Content check — Calculus Vol. 1, Chapter 4: Applications of Derivatives

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 4: Applications of Derivatives

327 problems · 37 figures

Checkpoint 4.1 §4.1 source ↗

What is the instantaneous rate of change of the radius when

r = 6 cm ?

Answer

\frac{1}{72 π} cm/sec,

or approximately 0.0044 cm/sec

Checkpoint 4.2 §4.1 source ↗

What is the speed of the plane if the distance between the person and the plane is increasing at the rate of

300 ft/sec ?

Answer

500 ft/sec

Checkpoint 4.3 §4.1 source ↗

What rate of change is necessary for the elevation angle of the camera if the camera is placed on the ground at a distance of

4000 ft

from the launch pad and the velocity of the rocket is 500 ft/sec when the rocket is

2000 ft

off the ground?

Answer

\frac{1}{10} rad/sec

Checkpoint 4.4 §4.1 source ↗

At what rate is the height of the water changing when the height of the water is

\frac{1}{4} ft ?

Answer

−0.61 ft/sec

4-1 §4.1 source ↗

For the following exercises, find the quantities for the given equation.

Find $\frac{d y}{d t}$ at $x = 1$ and $y = x^{2} + 3$ if $\frac{d x}{d t} = 4 .$

Answer

8

4-3 §4.1 source ↗

For the following exercises, find the quantities for the given equation.

Find $\frac{d z}{d t}$ at $(x, y) = (1, 3)$ and $z^{2} = x^{2} + y^{2}$ if $\frac{d x}{d t} = 4$ and $\frac{d y}{d t} = 3 .$

Answer

\pm \frac{13}{\sqrt{10}}

4-5 §4.1 source ↗

For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities.

A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?

A right triangle is formed by a ladder leaning up against a brick wall. The ladder forms the hypotenuse and is 10 ft long.

Answer

2 \sqrt{3}

ft/sec

4-7 §4.1 source ↗

For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities.

Two airplanes are flying in the air at the same height: airplane A is flying east at 250 mi/h and airplane B is flying north at $300 mi/h .$ If they are both heading to the same airport, located 30 miles east of airplane A and 40 miles north of airplane B, at what rate is the distance between the airplanes changing?

A right triangle is formed by two airplanes A and B moving perpendicularly to each other. The hypotenuse is the distance between planes A and B. The other sides are extensions of each plane’s path until they meet.

Answer

The distance is decreasing at

390 mi/h .

4-9 §4.1 source ↗

For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities.

Two buses are driving along parallel freeways that are $5 mi$ apart, one heading east and the other heading west. Assuming that each bus drives a constant $55 mph,$ find the rate at which the distance between the buses is changing when they are $13 mi$ apart, heading toward each other.

Answer

The distance between them shrinks at a rate of

\frac{1320}{13} \approx

101.5 mph .

4-11 §4.1 source ↗

For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities.

Using the previous problem, what is the rate at which the tip of the shadow moves away from the person when the person is 10 ft from the pole?

Answer

\frac{9}{2}

ft/sec

4-13 §4.1 source ↗

For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities.

Using the previous problem, what is the rate at which the shadow changes when the person is 10 ft from the wall, if the person is walking away from the wall at a rate of 2 ft/sec?

Answer

It grows at a rate

\frac{4}{9}

ft/sec

4-15 §4.1 source ↗

For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities.

Using the previous problem, what is the rate at which the distance between you and the helicopter is changing when the helicopter has risen to a height of 60 ft in the air, assuming that, initially, it was 30 ft above you?

Answer

The distance is increasing at

\frac{(135 \sqrt{26})}{26}

ft/sec

4-17 §4.1 source ↗

For the following exercises, draw and label diagrams to help solve the related-rates problems.

The volume of a cube decreases at a rate of 10 m³/s. Find the rate at which the side of the cube changes when the side of the cube is 2 m.

Answer

- \frac{5}{6}

m/sec

4-19 §4.1 source ↗

For the following exercises, draw and label diagrams to help solve the related-rates problems.

The radius of a sphere decreases at a rate of $3$ m/sec. Find the rate at which the surface area decreases when the radius is 10 m.

Answer

240 π

m²/sec

4-21 §4.1 source ↗

For the following exercises, draw and label diagrams to help solve the related-rates problems.

The radius of a sphere is increasing at a rate of 9 cm/sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.

Answer

\frac{1}{2 \sqrt{π}}

4-23 §4.1 source ↗

For the following exercises, draw and label diagrams to help solve the related-rates problems.

A triangle has two constant sides of length 3 ft and 5 ft. The angle between these two sides is increasing at a rate of 0.1 rad/sec. Find the rate at which the area of the triangle is changing when the angle between the two sides is $π / 6 .$

Answer

The area is increasing at a rate

\frac{(3 \sqrt{3})}{8} {ft}^{2} /sec.

4-25 §4.1 source ↗

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.

How fast does the depth of the water change when the water is 10 ft high if the cone leaks water at a rate of 10 ft³/min?

Answer

The depth of the water decreases at

\frac{128}{125 π}

ft/min.

4-27 §4.1 source ↗

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.

If the water level is decreasing at a rate of 3 in/min when the depth of the water is 8 ft, determine the rate at which water is leaking out of the cone.

Answer

The volume is decreasing at a rate of

\frac{(25 π)}{16} {ft}^{3} /min .

4-29 §4.1 source ↗

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.

A vertical cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of 2 m and a radius of 2 m. Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is 10 cm/min when the height is 1 m.

Answer

The water flows out at rate

\frac{(2 π)}{5} m^{3} /min.

4-31 §4.1 source ↗

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.

A tank is shaped like an upside-down square pyramid, with base of 4 m by 4 m and a height of 12 m (see the following figure). How fast does the height increase when the water is 2 m deep if water is being pumped in at a rate of $\frac{2}{3}$ m³/sec?

An upside-down square pyramid is shown with square side lengths 4 and height 12. There is an unspecified amount of water inside the shape.

Answer

\frac{3}{2}

m/sec

4-33 §4.1 source ↗

For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft³/min. The radius of the pool is 10 ft. The formula for the volume of a partial hemisphere is $V =$ $\frac{πh}{6} (3 r^{2} + h^{2})$ where $h$ is the height of the water and $r$ is the radius of the water.

Find the rate at which the depth of the water is changing when the water has a depth of 1 ft.

Answer

\frac{25}{19 π}

ft/min

4-35 §4.1 source ↗

Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of 10 ft³/min. The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of 5 ft.

Answer

\frac{2}{45 π}

ft/min

4-37 §4.1 source ↗

For the following exercises, solve the related-rates problems. Consider making a sketch that represents the situation to help you understand each problem.

You are stationary on the ground and are watching a bird fly horizontally at a rate of $10$ m/sec. The bird is located 40 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m?

Answer

The angle decreases at

\frac{400}{1681} rad/sec .

4-39 §4.1 source ↗

For the following exercises, solve the related-rates problems. Consider making a sketch that represents the situation to help you understand each problem.

A lighthouse, L, is on an island 4 mi away from the closest point, P, on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of 10 revolutions/min, how fast does the beam of light move across the beach 2 mi away from the closest point on the beach?

A right triangle is formed by a lighthouse L, a point P on the shore that is perpendicular to the line from the lighthouse to the shore, and a point 2 miles to the right of the point P. The distance from P to L is 4 miles.

Answer

100 π mi/min

4-41 §4.1 source ↗

For the following exercises, solve the related-rates problems. Consider making a sketch that represents the situation to help you understand each problem.

You are walking to a bus stop at a right-angle corner. You move north at a rate of 2 m/sec and are 20 m south of the intersection. The bus travels west at a rate of 10 m/sec away from the intersection – you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are 20 m south of the intersection and the bus is 10 m west of the intersection?

Answer

The angle is changing at a rate of

\frac{11}{25} rad/sec .

