Deck 6: Constructing Antiderivatives

Question 1

Find a function F such that $F^{\prime}(x)=x^{5}-\frac{4}{\sqrt{x}}$ .

A) $\frac{1}{6} x^{6}+4 \sqrt{x}+C$
B) $\frac{1}{6} x^{6}-4 \sqrt{x}+C$
C) $\frac{1}{6} x^{6}-8 \sqrt{x}+C$
D) $\frac{1}{6} x^{6}+8 \sqrt{x}+C$

Answer 1

$\frac{1}{6} x^{6}-8 \sqrt{x}+C$

Question 2

Given the values of the derivative f '(x)in the table and that f (0)= 60, estimate f (4).(Average left- and right-hand sums).
$\begin{array}{ccccc}x & 0 & 2 & 4 & 6 \\f^{\prime}(x) & 3 & 15 & 27 & 39\end{array}$

Answer 2

120

Question 3

A young girl who aspires to be a rocket scientist launches a model rocket from the ground at time t = 0.The rocket travels straight up in the air, and the following graph shows the upward velocity of the rocket as a function of time: Let v(t)be the function that gives the velocity of the rocket at time t.From the graph of the rocket's velocity, which of the following is larger?

A) $\left|\int_{0}^{t_{2}} v(t) d t\right|$
B) $\left|\int_{t_{2}}^{t_{3}} v(t) d t\right|$

Answer 3

$\left|\int_{0}^{t_{2}} v(t) d t\right|$

Question 4

If $f(x)$ is as shown in the first graph, could the function in the second graph represent the total area (i.e., not necessarily the definite integral)between f(x)and the x-axis from 0 to a?

Question 5

You decide to take a trip down a stretch of road that runs straight east and west.The following table gives your eastward velocity (in miles per minute)measured at one-minute intervals for the first ten minutes of your trip.
What is your best estimate of the total eastward distance of your car from your starting position after ten minutes? Round to the nearest whole number.

Question 6

Find a function F such that $F^{\prime}(x)=\sin x+\frac{7}{x}$ .

A) $\cos x+7 \ln x+C$
B) $-\cos x-7 \ln x+C$
C) $\cos x-7 \ln |x|+C$
D) $-\cos x+7 \ln |x|+C$

Question 7

Find an antiderivative of $\frac{5}{\left(a^{2}+x^{2}\right)^{3 / 2}}$ .

A) $\frac{10 x}{5\left(a^{2}+x^{2}\right)^{5 / 2}}+C$
B) $\frac{5 x}{a^{2} \sqrt{a^{2}+x^{2}}}+C$
C) $\frac{5}{\sqrt{a^{2}+x^{2}}}+C$
D) $\frac{10 x a^{2}}{\sqrt{a^{2}+x^{2}}}+C$

Question 8

The graph of is shown in the following figure.If G is an antiderivative of g such that , what is ?

Question 9

Estimate for the figure given.

Question 10

Find the total area bounded between the curve and the x-axis.Round to 3 decimal places.

Question 11

A young girl who aspires to be a rocket scientist launches a model rocket from the ground at time t = 0.The rocket travels straight up in the air, and the following graph shows the upward velocity of the rocket as a function of time: Let v(t)be the function that gives the velocity of the rocket at time t.From the graph of the rocket's velocity, what does the sign of $\int_{0}^{f_{3}} v(t) d t$ mean physically?

A)The rocket came to rest somewhere above the ground.
B)The rocket came to rest somewhere below the ground.
C)The rocket was accelerating when it hit the ground.
D)The rocket was decelerating when it hit the ground.

Question 12

Could the second function be an antiderivative of the first function?

Question 13

You decide to take a trip down a stretch of road that runs straight east and west.The following table gives your eastward velocity (in miles per minute)measured at one-minute intervals for the first ten minutes of your trip. $\begin{array}{cccccccccccc}\text { Time (min) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\text { Velocity } & 0.00 & 0.49 & 0.84 & 1.05 & 1.12 & 1.05 & 0.84 & 0.49 & 0.00 & -0.63 & -1.4\\\text { (mi/min)}\end{array}$ Does the following graph give your acceleration or your eastward distance from your starting place as a function of time?

