}}+\frac{1}{\sqrt{x}}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 47

Find the indefinite integral.
$$\int \frac{x^{3}+x^{2}-x+1}{x^{2}} d x$$

Nishant Tyagi

Numerade Educator

Problem 48

Find the indefinite integral.
$$\int \frac{t^{3}+\sqrt[3]{t}}{t^{2}} d t$$

Nishant Tyagi

Numerade Educator

Problem 49

Find the indefinite integral.
$$\int \frac{(\sqrt{x}-1)^{2}}{x^{2}} d x$$

Nishant Tyagi

Numerade Educator

Problem 50

Find the indefinite integral.
$$\int(x+1)^{2}\left(1-\frac{1}{x}\right) d x$$

Nishant Tyagi

Numerade Educator

Problem 51

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=3 x+1 ; f(1)=3$$

Nishant Tyagi

Numerade Educator

Problem 52

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=3 x^{2}-6 x ; f(2)=4$$

Nishant Tyagi

Numerade Educator

Problem 53

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=3 x^{2}+4 x-1 ; f(2)=9$$

Nishant Tyagi

Numerade Educator

Problem 54

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=\frac{1}{\sqrt{x}} ; f(4)=2$$

Nishant Tyagi

Numerade Educator

Problem 55

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=1+\frac{1}{x^{2}} ; f(1)=3$$

Nishant Tyagi

Numerade Educator

Problem 56

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=e^{x}-2 x ; f(0)=2$$

Nishant Tyagi

Numerade Educator

Problem 57

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=\frac{x+1}{x} ; f(1)=1$$

Nishant Tyagi

Numerade Educator

Problem 58

Find $f(x)$ by solving the initial value problem.
$$f^{\prime}(x)=1+e^{x}+\frac{1}{x} ; f(1)=3+e$$

Nishant Tyagi

Numerade Educator

Problem 59

Find the function $f$ given that the slope of the tangent line to the graph of fat any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=\frac{1}{2} x^{-1 / 2} ;(2, \sqrt{2})$$

Lucas Finney

Numerade Educator

Problem 60

Find the function $f$ given that the slope of the tangent line to the graph of fat any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Lucas Finney

Numerade Educator

Problem 61

Find the function $f$ given that the slope of the tangent line to the graph of fat any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=e^{x}+x ;(0,3)$$

Lucas Finney

Numerade Educator

Problem 62

Find the function $f$ given that the slope of the tangent line to the graph of fat any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=\frac{2}{x}+1 ;(1,2)$$

Lucas Finney

Numerade Educator

Problem 63

Madison Finance opened two branches on September 1 ( $t=0$ ). Branch $A$ is located in an established industrial park, and Branch $B$ is located in a fast-growing new development. The net rate at which money was deposited into Branch $A$ and Branch $B$ in the first 180 business days is given by the graphs of $f$ and $g$, respectively (see the figure). Which branch has the larger amount on deposit at the end of 180 business days? Justify your answer.

Lucas Finney

Numerade Educator

Problem 64

Two cars, side by side, start from rest and travel along a straight road. The velocity of Car $A$ is given by $v=f(t),$ and the velocity of $\operatorname{Car} B$ is given by $v=g(t) .$ The graphs of $f$ and $g$ are shown in the figure. Are the cars still side by side after $T$ sec? If not, which car is ahead of the other? Justify your answer.

Lucas Finney

Numerade Educator

Problem 65

U.S. Smartphone Users The number of smartphone users and penetration in the United States continues to grow steadily. The number of users (in millions) from 2011 through 2015 is projected to grow at the rate of
$$R(t)=14.3 \quad(0 \leq t \leq 4)$$
million/year. The number of users in $2011(t=0)$ was 90.1 million. Find an expression giving the projected number of smartphone users in year $t$. What is the estimated number of smartphone users in $2015 ?$

Lucas Finney

Numerade Educator

Problem 66

Households Owning More than One TV Television is the most popular mass medium in the United States. The percentage of households with multiple TV sets from the year 2000 through 2010 grew at the approximate rate of $$R(t)=0.7 \quad(0 \leq t \leq 10)$$ percent/year in year $t$. The percentage of American households owning multiple TV sets stood at 75\% in the year $2000(t=0)$. Find an expression giving the approximate percentage of households owning multiple TV sets in year $t$. What percentage of households owned multiple TV sets in 2010 ?

Lucas Finney

Numerade Educator

Problem 67

The velocity (in feet per second) of a maglev is
$$v(t)=0.2 t+3 \quad(0 \leq t \leq 120)$$ At $t=0,$ it is at the station. Find the function giving the position of the maglev at time $t,$ assuming that the motion takes place along a straight stretch of track.