4-43 §4.1 source ↗

For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft.

A baseball field is shown, with the bases labeled Home, 1st, 2nd, and 3rd making a square with side lengths 90 ft.

[T] A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)

Answer

The distance is increasing at a rate of

62.50

ft/sec.

4-45 §4.1 source ↗

For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft.

[T] Runners start at first and second base. When the baseball is hit, the runner at first base runs at a speed of 18 ft/sec toward second base and the runner at second base runs at a speed of 20 ft/sec toward third base. How fast is the distance between runners changing 1 sec after the ball is hit?

Answer

The distance is decreasing at a rate of

11.99

ft/sec.

Checkpoint 4.5 §4.2 source ↗

Find the local linear approximation to

f (x) = \sqrt[3]{x}

x = 8 .

Use it to approximate

\sqrt[3]{8.1}

to five decimal places.

Answer

L (x) =

2 +

\frac{1}{12} (x - 8);

2.00833

Checkpoint 4.6 §4.2 source ↗

Find the linear approximation for

f (x) = cos x

x = \frac{π}{2} .

Answer

L (x) = − x + \frac{π}{2}

Checkpoint 4.7 §4.2 source ↗

Find the linear approximation of

f (x) = {(1 + x)}^{4}

x = 0

without using the result from the preceding example.

Answer

L (x) = 1 + 4 x

Checkpoint 4.8 §4.2 source ↗

For

y = e^{x^{2}},

find

d y .

Answer

d y = 2 x e^{x^{2}} d x

Checkpoint 4.9 §4.2 source ↗

For

y = x^{2} + 2 x,

find

Δ y

and

d y

x = 3

d x = 0.2 .

Answer

d y = 1.6,

Δ y = 1.64

Checkpoint 4.10 §4.2 source ↗

Estimate the error in the computed volume of a cube if the side length is measured to be 6 cm with an accuracy of 0.2 cm.

Answer

The volume measurement is accurate to within

21.6 {cm}^{3} .

Checkpoint 4.11 §4.2 source ↗

Determine the percentage error if the radius of Earth is measured to be 3950 mi with an error of

± 100

mi.

Answer

7.6%

4-47 §4.2 source ↗

Determine the necessary conditions such that the linear approximation function is constant. Use a graph to prove your result.

Answer

f^{'} (a) = 0

4-49 §4.2 source ↗

When is the linear approximation exact?

Answer

The linear approximation exact when

y = f (x)

is linear or constant.

4-51 §4.2 source ↗

For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.

$f (x) = \frac{1}{x}, a = 2$

Answer

L (x) =

\frac{1}{2} -

\frac{1}{4} (x - 2)

4-53 §4.2 source ↗

For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.

$f (x) =$ $sin x,$ $a =$ $\frac{π}{2}$

Answer

L (x) = 1

4-55 §4.2 source ↗

For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.

$f (x) =$ ${sin}^{2} x,$ $a =$ $0$

Answer

L (x) = 0

4-57 §4.2 source ↗

For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f (x)$ and $a,$ and evaluating $L (x) =$ $f (a) +$ $f^{'} (a) (x - a) .$ Check your answer using a calculator.

[T] $sin (0.02)$

Answer

0.02

4-59 §4.2 source ↗

[T] ${(15.99)}^{1 / 4}$

Answer

1.9996875

4-61 §4.2 source ↗

[T] $sin (3.14)$

Answer

0.001593

4-63 §4.2 source ↗

For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) =$ $f (a) +$ $f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.

[T] $cos (0.01)$

Answer

1;

error,

~0.00005

4-65 §4.2 source ↗

For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) =$ $f (a) +$ $f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.

[T] ${(1.01)}^{−3}$

Answer

0.97;

error,

~0.0006

4-67 §4.2 source ↗

For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) =$ $f (a) +$ $f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.

[T] $\sqrt{8.99}$

Answer

3 - \frac{1}{600};

error,

~4.632 \times 10^{−7}

4-69 §4.2 source ↗

For the following exercises, find the differential of the function.

$y = x cos x$

Answer

d y =

(cos x - x sin x) d x

4-71 §4.2 source ↗

For the following exercises, find the differential of the function.

$y = \frac{x^{2} + 2}{x - 1}$

Answer

d y =

(\frac{x^{2} - 2 x - 2}{{(x - 1)}^{2}}) d x

4-73 §4.2 source ↗

For the following exercises, find the differential and evaluate for the given $x$ and $d x .$

$y = \frac{1}{x + 1},$ $x = 1,$ $d x = 0.25$

Answer

d y =

-

\frac{1}{{(x + 1)}^{2}} d x,

- \frac{1}{16}

4-75 §4.2 source ↗

For the following exercises, find the differential and evaluate for the given $x$ and $d x .$

$y = \frac{3 x^{2} + 2}{\sqrt{x + 1}},$ $x = 0,$ $d x = 0.1$

Answer

d y =

\frac{9 x^{2} + 12 x - 2}{2 {(x + 1)}^{3 / 2}} d x,

−0.1

4-77 §4.2 source ↗

For the following exercises, find the differential and evaluate for the given $x$ and $d x .$

$y = x^{3} + 2 x + \frac{1}{x},$ $x = 1,$ $d x = 0.05$

Answer

d y =

(3 x^{2} + 2 - \frac{1}{x^{2}}) d x,

0.2

4-79 §4.2 source ↗

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

$d A$ if the sides of a cube change from $x$ to $x + d x .$

Answer

12 x d x

4-81 §4.2 source ↗

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

$d V$ if the radius of a sphere changes from $r$ by $d r .$

Answer

4 π r^{2} d r

4-83 §4.2 source ↗

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

$d V$ if a circular cylinder of height 3 changes from $r = 2$ to $r = 1.9 cm .$

Answer

−1.2 π {cm}^{3}

4-85 §4.2 source ↗

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

A pool has a rectangular base of 10 ft by 20 ft and a depth of 6 ft. What is the change in volume if you only fill it up to 5.5 ft?

Answer

−100

ft³

Checkpoint 4.12 §4.3 source ↗

Find all critical points for

f (x) =

x^{3} -

\frac{1}{2} x^{2} -

2 x +

1 .

Answer

x = - \frac{2}{3},

x = 1

Checkpoint 4.13 §4.3 source ↗

Find the absolute maximum and absolute minimum of

f (x) =

x^{2} -

4 x +

3

over the interval

[1, 4] .

Answer

The absolute maximum is

3

and it occurs at

x = 4 .

The absolute minimum is

−1

and it occurs at

x = 2 .

4-91 §4.3 source ↗

If you are finding an absolute minimum over an interval

[a, b],

why do you need to check the endpoints? Draw a graph that supports your hypothesis.

Answer

Answers may vary

4-93 §4.3 source ↗

When you are checking for critical points, explain why you also need to determine points where

f' (x)

is undefined. Draw a graph to support your explanation.

Answer

Answers will vary

4-95 §4.3 source ↗

Can you have a finite absolute maximum for

y =

a x^{3} +

b x^{2} +

c x +

d

over

(− \infty, \infty)

assuming a is non-zero? Explain why or why not using graphical arguments.

Answer

No; answers will vary

4-97 §4.3 source ↗

Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis.

Answer

Since the absolute maximum is the function (output) value rather than the x value, the answer is no; answers will vary

4-99 §4.3 source ↗

[T] Graph the function

y = e^{a x} .

For which values of

a,

on any infinite domain, will you have an absolute minimum and absolute maximum?

Answer

When

a = 0

4-101 §4.3 source ↗

For the following exercises, determine where the local and absolute maxima and minima occur on the graph given. Assume the graph represents the entirety of each function. For any extrema located at an endpoint, approximate the x-value.

The function graphed starts at (−2.2, 10), decreases rapidly to (−2, −11), increases to (−1, 5) before decreasing slowly to (1, 3), at which point it increases to (2, 7), and then decreases to (3, −20).