A)Your acceleration
B)Your eastward distance

Question 14

The rate of change in concentration of a certain medication in a person's body, H'(t), in micrograms per milliliter per minute, is -1 for the first 2 minutes.Then it increases at a constant rate for 2 minutes, reaching 1 at t = 4.Then it remains constant for 1 minute.Sketch H(t), assuming H(0)= 9.

Question 15

Sketch the graphs of and y = ex.Find the average value of the difference between and ex on the interval between x = 0 and x = 1.5.Round to 2 decimal places.

Question 16

A young girl who aspires to be a rocket scientist launches a model rocket from the ground at time t = 0.The rocket travels straight up in the air, and the following graph shows the upward velocity of the rocket as a function of time: Let h be the height (or vertical displacement)of the rocket, let a be the acceleration of the rocket, and let h = 0 be the ground level from which the rocket was launched.Recall that the first derivative of displacement is velocity and that the first derivative of velocity is acceleration.Is the following a graph of the acceleration or the height of the rocket as a function of time?

A)The acceleration.
B)The height.

Question 17

Sketch the graphs of $y=e^{x}$ and y = ex.For which values of x is $e^{x}>e x$ ? Mark all that apply.

A) $x=0$
B) $0<$$x<1$
C) $x=1$
D) $1<$$x< e$
E) $x=e$

Question 18

If f is shown in the first graph, which of the two functions F could have F' = f ?

A)The first one
B)The second one
C)Both of them
D)Neither of them

Question 19

Sketch a graph of the antiderivative of g(x), given the graph g'(x)and g(0)=1.

Question 20

Is $x^{4} \sin x$ an antiderivative of $4 x^{3} \cos x$ ?

Question 21

On planet Janet the gravitational constant g is -10 feet per second per second: that is, for every second an object falls it picks up an extra 10 feet per second of velocity downward.A ball is thrown upward (from height zero)at time t = 0 at 40 feet per second.What is the peak height of the ball?

Question 22

A factory is dumping pollutants into a lake continuously at the rate of tons per week, where t is the time in weeks since the factory commenced operations.
Assume that natural processes can remove up to 0.138 tons of pollutant per week from the lake and that there was no pollution in the lake when the factory commenced operations one year ago.How many tons of pollutant have now accumulated in the lake? (Note: The amount of pollutant being dumped into the lake is never negative.)Round to 2 decimal places.

Question 23

Find the area above $y=3$ and below $f(x)=9-x^{2}$ .

A)9.7980
B)29.3939
C)19.5959
D)-19.5959
E)2.2020

Question 24

A car is going 64 feet per second and the driver puts on the brakes, bringing the car to a stop in 4 seconds.Assume the deceleration of the car is constant while the brakes are on.The acceleration (really deceleration)of the car is _____ ft/sec.

Question 25

Find an antiderivative of $\cos (7 \theta)$ .

A) $7 \sin (7 \theta)+C$
B) $-7 \sin (7 \theta)+C$
C) $\frac{1}{7} \sin (7 \theta)+C$
D) $-\frac{1}{7} \sin (7 \theta)+C$

Question 26

Find an antiderivative of $\pi+x^{6}+\frac{1}{\pi x^{6}}$ .

A) $\pi x+\frac{x^{7}}{7}-\frac{1}{5 \pi x^{5}}+C$
B) $\pi x+\frac{x^{7}}{7}+\frac{1}{7 \pi x^{7}}+C$
C) $\pi x+\frac{x^{7}}{7}+\frac{7}{\pi x^{7}}+C$
D) $\frac{\pi^{2}}{2}+\frac{x^{7}}{7}-\frac{7}{\pi x^{7}}+C$

Question 27

A factory is dumping pollutants into a lake continuously at the rate of tons per week, where t is the time in weeks since the factory commenced operations.After one year of operation, how many tons of pollutant has the factory dumped into the lake? Round to 2 decimal places.

Question 28

Find an antiderivative of $\frac{5}{x}+\frac{x}{5}$ .

A) $5 \ln |x|+\frac{x^{2}}{10}+C$
B) $5 \ln |x|+x \ln |5|+C$
C) $\frac{10}{x^{2}}+\frac{x^{2}}{10}+C$
D) $\frac{10}{x^{2}}+\frac{\ln |x|}{5}+C$

Question 29

Determine:

Question 30

Suppose $F(x)=5 \sin x+x+7$ .Find the total area bounded by F(x), x = 0, x = $\pi$ and y = 0.Round to 2 decimal places.