Gregory Higby

Numerade Educator

Problem 68

The management of Lorimar Watch Company has determined that the daily marginal revenue function associated with producing and selling their travel clocks is given by $$R^{\prime}(x)=-0.009 x+12$$ where $x$ denotes the number of units produced and sold and $R^{\prime}(x)$ is measured in dollars per unit.
a. Determine the revenue function $R(x)$ associated with producing and selling these clocks.
b. What is the demand equation that relates the wholesale unit price with the quantity of travel clocks demanded?

Lucas Finney

Numerade Educator

Problem 69

Cannon Precision Instruments makes an automatic electronic flash with Thyrister circuitry. The estimated marginal profit associated with producing and selling these electronic flashes is
$$P^{\prime}(x)=-0.004 x+20$$
dollars per unit per month when the production level is $x$ units per month. Cannon's fixed cost for producing and selling these electronic flashes is $\$ 16,000$ /month. At what level of production does Cannon realize a maximum profit? What is the maximum monthly profit?

Lucas Finney

Numerade Educator

Problem 70

Carlota Music Company estimates that the marginal cost of manufacturing its Professional Series guitars is
$$C^{\prime}(x)=0.002 x+100$$
dollars/month when the level of production is $x$ guitars/month. The fixed costs incurred by Carlota are $\ 4000 /$ month. Find the total monthly cost incurred by Carlota in manufacturing $x$ guitars/month.

Chris Trentman

Numerade Educator

Problem 71

China's push to install more wind energy capacity has started paying off. As of 2012 , wind energy had become the country's third largest source of energy, after coal and hydroelectric power. The amount of wind energy generated in China from 2005 through 2012 grew at the rate of
$$r(t)=5.018 t-3.204 \quad(0 \leq t \leq 7)$$
terawatt-hours/year, in year $t,$ where $t=0$ corresponds to $2005 .$ The amount of wind energy generated in 2005 stood at 1.8 terawatt-hours.
a. Find an expression giving the amount of wind energy generated in year $t$
b. How much wind energy was generated in $2012 ?$
c. If the trend continued into 2013 , how much wind energy was generated in China in that year?

Lucas Finney

Numerade Educator

Problem 72

National health expenditures are projected to grow at the rate of
$$r(t)=0.0058 t+0.159 \quad(0 \leq t \leq 13)$$
trillion dollars/year from 2002 through 2015. Here, $t=0$ corresponds to 2002 . The expenditure in 2002 was $\1.60$ trillion.
a. Find a function $f$ giving the projected national health expenditures in year $f$
b. What does your model project the national health expenditure to be in $2015 ?$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 73

The total number of acres of genetically modified crops grown in developing countries from 2006 through 2012 was changing at the rate of
$$R(t)=150 t+14.82 \quad(0 \leq t \leq 6)$$
million acres/year. The total number of acres of such crops grown in $2006(t=0)$ was 27.2 million acres. How many acres of genetically modified crops were grown in developing countries in $2012 ?$

Lucas Finney

Numerade Educator

Problem 74

The velocity of a car (in feet per second) $t$ sec after starting from rest is given by the function
$$f(t)=2 \sqrt{t} \quad(0 \leq t \leq 30)$$
Find the car's position, $s(t),$ at any time $t .$ Assume that $s(0)=0$

Lucas Finney

Numerade Educator

Problem 75

A ballast is dropped from a stationary hot-air balloon that is hovering at an altitude of $400 \mathrm{ft}$. The velocity of the ballast after $t$ sec is $(-32 t+4) \mathrm{ft} / \mathrm{sec}$.
a. Find the height $h(t)$ of the ballast from the ground at time $t$
b. When will the ballast strike the ground?
c. Find the velocity of the ballast when it hits the ground.

Lucas Finney

Numerade Educator

Problem 76

The development of AstroWorld ("The Amusement Park of the Future") on the outskirts of a city will increase the city's population at the rate of $$4500 \sqrt{t}+1000$$
people/year, $t$ years from the start of construction. The population before construction is 30,000 . Determine the projected population 9 years after construction of the park has begun.

Lucas Finney

Numerade Educator

Problem 77

A study conducted by Telecable estimates that the number of cable TV subscribers will grow at the rate of
$$100+210 t^{3 / 4}$$
new subscribers/month, $t$ months from the start date of the service. If 5000 subscribers signed up for the service before the starting date, how many subscribers will there be 16 months from that date?

Lucas Finney

Numerade Educator

Problem 78

Nineteenth-century physician Jean Louis Marie Poiseuille discovered that the rate of change of the velocity of blood $r$ cm from the central axis of an artery (in centimeters per second per centimeter) is given by
$$a(r)=-k r$$ where $k$ is a constant. If the radius of an artery is $R \mathrm{cm}$, find an expression for the velocity of blood as a function of $r$ (see the accompanying figure).