Answer

Absolute minimum at 3; Absolute maximum at −2.2; local minima at −2, 1; local maxima at −1, 2

4-103 §4.3 source ↗

The function graphed starts at (−2.5, 1), decreases rapidly to (−2, −1.25), increases to (−1, 0.25) before decreasing slowly to (0, 0.2), at which point it increases slowly to (1, 0.25), then decreases rapidly to (2, −1.25), and finally increases to (2.5, 1).

Answer

Absolute minima at −2, 2; absolute maxima at −2.5, 2.5; minima at –2, 0, and 2 (Note: –2 and 2 are both local and absolute minima); local maxima at −1, 1

4-105 §4.3 source ↗

For the following problems, draw graphs of $f (x),$ which is continuous, over the interval $[−4, 4]$ with the following properties:

Absolute minimum at $x = 1$ and absolute maximum at $x = 2$

Answer

Answers may vary.

4-107 §4.3 source ↗

For the following problems, draw graphs of $f (x),$ which is continuous, over the interval $[−4, 4]$ with the following properties:

Absolute maxima at $x = 2$ and $x = −3,$ local minimum at $x = 1,$ and absolute minimum at $x = 4$

Answer

Answers may vary.

4-109 §4.3 source ↗

For the following exercises, find the critical numbers in the domains of the following functions.

$y = 4 \sqrt{x} - x^{2}$

Answer

x = 1

4-111 §4.3 source ↗

For the following exercises, find the critical numbers in the domains of the following functions.

$y = ln (x - 2)$

Answer

None

4-113 §4.3 source ↗

For the following exercises, find the critical numbers in the domains of the following functions.

$y = \sqrt{4 - x^{2}}$

Answer

x = 0; x = \pm 2

4-115 §4.3 source ↗

For the following exercises, find the critical numbers in the domains of the following functions.

$y =$ $\frac{x^{2} - 1}{x^{2} + 2 x - 3}$

Answer

None

4-117 §4.3 source ↗

For the following exercises, find the critical numbers in the domains of the following functions.

$y = x + \frac{1}{x}$

Answer

x = −1, 1

4-119 §4.3 source ↗

For the following exercises, find the local and/or absolute extrema for the functions over the specified domain.

$y = x^{2} + \frac{2}{x}$ over $[1, 4]$

Answer

Absolute maximum:

x = 4,

y = \frac{33}{2};

absolute minimum:

x = 1,

y = 3

4-121 §4.3 source ↗

For the following exercises, find the local and/or absolute extrema for the functions over the specified domain.

$y = \frac{1}{(x - x^{2})}$ over $(0, 1)$

Answer

Absolute minimum:

x = \frac{1}{2},

y = 4

4-123 §4.3 source ↗

For the following exercises, find the local and/or absolute extrema for the functions over the specified domain.

$y = x + sin (x)$ over $[0, 2 π]$

Answer

Absolute maximum:

x = 2 π,

y = 2 π;

absolute minimum:

x = 0,

y = 0

4-125 §4.3 source ↗

For the following exercises, find the local and/or absolute extrema for the functions over the specified domain.

$y =$ $| x +$ $1 | +$ $| x -$ $1 |$ over $[−3, 2]$

Answer

Absolute maximum:

x = −3

and

y = 6;;

absolute minimum:

−1 \leq x \leq 1,

y = 2

4-127 §4.3 source ↗

For the following exercises, find the local and/or absolute extrema for the functions over the specified domain.

$y = sin x + cos x$ over $[0, 2 π]$

Answer

Absolute maximum:

x = \frac{π}{4},

y = \sqrt{2};

absolute minimum:

x = \frac{5 π}{4},

y = − \sqrt{2}

4-129 §4.3 source ↗

For the following exercises, find the local and absolute minima and maxima for the functions over $(− \infty, \infty) .$

$y = x^{2} + 4 x + 5$

Answer

Absolute minimum:

x = −2,

y = 1

4-131 §4.3 source ↗

For the following exercises, find the local and absolute minima and maxima for the functions over $(− \infty, \infty) .$

$y =$ $3 x^{4} +$ $8 x^{3} -$ $18 x^{2}$

Answer

Absolute minimum:

x = −3,

y = −135;

local maximum:

x = 0,

y = 0;

local minimum:

x = 1,

y = −7

4-133 §4.3 source ↗

For the following exercises, find the local and absolute minima and maxima for the functions over $(− \infty, \infty) .$

$y = \frac{x^{2} + x + 6}{x - 1}$

Answer

Local maximum:

x = 1 - 2 \sqrt{2},

y = 3 - 4 \sqrt{2};

local minimum:

x = 1 + 2 \sqrt{2},

y = 3 + 4 \sqrt{2}

4-135 §4.3 source ↗

For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly.

[T] $y = 3 x \sqrt{1 - x^{2}}$

Answer

Absolute maximum:

x = \frac{\sqrt{2}}{2},

y = \frac{3}{2};

absolute minimum:

x = - \frac{\sqrt{2}}{2},

y = - \frac{3}{2}

4-137 §4.3 source ↗

For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly.

[T] $y =$ $12 x^{5} +$ $45 x^{4} +$ $20 x^{3} -$ $90 x^{2} -$ $120 x +$ $3$

Answer

Local maximum:

x = −2,

y = 59;

local minimum:

x = 1,

y = −130

4-139 §4.3 source ↗

For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly.

[T] $y = \frac{\sqrt{4 - x^{2}}}{\sqrt{4 + x^{2}}}$

Answer

Absolute maximum:

x = 0,

y = 1;

absolute minimum:

x = −2, 2,

y = 0

4-141 §4.3 source ↗

For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly.

A ball is thrown into the air and its position is given by $h (t) =$ $−4.9 t^{2} +$ $60 t +$ $5 m .$ Find the height at which the ball stops ascending. How long after it is thrown does this happen?

Answer

h = \frac{9245}{49} m,

t = \frac{300}{49} s

4-143 §4.3 source ↗

For the following exercises, consider the production of gold during the California gold rush (1848–1888). The production of gold can be modeled by $G (t) =$ $\frac{(25 t)}{(t^{2} + 16)},$ where $t$ is the number of years since the rush began $(0 \leq t \leq 40)$ and $G$ is ounces of gold produced (in millions). A summary of the data is shown in the following figure.

The bar graph shows gold (in millions of troy ounces) per year, starting in 1848 and ending in 1888. In 1848, the bar graph shows 0.05; in 1849, 0.5; in 1850, 2; in 1851, 3.6; in 1852, 3.9; in 1853, 3.3; in 1854, 3.4; in 1855, 2.6; in 1856, 2.75; in 1857, 2.1; in 1858, 2.2; in 1859, 2.15; in 1860, 2.1; in 1861, 2; in 1862, 1.8; in 1863, 1.1; in 1864, 1.15; in 1865, 0.9; in 1866, 0.85; in 1867, 0.9; in 1868, 0.85; in 1869, 0.9; in 1870, 0.85; in 1871, 0.85; in 1872, 0.75; in 1873, 0.7; in 1874, 0.8; in 1875, 0.75; in 1876, 0.7; in 1877, 0.73; in 1878, 0.9; in 1879, 0.95; in 1880, 1; in 1881, 0.95; in 1882, 0.85; in 1883, 1.1; in 1884, 0.6; in 1885, 0.55; in 1886, 0.65; in 1887, 0.6; and in 1888, 0.55.

Find when the minimum (local and absolute) gold production occurred. What was the amount of gold produced during this minimum?

Answer

The absolute minimum was in 1848, when no gold was produced.

4-145 §4.3 source ↗

Find the critical points, maxima, and minima for the following piecewise functions.