Question 31

Find the general antiderivative of $P(x)=\frac{1}{2 x}$ .

A) $2 \ln |x|+C$
B) $\frac{1}{2} \ln |x|+C$
C) $\frac{-2}{2} x^{-2}+C$
D) $-(2 x)^{-2}+C$
E) $\ln |2 x|+C$

Question 32

A ball is dropped from a window 80 feet above the ground.Assume that its acceleration is a(t)= -32 ft/sec² for t $\ge$ 0.After how many seconds does it hit the ground? Round to 2 decimal places.

Question 33

Suppose the rate at which ice in a skating pond is melting is given by , where V is the volume of the ice in cubic feet, and t is the time in minutes.Use a definite integral to find how many cubic feet of ice have melted in the first 2 minutes.

Question 34

A car is going 56 feet per second and the driver puts on the brakes, bringing the car to a stop in 4 seconds.Assume the deceleration of the car is constant while the brakes are on.How many feet does the car travel from the time the brakes are applied until it stops?

Question 35

Find an antiderivative of $e^{t}+e^{7}$ .

A) $\frac{e^{t+1}}{t+1}+e^{7} t+C$
B) $\frac{e^{t}}{t}+e^{7} t+C$
C) $e^{t}+C$
D) $e^{t}+e^{7} t+C$

Question 36

A ball is dropped from a window 140 feet above the ground.Assume that its acceleration is a(t)= -32 ft/sec² for t $\ge$ 0.Find the velocity of the ball as a function of time t.(All answers are in ft/sec.)

A) $-16 t^{2}+140$
B) $-32 t$
C) $-32+140$
D) $-16 t^{2}$

Question 37

On planet Janet the gravitational constant g is -11 feet per second per second: that is, for every second an object falls it picks up an extra 11 feet per second of velocity downward.A ball is thrown upward at time t = 0 at 44 feet per second.After how many seconds does the ball reach the peak of its flight?

Question 38

Find the exact area between the graphs of $y=x^{3}+9$ and $y=-x+9$ for 0 $\le$ x $\le$ 5.

Question 39

Find an antiderivative of $\sqrt{7 x}-\frac{6}{x \sqrt{x}}$ .

A) $\frac{3}{14}(7 x)^{3 / 2}-6 x^{-1 / 2}+C$
B) $\frac{2}{21}(7 x)^{3 / 2}+12 x^{-1 / 2}+C$
C) $\frac{3}{14}(7 x)^{3 / 2}-12 x^{-5 / 2}+C$
D) $\frac{2}{21}(7 x)^{3 / 2}+\frac{5}{12} x^{-5 / 2}+C$

Question 40

A car is going 85 feet per second and the driver puts on the brakes, bringing the car to a stop in 5 seconds.Assume the deceleration of the car is constant while the brakes are on.Suppose a second car is traveling the same speed and the brakes are twice as strong (can stop the car twice as fast)as those in the first car.How far does the second car travel before it stops?

A)Half as far as the first car.
B)Twice as far as the first car.
C)The same distance as the first car.
D)Four times as far as the first car.
E)One-fourth as far as the first car.

Question 41

Below are the graphs of (i) $f(x)$ , and (ii) $\int_{0}^{x} f(t) d t$ (not necessarily in that order). Which one is the graph of (ii)?

A)The first one.
B)The second one.

Question 42

The police observe that the skid marks made by a stopping car are 240 ft long.Assuming the car decelerated at a constant rate of 21 ft/ sec², skidding all the way, how fast was the car going when the brakes were applied? Round to 2 decimal places.

Question 43

Let and .If A, B, C, D, J, and K represent positive areas as shown in the graph, what combination of these areas represent G on the graph?

Question 44

A dog's bone is tossed in a yard traveling with a vertical velocity of feet/sec.Determine the maximum height reached by the bone.

Question 45

Evaluate $\int\left(x^{2}+e^{3 x}\right)\left(x^{3}+e^{3 x}\right)^{2 / 3} d x$ .Some of the coefficients may not be reduced.

A) $\frac{3}{15}\left(x^{3}+e^{3 x}\right)^{5 / 3}+C$
B) $\frac{3}{5}\left(x^{3}+e^{3 x}\right)^{5 / 3}+C$
C) $\frac{9}{5}\left(x^{3}+e^{3 x}\right)\left(x^{4}+e^{3 x}\right)^{5 / 3}+C$
D) $\frac{3}{60}\left(x^{3}+e^{3 x}\right)\left(x^{4}+e^{3 x}\right)^{5 / 3}+C$

Question 46

Find the general solution of the differential equation .