Lucas Finney

Numerade Educator

Problem 79

The velocity, in feet per second, of a rocket $t$ sec into vertical flight is given by
$$v(t)=-3 t^{2}+192 t$$
Find an expression $h(t)$ that gives the rocket's altitude, in feet, $t$ sec after liftoff. What is the altitude of the rocket 30 sec after liftoff?

Lucas Finney

Numerade Educator

Problem 80

As part of a quality-control program, the chess sets manufactured by Jones Brothers are subjected to a final inspection before packing. The rate of increase in the number of sets checked per hour by an inspector $t$ hr into the 8 A.M. to 12 noon shift is approximately
$$N^{\prime}(t)=-3 t^{2}+12 t+45 \quad(0 \leq t \leq 4)$$
a. Find an expression $N(t)$ that approximates the number of sets inspected at the end of $t$ hours.
b. How many sets does the average inspector check during a morning shift?

Lucas Finney

Numerade Educator

Problem 81

Lorimar Watch Company manufactures travel clocks. The daily marginal cost function associated with producing these clocks is
$$C^{\prime}(x)=0.000009 x^{2}-0.009 x+8$$
where $C^{\prime}(x)$ is measured in dollars per unit and $x$ denotes the number of units produced. Management has determined that the daily fixed cost incurred in producing these clocks is $\ 120 .$ Find the total cost incurred by Lorimar in producing the first 500 travel clocks per day.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 82

The rate at which the risk of Down syndrome is changing is approximated by the function
$$r(x)=0.004641 x^{2}-0.3012 x+4.9 \quad(20 \leq x \leq 45)$$
where $r(x)$ is measured in percentage of all births per year and $x$ is the maternal age at delivery.
a. Find a function $f$ giving the risk as a percentage of all births when the maternal age at delivery is $x$ years, given that the risk of Down syndrome at age 30 is $0.14 \%$ of all births.
b. Based on this model, what is the risk of Down syndrome when the maternal age at delivery is 40 years? 45 years?

Lucas Finney

Numerade Educator

Problem 83

During a thunderstorm, rain was falling at the rate of$$8 \sqrt{2 t}-32 t^{3} \quad(0 \leq t \leq 0.6)$$ in. /hr.
a. Find an expression giving the total amount of rainfall after $t$ hr.
b. How much rain had fallen after $\frac{1}{2} \mathrm{hr} ?$

Lucas Finney

Numerade Educator

Problem 84

The percentage of married women who earned more than their husbands over the period from $1960(t=0)$ through $2011(t=51)$ was growing at the rate of approximately
$$r(t)=-0.00025142 t^{2}+0.02116 t+0.0328 \quad(0 \leq t \leq 51)$$
percent/year in year $t$. The percentage of married women earning more than their husbands in 1960 stood at 6.2\%.
a. Find an expression giving the approximate percentage of married women who earned more than their husbands in year $t$
b. Use the result of part (a) to estimate the percentage of married women who earned more than their husbands in 2013 , assuming that the trend continued.

Lucas Finney

Numerade Educator

Problem 85

The percentage of people age 12 years and older using social network sites and/or services "several" times a day from 2009 through 2013 grew at the rate of
$$R(t)=5.92 t^{-0.168} \quad(1 \leq t \leq 5)$$
percent/year in year $t,$ with $t=1$ corresponding to $2009 .$ The percentage of people age 12 years and older using social network sites and/or services in 2009 was $7 \%$.
a. Find an expression giving the percentage of people age 12 years and older using social network sites and/or services in year $t(1 \leq t \leq 5)$
b. According to this model, what percentage of people age 12 years and older used social network sites and/or services in $2013 ?$

Lucas Finney

Numerade Educator

Problem 86

The U.S. coal industry, under increasing pressure from tougher antipollution rules, is ratcheting up its export business. The rate of change of coal exports from 2010 through
2012 is given by
$$f(t)=31.863 t^{-0.61} \quad(0 \leq t \leq 2)$$
million short tons (2000 lb) per year in year $t,$ where $t$ is measured in years, with $t=0$ corresponding to $2010 .$ U.S. coal exports in 2010 were 81.7 million short tons.
a. Find an expression for U.S. coal exports in year $t$
b. Assuming that the trend continued through 2013 , what were U.S. coal exports for that year?

Lucas Finney

Numerade Educator

Problem 87

Empirical data suggest that the surface area of a $180-$ cm-tall human body changes at the rate of
$$S^{\prime}(W)=0.131773 W^{-0.575}$$
$\mathrm{m}^{2} / \mathrm{kg},$ where $W$ is the mass of the body in kilograms. If the surface area of a $180-\mathrm{cm}$ -tall human body weighing 70 kg is 1.886277 m^?2, what is the surface area of a human body of the same height weighing 75 kg?