$y = {\begin{matrix} x^{2} + 1 & x \leq 1 \\ x^{2} - 4 x + 5 & x > 1 \end{matrix}$

Answer

Absolute minima:

x = 0,

x = 2,

y = 1;

local maximum at

x = 1,

y = 2

4-147 §4.3 source ↗

For the following exercises, find the critical points of the following generic functions. Are they maxima, minima, or neither? State the necessary conditions.

$y = {(x - 1)}^{a},$ given that $a > 1$ and a is an integer.

Answer

No maxima/minima if

a

is odd, minimum at

x = 1

a

is even

Checkpoint 4.14 §4.4 source ↗

Verify that the function

f (x) =

2 x^{2} -

8 x +

6

defined over the interval

[1, 3]

satisfies the conditions of Rolle’s theorem. Find all points

c

guaranteed by Rolle’s theorem.

Answer

c = 2

Checkpoint 4.15 §4.4 source ↗

Suppose a ball is dropped from a height of 200 ft. Its position at time

t

s (t) =

−16 t^{2} +

200 .

Find the time

t

when the instantaneous velocity of the ball equals its average velocity.

Answer

\frac{5}{2 \sqrt{2}}

sec

4-149 §4.4 source ↗

Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.

Answer

One example is

f (x) =

| x | +

3,

−2 \leq x \leq 2

4-151 §4.4 source ↗

If you have a function with a discontinuity, is it still possible to have

f^{'} (c) (b - a) =

f (b) -

f (a) ?

Draw such an example or prove why not.

Answer

Yes, but the Mean Value Theorem still does not apply

4-153 §4.4 source ↗

For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

$y = \frac{1}{x^{3}}$

Answer

Any closed interval in

(− \infty, 0) or (0, \infty)

4-155 §4.4 source ↗

For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

$y = \sqrt{x^{2} - 4}$

Answer

Any closed interval in

(− \infty, −2) or (2, \infty)

4-157 §4.4 source ↗

For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points $c$ such that $f^{'} (c) (b - a) =$ $f (b) -$ $f (a) .$

[T] $y = 3 x^{3} + 2 x + 1$ over $[−1, 1]$

Answer

2 points

4-159 §4.4 source ↗

[T] $y = x^{2} cos (π x)$ over $[−2, 2]$

Answer

5 points

4-161 §4.4 source ↗

For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) -$ $f (0) =$ $f^{'} (c) (2 - 0) .$

$f (x) = x^{3}$

Answer

c = \frac{2 \sqrt{3}}{3}

4-163 §4.4 source ↗

For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) -$ $f (0) =$ $f^{'} (c) (2 - 0) .$

$f (x) =$ $cos (2 π x)$

Answer

c = \frac{1}{2}, 1, \frac{3}{2}

4-165 §4.4 source ↗

For the following exercises, find all points $0 < c < 2$ predicted by the Mean Value Theorem satisfying $f (2) -$ $f (0) =$ $f^{'} (c) (2 - 0) .$

$f (x) =$ ${(x - 1)}^{10}$

Answer

c = 1

4-167 §4.4 source ↗

For the following exercises, show there is no $c$ such that $f (1) -$ $f (−1) =$ $f^{'} (c) (2) .$ Explain why the Mean Value Theorem does not apply over the interval $[−1, 1] .$

$f (x) = | x - \frac{1}{2} |$

Answer

Not differentiable

4-169 §4.4 source ↗

For the following exercises, show there is no $c$ such that $f (1) -$ $f (−1) =$ $f^{'} (c) (2) .$ Explain why the Mean Value Theorem does not apply over the interval $[−1, 1] .$

$f (x) = \sqrt{| x |}$

Answer

Not differentiable

4-171 §4.4 source ↗

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.

$y = e^{x}$ over $[0, 1]$

Answer

Yes

4-173 §4.4 source ↗

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.

$f (x) =$ $tan (2 π x)$ over $[0, 2]$

Answer

The Mean Value Theorem does not apply since the function is discontinuous at

x =

\frac{1}{4},

\frac{3}{4},

\frac{5}{4},

\frac{7}{4} .

4-175 §4.4 source ↗

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.

$y = \frac{1}{| x + 1 |}$ over $[0, 3]$

Answer

Yes

4-177 §4.4 source ↗

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.

$y = \frac{x^{2} + 3 x + 2}{x}$ over $[−1, 1]$

Answer

The Mean Value Theorem does not apply; discontinuous at

x = 0 .

4-179 §4.4 source ↗

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.

$y = ln (x + 1)$ over $[0, e - 1]$

Answer

Yes

4-181 §4.4 source ↗

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $[a, b] .$ Justify your answer.

$y = 5 + | x |$ over $[−1, 1]$

Answer

The Mean Value Theorem does not apply; not differentiable at

x = 0 .

4-183 §4.4 source ↗

For the following exercises, consider the roots of the equation.

Find the conditions for exactly one root (double root) for the equation $y = x^{2} + b x + c$

Answer

b = ± 2 \sqrt{c}

4-185 §4.4 source ↗

For the following exercises, use a calculator to graph the function over the interval $[a, b]$ and graph the secant line from $a$ to $b .$ Use the calculator to estimate all values of $c$ as guaranteed by the Mean Value Theorem. Then, find the exact value of $c,$ if possible, or write the final equation and use a calculator to estimate to four digits.

[T] $y = tan (π x)$ over $[- \frac{1}{4}, \frac{1}{4}]$

Answer

c =

± \frac{1}{π} {cos}^{−1} (\frac{\sqrt{π}}{2}),

c = ± 0.1533

4-187 §4.4 source ↗

[T] $y = | x^{2} + 2 x - 4 |$ over $[−4, 0]$

Answer

The Mean Value Theorem does not apply.

4-189 §4.4 source ↗

[T] $y = \sqrt{x + 1} + \frac{1}{x^{2}}$ over $[3, 8]$

Answer

\frac{1}{2 \sqrt{c + 1}} -

\frac{2}{c^{3}} =

\frac{521}{2880};

c =

3.133,

5.867

4-191 §4.4 source ↗

Two cars drive from one stoplight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.

Answer

Yes

4-193 §4.4 source ↗

Show that $y = {csc}^{2} x$ and $y = {cot}^{2} x$ have the same derivative. What can you say about $y =$ ${csc}^{2} x -$ ${cot}^{2} x ?$

Answer

Let $s_{1} (t)$ represent the distance of Car 1 from Stoplight 1 at time $t$ . Let $s_{2} (t)$ represent the distance of Car 2 from Stoplight 1 at time $t$ . Then $s_{1} (0) =$ $0 =$ $s_{2} (0)$ . Let $t = m$ represent the time that the cars arrive at Stoplight 2. Then $s_{1} (m) =$ $d =$ $s_{2} (m)$ where $d$ is the distance between the stoplights.

Define a new function $f (t) =$ $s_{1} (t) -$ $s_{2} (t)$ . Then $f (0) = 0 = f (m)$ . By the Mean Value Theorem, there is a time $0 < c < m$ such that

f^{'} (c) = \frac{f (m) - f (0)}{m - 0} = \frac{0}{m} = 0 .

At time c,

f^{'} (c) = 0

s_{1}^{'} (c) - s_{2}^{'} (c) = 0

s_{1}^{'} (c) = s_{2}^{'} (c)

So, at time c, Car 1 and Car 2 are traveling at the same speed.

Checkpoint 4.16 §4.5 source ↗

Use the first derivative test to locate all local extrema for

f (x) =

− x^{3} +

\frac{3}{2} x^{2} +

18 x .

Answer

f

has a local minimum at

−2

and a local maximum at

3 .

Checkpoint 4.17 §4.5 source ↗

Use the first derivative test to find all local extrema for

f (x) = \sqrt[3]{x - 1} .

Answer

f

has no local extrema because

f^{'}

does not change sign at

x = 1 .

Checkpoint 4.18 §4.5 source ↗

For

f (x) =

− x^{3} +

\frac{3}{2} x^{2} +

18 x,

find all intervals where

f

is concave up and all intervals where

f

is concave down.