Question 47

Assuming the 440 feet is accurate and you neglect air resistance, determine the accuracy of the following paragraph: MY JOURNEY BENEATH THE EARTH
Condensed from "A Wolverine is Eating My Leg"
Tim Cahill:
I am in Ellison's Cave, about to rappel down Incredible Pit, the second-deepest cave pit in the continental United States.The drop is 440 feet, about what you'd experience from the top of a 40-story building.If you took the shaft in a free fall, you'd accelerate to more than 100 miles an hour and then--about five seconds into the experience--you'd decelerate to zero.And die.

A)The time (about 5 seconds)is fairly accurate, but the speed (more than 100 mph)is not.
B)The speed (more than 100 mph)is fairly accurate, but the time (about 5 seconds)is not.
C)Both of the numbers are fairly accurate.
D)Neither of the numbers are fairly accurate

Question 48

Evaluate $\int \frac{6 e^{-2 w}}{1-5 e^{-2 w}} d w$ .

A) $\frac{12 e^{-2 w}}{1 t-10 e^{-2 w}}+C$
B) $\frac{6 e^{-2 w}}{2 t-5 e^{-2 w}}+C$
C) $\frac{3}{5} \ln \left|1-5 e^{-2 w}\right|+C$
D) $\frac{1}{10} \ln \left|1-5 e^{-2 w}\right|+C$

Question 49

On planet Janet the gravitational constant g is -10 feet per second per second: that is, for every second an object falls it picks up an extra 10 feet per second of velocity downward.A ball is thrown upward at time t = 0 at 25 feet per second.On planet Nanette, g is one-fourth as great as on Janet.What is the peak height of the ball on planet Nanette?

Question 50

A ball is thrown vertically upwards from the top of a 256-foot cliff with initial velocity of 96 feet per second.Find its maximum height (in ft).

Question 51

A function g is known to be linear on the interval from $-\infty$ to 2 (inclusive)and also linear on the interval from 2 to $\infty$ (again inclusive).
Furthermore, g(1)= 2, g(2)= 0, g(4)= 8.Another function f satisfies f (0)= 0 and f ' = g.What is $f(3)$ ?

Question 52

The function f(t)is graphed below and we define $F(x)=\int_{0}^{x} f(t) d t$ . Is $F(x)$ concave down for x =1/2?

Question 53

For , define .What is the value of ? Round to 2 decimal places.

Question 54

For -1 $\le$ x $\le$ 1, define $F(x)=\int_{-1}^{x} \sqrt{1-t^{2}} d t$ .What does F(1)represent geometrically?

A)The area of a quarter circle of radius 1.
B)The area of a circle of radius 1.
C)The area of a semicircle of radius 1.
D)None of the above

Question 55

Find the solution of the differential equation satisfying .

Question 56

Below are the graphs of (i) $f(x)$ , (ii) $f^{\prime}(x)$ , and (iii) $\int_{0}^{x} f(t) d t$ (not necessarily in that order). Which one is the graph of (iii)?

A)The first one.
B)The third one.
C)The second one.

Question 57

Evaluate $\int \frac{\cos (\ln x)}{x} d x$ .

A) $-\sin (\ln x)+C$
B) $\sin (\ln x)+C$
C) $\frac{-\sin (\ln x)}{x}+C$
D) $\frac{2 \sin (\ln x)}{x^{2}}+C$

Question 58

Evaluate .

Question 59

For $-4 \leq x \leq 4$ , define $F(x)=\int_{-4}^{x} \sqrt{16-t^{2}} d t$ .Find F'(x).

A) $\sqrt{2 x}$
B) $-\sqrt{2 x}$
C) $-\sqrt{16-x^{2}}$
D) $\sqrt{16-x^{2}}$

Question 60

Find the solution of the initial value problems $\frac{d K}{d t}=2-\cos 5 t$ when $K(0)=-12$ .

A) $2 t-\frac{\sin 5 t}{5}-12$
B) $2 t-\frac{\sin 5 t}{5}+12$
C) $2 t+\frac{\sin 5 t}{5}-12$
D) $2 t+\frac{\sin 5 t}{5}+12$

Question 61

For , find .