Breanna Ollech

Numerade Educator

Problem 88

The rate of change of the level of ozone, an invisible gas that is an irritant and impairs breathing, present in the atmosphere on a certain May day in the city of Riverside is given by
$$R(t)=3.2922 t^{2}-0.366 t^{3} \quad(0<t<11)$$
(measured in pollutant standard index per hour). Here, $t$ is measured in hours, with $t=0$ corresponding to 7 A.M. Find the ozone level $A(t)$ at any time $t,$ assuming that at 7 A.M. it is zero.

Lucas Finney

Numerade Educator

Problem 89

According to the Jenss model for predicting the height of preschool children, the rate of growth of a typical preschool child is
$$R(t)=1.0490 t^{4}-4.2255 t^{3}+12.7659 t^{2}-25.7119 t+32.28 \quad\left(\frac{1}{4} \leq t \leq 1\right)$$
cm/year, where $t$ is measured in years. The height of a typical 3-month-old preschool child is $60.30 \mathrm{cm}$
a. What is the height of a typical infant at age $t ?$
b. Use the result of part (a) to estimate the height of a typical 1-year-old child.

Lucas Finney

Numerade Educator

Problem 90

A car traveling along a straight road at $66 \mathrm{ft} /$ sec accelerated to a speed of $88 \mathrm{ft} / \mathrm{sec}$ over a distance of $440 \mathrm{ft}$. What was the acceleration of the car, assuming that it was constant?

Lucas Finney

Numerade Educator

Problem 91

What constant deceleration would a car moving along a straight road have to be subjected to if it were brought to rest from a speed of $88 \mathrm{ft} / \mathrm{sec}$ in 9 sec? What would be the stopping distance?

Lucas Finney

Numerade Educator

Problem 92

A pilot lands a fighter aircraft on an aircraft carrier. At the moment of touchdown, the speed of the aircraft is $160 \mathrm{mph.}$. If the aircraft is brought to a complete stop in $1 \mathrm{sec}$ and the deceleration is assumed to be constant, find the number of g's the pilot is subjected to during landing $\left(1 \mathrm{g}=32 \mathrm{ft} / \mathrm{sec}^{2}\right)$

Lucas Finney

Numerade Educator

Problem 93

After rounding the final turn in the bell lap, two runners emerge ahead of the pack. When Runner $A$ is 200 ft from the finish line, his speed is $22 \mathrm{ft} / \mathrm{sec}$, a speed that he maintains until he crosses the line. At that instant of time, Runner $B$, who is $20 \mathrm{ft}$ behind Runner $A$ and running at a speed of $20 \mathrm{ft} / \mathrm{sec}$, begins to sprint. Assuming that Runner $B$ sprints with a constant acceleration, what minimum acceleration will enable him to cross the finish line ahead of Runner $A$ ?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 94

A tank has a constant cross-sectional area of $50 \mathrm{ft}^{2}$ and an orifice of constant cross-sectional area of $\frac{1}{2} \mathrm{ft}^{2}$ located at the bottom of the tank (see the accompanying figure). If the tank is filled with water to a height of $h$ ft and allowed to drain, then the height of the water decreases at a rate that is described by the equation $h$ $$\frac{d h}{d t}=-\frac{1}{25}\left(\sqrt{20}-\frac{t}{50}\right) \quad(0 \leq t \leq 50 \sqrt{20})$$ Find an expression for the height of the water at any time $t$ if its height initially is $20 \mathrm{ft}$.

Lucas Finney

Numerade Educator

Problem 95

A fighter aircraft is launched from the deck of a Nimitz-class aircraft carrier with the help of a steam catapult. If the aircraft is to attain a takeoff speed of at least 240 ft/sec after traveling 800 ft along the flight deck, find the minimum acceleration to which it must be subjected, assuming that it is constant.

Lucas Finney

Numerade Educator

Problem 96

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $F$ and $G$ are antiderivatives of $f$ on an interval $I$ then $F(x)=G(x)+C$ on $I$

Lucas Finney

Numerade Educator

Problem 97

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $F$ is an antiderivative of $f$ on an interval $I,$ then $\int f(x) d x=F(x)$

Lucas Finney

Numerade Educator

Problem 98

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are integrable, then $$\int[2 f(x)-3 g(x)] d x=2 \int f(x) d x-3 \int g(x) d x$$

Gregory Higby

Numerade Educator

Problem 99

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$$\int \frac{d}{d x}[f(x)] d x=f(x)$$

Gregory Higby

Numerade Educator

Problem 100

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are integrable, then $$\int f(x) g(x) d x=\left[\int f(x) d x\right]\left[\int g(x) d x\right]$$

Gregory Higby

Numerade Educator