Answer

f

is concave up over the interval

(− \infty, \frac{1}{2})

and concave down over the interval

(\frac{1}{2}, \infty)

Checkpoint 4.19 §4.5 source ↗

Consider the function

f (x) =

x^{3} -

(\frac{3}{2}) x^{2} -

18 x .

The points

c = 3, −2

satisfy

f^{'} (c) = 0 .

Use the second derivative test to determine whether

f

has a local maximum or local minimum at those points.

Answer

f

has a local maximum at

−2

and a local minimum at

3 .

4-195 §4.5 source ↗

For the function

y = x^{3},

x = 0

both an inflection point and a local maximum/minimum?

Answer

It is not a local maximum/minimum because

f^{'}

does not change sign

4-197 §4.5 source ↗

Is it possible for a point

c

to be both an inflection point and a local extremum of a twice differentiable function?

Answer

4-199 §4.5 source ↗

Explain whether a concave-down function has to cross

y = 0

for some value of

x .

Answer

False; for example,

y = \sqrt{x} .

4-201 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−2, 0). Then it continues increasing a little before decreasing and crossing the x axis at (−1, 0). It achieves a local minimum at (1, −6) before increasing and crossing the x axis at (2, 0).

Answer

Increasing for

−2 <

x <

−1

and

x > 2;

decreasing for

x < −2

and

−1 <

x <

2

4-203 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts negative and touches the x axis at the origin. Then it decreases a little before increasing to cross the x axis at (1, 0) and continuing to increase.

Answer

Decreasing for

x < 1,

increasing for

x > 1

4-205 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts at (−2, 0), decreases to (−1.5, −1.5), increases to (−1, 0), and continues increasing before decreasing to the origin. Then the other side is symmetric: that is, the function increases and then decreases to pass through (1, 0). It continues decreasing to (1.5, −1.5), and then increase to (2, 0).

Answer

Decreasing for

−2 <

x <

−1

and

1 <

x <

2;

increasing for

−1 <

x <

1

and

x < −2

and

x > 2

4-207 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where

The function f’(x) is graphed. The function starts at (−2, 0), increases and then decreases to (−1, 0), decreases and then increases to an inflection point at the origin. Then the function increases and decreases to cross (1, 0). It continues decreasing and then increases to (2, 0).

Answer

a. Increasing over

−2 <

x <

−1,

0 <

x <

1,

x >

2,

decreasing over

x < −2,

−1 <

x <

0,

1 <

x <

2;

b. maxima at

x = −1

and

x = 1,

minima at

x = −2

and

x = 0

and

x = 2

4-209 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where

The function f’(x) is graphed. The function starts negative and crosses the x axis at the origin, which is an inflection point. Then it continues increasing.

Answer

a. Increasing over

x > 0,

decreasing over

x < 0;

b. Minimum at

x = 0

4-211 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals where $f$ is concave up and concave down.

The function f’(x) is graphed. The function is linear and starts negative. It crosses the x axis at the origin.

Answer

Concave up on all

x,

no inflection points

4-213 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals where $f$ is concave up and concave down.

The function f’(x) is graphed. The function resembles the graph of x3: that is, it starts negative and crosses the x axis at the origin. Then it continues increasing.

Answer

Concave up on all

x,

no inflection points

4-215 §4.5 source ↗

For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals where $f$ is concave up and concave down.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing to a local maximum at (0, 1), at which point it decreases and touches the x axis at (1, 0). It then increases.

Answer

Concave up for

x < 0

and

x > 1,

concave down for

0 <

x <

1,

inflection points at

x = 0

and

x = 1

4-217 §4.5 source ↗

For the following exercises, draw a graph that satisfies the given specifications for the domain $x ϵ [−3, 3] .$ The function does not have to be continuous or differentiable.

$f^{'} (x) > 0$ over $x >$ $2,$ $−3 <$ $x <$ $−1,$ $f^{'} (x) <$ $0$ over $−1 <$ $x <$ $2,$ $f^{″} (x) <$ $0$ for all $x$

Answer

Answers will vary

4-219 §4.5 source ↗

For the following exercises, draw a graph that satisfies the given specifications for the domain $x ϵ [−3, 3] .$ The function does not have to be continuous or differentiable.

There is a local maximum at $x = 2,$ local minimum at $x = 1,$ and the graph is neither concave up nor concave down.

Answer

Answers will vary

4-221 §4.5 source ↗

For the following exercises, determine

If needed, use a calculator to graph the functions, but show your work.

$f (x) =$ $sin x +$ ${sin}^{3} x$ over $− π <$ $x <$ $π$

Answer

a. Increasing over

-

\frac{π}{2} <

x <

\frac{π}{2},

decreasing over

x <

-

\frac{π}{2},

x >

\frac{π}{2}

b. Local maximum at

x = \frac{π}{2};

local minimum at

x = - \frac{π}{2}

4-223 §4.5 source ↗

For the following exercises, determine a. intervals where $f$ is concave up or concave down, and b. the inflection points of $f .$

$f (x) =$ $x^{3} -$ $4 x^{2} +$ $x +$ $2$

Answer

a. Concave up for

x > \frac{4}{3},

concave down for

x < \frac{4}{3}

b. Inflection point at

x = \frac{4}{3}

4-225 §4.5 source ↗

For the following exercises, determine

$f (x) = x^{3} - 6 x^{2}$

Answer

a. Increasing over

x < 0

and

x > 4,

decreasing over

0 < x < 4

b. Maximum at

x = 0,

minimum at

x = 4

c. Concave up for

x > 2,

concave down for

x < 2

d. Infection point at

x = 2

4-227 §4.5 source ↗

For the following exercises, determine

$f (x) =$ $x^{11} -$ $6 x^{10}$

Answer

a. Increasing over

x < 0

and

x > \frac{60}{11},

decreasing over

0 <

x <

\frac{60}{11}

b. Minimum at

x = \frac{60}{11}

, local maximum at x = 0 c. Concave down for

x < \frac{54}{11},

concave up for

x > \frac{54}{11}

d. Inflection point at

x = \frac{54}{11}

4-229 §4.5 source ↗

For the following exercises, determine

$f (x) = x^{2} + x + 1$

Answer

a. Increasing over

x > - \frac{1}{2},

decreasing over

x < - \frac{1}{2}

b. Minimum at

x = - \frac{1}{2}

c. Concave up for all

x

d. No inflection points

4-231 §4.5 source ↗

For the following exercises, determine

[T] $f (x) =$ $sin (π x) -$ $cos (π x)$ over $x = [−1, 1]$

Answer

a. Increases over

-

\frac{1}{4} <

x <

\frac{3}{4},

decreases over

x > \frac{3}{4}

and

x < - \frac{1}{4}

b. Minimum at

x = - \frac{1}{4},

maximum at

x = \frac{3}{4}

c. Concave up for

-

\frac{3}{4} <

x <

\frac{1}{4},

concave down for

x < - \frac{3}{4}

and

x > \frac{1}{4}

d. Inflection points at

x = - \frac{3}{4}, x = \frac{1}{4}

4-233 §4.5 source ↗

For the following exercises, determine

[T] $f (x) =$ $sin x +$ $tan x$ over $(- \frac{π}{2}, \frac{π}{2})$

Answer

a. Increasing for all

x

b. No local minimum or maximum c. Concave up for

x > 0,

concave down for

x < 0

d. Inflection point at

x = 0

4-235 §4.5 source ↗

For the following exercises, determine

[T] $f (x) =$ $\frac{1}{1 - x},$ $x \neq 1$

Answer

a. Increasing for all

x

where defined b. No local minima or maxima c. Concave up for

x < 1;

concave down for

x > 1

d. No inflection points in domain

4-237 §4.5 source ↗

For the following exercises, determine

$f (x) =$ $sin (x) e^{x}$ over $x = [− π, π]$

Answer

a. Increasing over

-

\frac{π}{4} <

x <

\frac{3 π}{4},

decreasing over

x >

\frac{3 π}{4},

x <

-

\frac{π}{4}

b. Minimum at

x = - \frac{π}{4},

maximum at

x = \frac{3 π}{4}

c. Concave up for

-

\frac{π}{2} <

x <

\frac{π}{2},

concave down for

x <

-

\frac{π}{2},

x >

\frac{π}{2}

d. Infection points at

x = ± \frac{π}{2}

4-239 §4.5 source ↗

For the following exercises, determine

$f (x) =$ $\frac{1}{4} \sqrt{x} +$ $\frac{1}{x},$ $x >$ $0$

Answer

a. Increasing over

x > 4,

decreasing over

0 < x < 4

b. Minimum at

x = 4

c. Concave up for

0 <

x <

8 \sqrt[3]{2},

concave down for

x > 8 \sqrt[3]{2}

d. Inflection point at

x = 8 \sqrt[3]{2}

4-241 §4.5 source ↗

For the following exercises, interpret the sentences in terms of $f, f^{'}, and f^{″} .$

The population is growing more slowly. Here $f$ is the population.