Question 62

Evaluate .

Question 63

Find an antiderivative F(x)with F'(x)= f (x)and F(0)= 3 when $f(x)=\sin x-\cos x$ .

Question 64

The area between , the x-axis, and x = b is approximately 79.9.Find the value of b using the Fundamental Theorem.Round to 1 decimal place.

Question 65

A boat has constant deceleration.It was initially moving at 80 mph and stopped in a distance of 300 feet.The rate of deceleration is _____ ft/ sec².(Note: 1 mph = 22/15 ft/ sec.)Round to 2 decimal places.

Question 66

A boulder is dropped from a cliff.A second boulder is dropped from a cliff that is half as high.How does the speed of the second boulder upon impact compare with that of the first?

A)Ths speed of the second is approximately 0.5 times the speed of the first.
B)Ths speed of the second is approximately 0.7 times the speed of the first.
C)Ths speed of the second is approximately 0.25 times the speed of the first.
D)Ths speed of the second is approximately 2 times the speed of the first.
E)Ths speed of the second is approximately the same as the speed of the first.

Question 67

Use the Fundamental Theorem of Calculus to evaluate .

Question 68

Evaluate .

Question 69

At time t = 0, a bowling ball rolls off a 250-meter ledge with velocity 30 meters/sec downward.Express its height, h(t), in meters above the ground as a function of time, t, in seconds.

A) $h(t)=-4.9 t^{2}-30 t+250$
B) $h(t)=-4.9 t^{2}-30 t-250$
C) $h(t)=4.9 t^{2}-30 t+250$
D) $h(t)=4.9 t^{2}-30 t-250$

Question 70

Find the area of the region between and , accurate to 2 decimal places.

Question 71

On the moon the acceleration due to gravity is 5 feet/ sec².A brick is dropped from the top of a tower on the moon and hits the ground in 16 seconds.How many feet high is the tower?

Question 72

The general solution of the differential equation $\frac{d y}{d x}=\frac{2}{x}+\sin 4 x$ is $2 \ln |x|+\frac{\cos 4 x}{4}+C$ .

Question 73

A boulder is dropped from a 150-foot cliff.How fast is it going when it hits the ground? Round to 2 decimal places.

Question 74

Find $\frac{d}{d x} \int_{x}^{1}(1+t)^{3} d t$ .

A) $(1+x)^{3}$
B) $-(1+x)^{3}$
C) $\frac{1}{3}(1+x)^{4}$
D) $\frac{1}{3}(1+x)^{4}-\frac{16}{3}$

Question 75

Find $\frac{d}{d x} \int_{0}^{x}\left(\cos \left(t^{4}\right)+\sin \left(t^{4}\right)\right) d t$ .

A) $\cos \left(x^{4}\right)+\sin \left(x^{4}\right)$
B) $\sin \left(x^{4}\right)-\cos \left(x^{4}\right)$
C) $\sin \left(4 x^{3}\right)-\cos \left(4 x^{3}\right)$
D) $\frac{x^{5}}{5}\left(\sin \left(x^{4}\right)-\cos \left(x^{4}\right)\right)$

Question 76

Write an expression for the function, f(x), with $f^{\prime}(x)=\sin ^{2} x+\sin x$ and $f(\pi / 3)=8$ .

A) $f(x)=8+\int_{0}^{x}\left(\sin ^{2} t+\sin t\right) d t$
B) $f(x)=\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t-8$
C) $f(x)=8+\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t$
D) $f(x)=8-\frac{\pi}{3}+\int_{0}^{x}\left(\sin ^{2} t+\sin t\right) d t$

Question 77

Suppose the acceleration due to gravity on Planet A is twice that of Planet B.A brick dropped from the top of a tower on Planet A takes 20 seconds to hit the ground.A brick dropped from the top of a tower on Planet B also takes 20 seconds to hit the ground.How does the height of the tower on Planet A compare with the height of the tower on Planet B?

A)The tower on Planet A is 4 times the height of the one on Planet B.
B)The tower on Planet A is twice the height of the one on Planet B.
C)The tower on Planet A is half the height of the one on Planet B.
D)The tower on Planet A is the same as the height of the one on Planet B.

Question 78

Evaluate .

Question 79

Evaluate .

Question 80

Find the value of G( $\pi$ /2)where G '(x)= 2 sin x cos x and G(0)= 1.