Answer

f >

0,

f^{'} >

0,

f^{″} <

0

4-243 §4.5 source ↗

For the following exercises, interpret the sentences in terms of $f, f^{'}, and f^{″} .$

The airplane lands smoothly. Here $f$ is the plane’s altitude.

Answer

f >

0,

f^{'} <

0,

f^{″} >

0

4-245 §4.5 source ↗

For the following exercises, interpret the sentences in terms of $f, f^{'}, and f^{″} .$

The economy is picking up speed. Here $f$ is a measure of the economy, such as GDP.

Answer

f >

0,

f^{'} >

0,

f^{″} >

0

4-247 §4.5 source ↗

For the following exercises, consider a third-degree polynomial $f (x),$ which has the properties $f^{'} (1) =$ $0,$ $f^{'} (3) =$ $0$ (not applicable to Exercises 249 and 250). Determine whether the following statements are true or false. Justify your answer.

$f^{″} (x) = 0$ for some $1 \leq x \leq 3$

Answer

True, by the Mean Value Theorem

4-249 §4.5 source ↗

If $f (x)$ has three roots, then it has $1$ inflection point.

Answer

True, examine derivative

Checkpoint 4.20 §4.6 source ↗

Evaluate

lim_{x \to − \infty} (3 + \frac{4}{x})

and

lim_{x \to \infty} (3 + \frac{4}{x}) .

Determine the horizontal asymptotes of

f (x) = 3 + \frac{4}{x},

if any.

Answer

Both limits are

3 .

The line

y = 3

is a horizontal asymptote.

Checkpoint 4.21 §4.6 source ↗

Use the formal definition of limit at infinity to prove that

lim_{x \to \infty} (3 - \frac{1}{x^{2}}) =

3 .

Answer

Let $ε > 0 .$ Let $N = \frac{1}{\sqrt{ε}} .$ Therefore, for all $x > N,$ we have

$| 3 -$ $\frac{1}{x^{2}} -$ $3 | =$ $\frac{1}{x^{2}} <$ $\frac{1}{N^{2}} =$ $ε$

Therefore, $lim_{x \to \infty} (3 - 1 / x^{2}) =$ $3 .$

Checkpoint 4.22 §4.6 source ↗

Use the formal definition of infinite limit at infinity to prove that

lim_{x \to \infty} 3 x^{2} =

\infty .

Answer

Let $M > 0 .$ Let $N = \sqrt{\frac{M}{3}} .$ Then, for all $x > N,$ we have

$3 x^{2} >$ $3 N^{2} =$ $3 {(\sqrt{\frac{M}{3}})}^{2}^{2} =$ $\frac{3 M}{3} =$ $M$

Checkpoint 4.23 §4.6 source ↗

Let

f (x) = −3 x^{4} .

Find

lim_{x \to \infty} f (x) .

Answer

− \infty

Checkpoint 4.24 §4.6 source ↗

Evaluate

lim_{x \to ± \infty} \frac{3 x^{2} + 2 x - 1}{5 x^{2} - 4 x + 7}

and use these limits to determine the end behavior of

f (x) =

\frac{3 x^{2} + 2 x - 1}{5 x^{2} - 4 x + 7} .

Answer

\frac{3}{5}

Checkpoint 4.25 §4.6 source ↗

Evaluate

lim_{x \to \infty} \frac{\sqrt{3 x^{2} + 4}}{x + 6} .

Answer

± \sqrt{3}

Checkpoint 4.26 §4.6 source ↗

Find the limits as

x \to \infty

and

x \to − \infty

for

f (x) =

\frac{(3 e^{x} - 4)}{(5 e^{x} + 2)} .

Answer

lim_{x \to \infty} f (x) =

\frac{3}{5},

lim_{x \to − \infty} f (x) =

−2

Checkpoint 4.27 §4.6 source ↗

Sketch a graph of

f (x) =

{(x - 1)}^{3} (x + 2) .

Answer

The function f(x) = (x −1)3(x + 2) is graphed.

Checkpoint 4.28 §4.6 source ↗

Sketch a graph of

f (x) =

\frac{(3 x + 5)}{(8 + 4 x)} .

Answer

The function f(x) = (3x + 5)/(8 + 4x) is graphed. It appears to have asymptotes at x = −2 and y = 1.

Checkpoint 4.29 §4.6 source ↗

Find the oblique asymptote for

f (x) =

\frac{(3 x^{3} - 2 x + 1)}{(2 x^{2} - 4)} .

Answer

y = \frac{3}{2} x

Checkpoint 4.30 §4.6 source ↗

Consider the function

f (x) =

5 -

x^{2 / 3} .

Determine the point on the graph where a cusp is located. Determine the end behavior of

f .

Answer

The function

f

has a cusp at

(0, 5)

lim_{x \to 0^{-}} f^{'} (x) =

\infty,

lim_{x \to 0^{+}} f^{'} (x) =

− \infty .

For end behavior,

lim_{x \to ± \infty} f (x) =

− \infty .

4-251 §4.6 source ↗

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

The function graphed decreases very rapidly as it approaches x = 1 from the left, and on the other side of x = 1, it seems to start near infinity and then decrease rapidly.

Answer

x = 1

4-253 §4.6 source ↗

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

The function graphed decreases very rapidly as it approaches x = −1 from the left, and on the other side of x = −1, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.

Answer

x = −1, x = 2

4-255 §4.6 source ↗

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side being a normal function that appears as if it will take the entirety of the values of the x-axis.

Answer

x = 0

4-257 §4.6 source ↗

For the following functions $f (x),$ determine whether there is an asymptote at $x = a .$ Justify your answer without graphing on a calculator.

$f (x) =$ $\frac{x}{x - 2},$ $a =$ $2$

Answer

Yes, there is a vertical asymptote

4-259 §4.6 source ↗

For the following functions $f (x),$ determine whether there is an asymptote at $x = a .$ Justify your answer without graphing on a calculator.

$f (x) =$ ${(x - 1)}^{−1 / 3},$ $a =$ $1$

Answer

Yes, there is vertical asymptote

4-261 §4.6 source ↗

For the following exercises, evaluate the limit.

$lim_{x \to \infty} \frac{1}{3 x + 6}$

Answer

0

4-263 §4.6 source ↗

For the following exercises, evaluate the limit.

$lim_{x \to \infty} \frac{x^{2} - 2 x + 5}{x + 2}$

Answer

\infty

4-265 §4.6 source ↗

For the following exercises, evaluate the limit.

$lim_{x \to − \infty} \frac{x^{4} - 4 x^{3} + 1}{2 - 2 x^{2} - 7 x^{4}}$

Answer

- \frac{1}{7}

4-267 §4.6 source ↗

For the following exercises, evaluate the limit.

$lim_{x \to − \infty} \frac{\sqrt{4 x^{2} - 1}}{x + 2}$

Answer

−2

4-269 §4.6 source ↗

For the following exercises, evaluate the limit.

$lim_{x \to − \infty} \frac{4 x}{\sqrt{x^{2} - 1}}$

Answer

−4

4-271 §4.6 source ↗

For the following exercises, find the horizontal and vertical asymptotes.

$f (x) = x - \frac{9}{x}$

Answer

Horizontal: none, vertical:

x = 0

4-273 §4.6 source ↗

For the following exercises, find the horizontal and vertical asymptotes.

$f (x) = \frac{x^{3}}{4 - x^{2}}$

Answer

Horizontal: none, vertical:

x = ± 2

4-275 §4.6 source ↗

For the following exercises, find the horizontal and vertical asymptotes.

$f (x) =$ $sin (x) sin (2 x)$

Answer

Horizontal: none, vertical: none

4-277 §4.6 source ↗

For the following exercises, find the horizontal and vertical asymptotes.

$f (x) =$ $\frac{x sin (x)}{x^{2} - 1}$

Answer

Horizontal:

y = 0,

vertical:

x = ± 1

4-279 §4.6 source ↗

For the following exercises, find the horizontal and vertical asymptotes.

$f (x) = \frac{1}{x^{3} + x^{2}}$

Answer

Horizontal:

y = 0,

vertical:

x = 0

and

x = −1

4-281 §4.6 source ↗

For the following exercises, find the horizontal and vertical asymptotes.

$f (x) =$ $\frac{x^{3} + 1}{x^{3} - 1}$

Answer

Horizontal:

y = 1,

vertical:

x = 1

4-283 §4.6 source ↗

For the following exercises, find the horizontal and vertical asymptotes.

$f (x) = x - sin x$

Answer

Horizontal: none, vertical: none

4-285 §4.6 source ↗

For the following exercises, construct a function $f (x)$ that has the given asymptotes.

$x = 1$ and $y = 2$

Answer

Answers will vary, for example:

y = \frac{2 x}{x - 1}

4-287 §4.6 source ↗

For the following exercises, construct a function $f (x)$ that has the given asymptotes.

$y = 4,$ $x = −1$

Answer

Answers will vary, for example:

y = \frac{4 x}{x + 1}

4-289 §4.6 source ↗

For the following exercises, graph the function on a graphing calculator on the window $x = [−5, 5]$ and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.

[T] $f (x) = \frac{1}{x + 10}$

Answer

y = 0

4-291 §4.6 source ↗

[T] $lim_{x \to − \infty} x^{2} +$ $10 x +$ $25$

Answer

\infty

4-293 §4.6 source ↗

[T] $lim_{x \to \infty} \frac{3 x + 2}{x + 5}$

Answer

y = 3

4-295 §4.6 source ↗

For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.

$y = x^{3} - 3 x^{2} + 4$

Answer

The function starts in the third quadrant, increases to pass through (−1, 0), increases to a maximum and y intercept at 4, decreases to touch (2, 0), and then increases to (4, 20).

4-297 §4.6 source ↗

$y =$ $\frac{x^{3} + 4 x^{2} + 3 x}{3 x + 9}$

Answer

An upward-facing parabola with minimum between x = 0 and x = −1 with y intercept between 0 and 1.

4-299 §4.6 source ↗

$y = \sqrt{x^{2} - 5 x + 4}$

Answer

This graph starts at (−2, 4) and decreases in a convex way to (1, 0). Then the graph starts again at (4, 0) and increases in a convex way to (6, 3).

4-301 §4.6 source ↗

$y = \frac{cos x}{x},$ on $x = [−2 π, 2 π]$

Answer

This graph has vertical asymptote at x = 0. The first part of the function occurs in the second and third quadrants and starts in the third quadrant just below (−2π, 0), increases and passes through the x axis at −3π/2, reaches a maximum and then decreases through the x axis at −π/2 before approaching the asymptote. On the other side of the asymptote, the function starts in the first quadrant, decreases quickly to pass through π/2, decreases to a local minimum and then increases through (3π/2, 0) before staying just above (2π, 0).

4-303 §4.6 source ↗

$y =$ $x tan x,$ $x =$ $[− π, π]$

Answer

This graph has vertical asymptotes at x = ±π/2. The graph is symmetric about the y axis, so describing the left hand side will be sufficient. The function starts at (−π, 0) and decreases quickly to the asymptote. Then it starts on the other side of the asymptote in the second quadrant and decreases to the the origin.

4-305 §4.6 source ↗

$y =$ $x^{2} sin (x),$ $x =$ $[−2 π, 2 π]$

Answer

This function starts at (−2π, 0), increases to near (−3π/2, 25), decreases through (−π, 0), achieves a local minimum and then increases through the origin. On the other side of the origin, the graph is the same but flipped, that is, it is congruent to the other half by a rotation of 180 degrees.

4-307 §4.6 source ↗

For $f (x) =$ $\frac{P (x)}{Q (x)}$ to have an asymptote at $x = 0,$ then the polynomials $P (x)$ and $Q (x) .$ must have what relation?

Answer

P (0) \neq 0 and Q (0) =

0

4-309 §4.6 source ↗

Both $f (x) = \frac{1}{(x - 1)}$ and $g (x) =$ $\frac{1}{{(x - 1)}^{2}}$ have asymptotes at $x = 1$ and $y = 0 .$ What is the most obvious difference between these two functions?

Answer

lim_{x \to 1^{-}} f (x) =

-∞ and lim_{x \to 1^{-}} g (x) =

\infty

Checkpoint 4.31 §4.7 source ↗

Determine the maximum area if we want to make the same rectangular garden as in Figure 4.63, but we have

200

ft of fencing.

The function A(x) = 100x – 2x is graphed. At its maximum there is an intersection of two dashed lines and text that reads “Maximum area is 1250 square feet when x = 25 feet.” — Figure 4.63 in the book ↗

Answer

The maximum area is

5000 {ft}^{2} .

Checkpoint 4.32 §4.7 source ↗

Suppose the dimensions of the cardboard in Example 4.33 are 20 in. by 30 in. Let

x

be the side length of each square and write the volume of the open-top box as a function of

x .

Determine the domain of consideration for

x .

Answer

V (x) =

x (20 - 2 x) (30 - 2 x) .

The domain is

[0, 10] .

Checkpoint 4.33 §4.7 source ↗

Suppose the island is

1

mi from shore, and the distance from the cabin to the point on the shore closest to the island is

15 mi .

Suppose a visitor swims at the rate of

2.5 mph

and runs at a rate of

6 mph .

Let

x

denote the distance the visitor will run before swimming, and find a function for the time it takes the visitor to get from the cabin to the island.

Answer

T (x) =

\frac{x}{6} +

\frac{\sqrt{{(15 - x)}^{2} + 1}}{2.5}

Checkpoint 4.34 §4.7 source ↗

A car rental company charges its customers

p

dollars per day, where

60 \leq p \leq 150 .

It has found that the number of cars rented per day can be modeled by the linear function

n (p) =

750 -

5 p .

How much should the company charge each customer to maximize revenue?

Answer

The company should charge

$ 75

per car per day.

Checkpoint 4.35 §4.7 source ↗

Modify the area function

A

if the rectangle is to be inscribed in the unit circle

x^{2} + y^{2} = 1 .

What is the domain of consideration?

Answer

A (x) =

4 x \sqrt{1 - x^{2}} .

The domain of consideration is

[0, 1] .

Checkpoint 4.36 §4.7 source ↗

Consider the same open-top box, which is to have volume

216 in .^{3} .

Suppose the cost of the material for the base is

20 ¢ / in .^{2}

and the cost of the material for the sides is

30 ¢ / in .^{2}

and we are trying to minimize the cost of this box. Write the cost as a function of the side lengths of the base. (Let

x

be the side length of the base and

y

be the height of the box.)

Answer

c (x) =

\frac{259.2}{x} +

0.2 x^{2}

dollars

4-311 §4.7 source ↗

For the following exercises, answer by proof, counterexample, or explanation.

When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points?

Answer

The critical points can be the minima, maxima, or neither.

4-313 §4.7 source ↗

For the following exercises, answer by proof, counterexample, or explanation.

True or False. For every continuous nonlinear function, you can find the value $x$ that maximizes the function.

Answer

False;

y = x^{2}

has a minimum only

4-315 §4.7 source ↗

For the following exercises, set up and evaluate each optimization problem.

To carry a suitcase on an airplane, the length $+ width +$ height of the box must be less than or equal to $62 in .$ Assuming the base of the suitcase is square, show that the volume is $V =$ $h {(31 - (\frac{1}{2}) h)}^{2} .$ What height allows you to have the largest volume?

Answer

h = \frac{62}{3}

in.

4-317 §4.7 source ↗

For the following exercises, set up and evaluate each optimization problem.

Find the positive integer that minimizes the sum of the number and its reciprocal.

Answer

1

4-319 §4.7 source ↗

For the following exercises, consider the construction of a pen to enclose an area.

You have $400 ft$ of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?

Answer

100 ft by 100 ft

4-321 §4.7 source ↗

For the following exercises, consider the construction of a pen to enclose an area.

You need to construct a fence around an area of $1600 {ft}^{2} .$ What are the dimensions of the rectangular pen to minimize the amount of material needed?

Answer

40 ft by 40 ft

4-323 §4.7 source ↗

For the following exercises, consider the construction of a pen to enclose an area.

[T] You are moving into a new apartment and notice there is a corner where the hallway narrows from $8 ft to 6 ft .$ What is the length of the longest item that can be carried horizontally around the corner?

An upside L-shaped figure is drawn with the _ part being 6 wide and the | part being 8 wide. There is a line drawn from the _ part to the | part that touches the near corner of the shape to form a hypotenuse for a right triangle the other sides being the the rest of the _ and | parts. This line is marked L.

Answer

19.73 ft .

4-325 §4.7 source ↗

For the following exercises, consider the construction of a pen to enclose an area.

In the previous problem, assume the patient was nervous during the third measurement, so we only weight that value half as much as the others. What is the value that minimizes ${(x - 70)}^{2} +$ ${(x - 80)}^{2} +$ $\frac{1}{2} {(x - 120)}^{2} ?$

Answer

84 bpm

4-327 §4.7 source ↗

For the following problems, consider a lifeguard at a circular pool with diameter $40 m .$ He must reach someone who is drowning on the exact opposite side of the pool, at position $C .$ The lifeguard swims with a speed $v$ and runs around the pool at speed $w = 3 v .$

A circle is drawn with points A and C on a diameter. There is a point B drawn on the circle such that angle BAC form an acute angle θ.

Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, $θ .$

Answer

T (θ) =

\frac{40 θ}{3 v} +

\frac{40 cos θ}{v}

4-329 §4.7 source ↗

A truck uses gas as $g (v) = a v + \frac{b}{v},$ where $v$ represents the speed of the truck and $g$ represents the gallons of fuel per mile. Assuming $a$ and $b$ are positive, at what speed is fuel consumption minimized?

Answer

v = \sqrt{\frac{b}{a}}

4-331 §4.7 source ↗

For the following exercises, consider a limousine that gets $m (v) =$ $\frac{(120 - 2 v)}{5} mi/gal$ at speed $v,$ the chauffeur costs $$15/h,$ and gas is $$ 3.5 / gal .$

Find the cheapest driving speed.

Answer

approximately

34 mph

4-333 §4.7 source ↗

For the following exercises, consider a pizzeria that sell pizzas for a revenue of $R (x) = a x$ and costs $C (x) =$ $b +$ $c x +$ $d x^{2},$ where $x$ represents the number of pizzas $;$ $a >$ $c$ .

Assume that $R (x) = 10 x$ and $C (x) = 2 x + x^{2} .$ How many pizzas sold maximizes the profit?

Answer

4

4-335 §4.7 source ↗

For the following exercises, consider a wire $4 ft$ long cut into two pieces. One piece forms a circle with radius $r$ and the other forms a square of side $x .$

Choose $x$ to maximize the sum of their areas.

Answer

0

4-337 §4.7 source ↗

For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x + y = 10 .$ Maximize and minimize the quantities.

$x y$

Answer

Maximal:

x = 5, y = 5;

minimal:

x = 0, y = 10

and

y = 0, x = 10

4-339 §4.7 source ↗

For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x + y = 10 .$ Maximize and minimize the quantities.

$y - \frac{1}{x}$

Answer

Maximal:

x = 1, y = 9;

minimal: none

4-341 §4.7 source ↗

For the following exercises, draw the given optimization problem and solve.

Find the volume of the largest right circular cylinder that fits in a sphere of radius $1 .$

Answer

\frac{4 π}{3 \sqrt{3}}

4-343 §4.7 source ↗

For the following exercises, draw the given optimization problem and solve.

Find the area of the largest rectangle that fits into the triangle with sides $x = 0, y = 0$ and $\frac{x}{4} + \frac{y}{6} = 1 .$

Answer

6

4-345 §4.7 source ↗

For the following exercises, draw the given optimization problem and solve.

Find the dimensions of the closed cylinder volume $V = 16 π$ that has the least amount of surface area.

Answer

r = 2, h = 4

4-347 §4.7 source ↗

For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.

[T] Where is the line $y = 5 - 2 x$ closest to the origin?

Answer

(2, 1)

4-349 §4.7 source ↗

For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.

[T] Where is the parabola $y = x^{2}$ closest to point $(2, 0) ?$

Answer

(0.8351, 0.6974)

4-351 §4.7 source ↗

For the following exercises, write an equation that models each related-rates situation. Do not solve.

A window is composed of a semicircle placed on top of a rectangle. If you have $20 ft$ of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use $r$ to represent the radius of the semicircle.

A semicircular window is drawn with radius r.

Answer

\begin{matrix} A & = & 20 \frac{20}{π + 4} - 2 {\frac{20}{π + 4}}^{2} - \frac{1}{2} π {\frac{20}{π + 4}}^{2} \end{matrix}

4-353 §4.7 source ↗

For the following exercises, write an equation that models each related-rates situation. Do not solve.

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $$ 5 / {ft}^{2}$ and the material for the sides costs $$ 2 / {ft}^{2} .$ You need a box with volume $4 {ft}^{3} .$ Find the dimensions of the box that minimize cost. Use $x$ to represent the length of the side of the box.

Answer

C (x) =

5 x^{2} +

\frac{32}{x}

4-355 §4.7 source ↗

For the following exercises, write an equation that models each related-rates situation. Do not solve.

You are the manager of an apartment complex with $50$ units. When you set rent at $$ 800 / month,$ all apartments are rented. As you increase rent by $$ 25 / month,$ one fewer apartment is rented. Maintenance costs run $$ 50 / month$ for each occupied unit. What is the rent that maximizes the total amount of profit?

Answer

P (x) =

(50 - x) (800 + 25 x - 50)

Checkpoint 4.37 §4.8 source ↗

Evaluate

lim_{x \to 0} \frac{x}{tan x} .

Answer

1

Checkpoint 4.38 §4.8 source ↗

Evaluate

lim_{x \to \infty} \frac{ln x}{5 x} .

Answer

0

Checkpoint 4.39 §4.8 source ↗

Explain why we cannot apply L’Hôpital’s rule to evaluate

lim_{x \to 0^{+}} \frac{cos x}{x} .

Evaluate

lim_{x \to 0^{+}} \frac{cos x}{x}

by other means.

Answer

lim_{x \to 0^{+}} cos x =

1 .

Therefore, we cannot apply L’Hôpital’s rule. The limit of the quotient is

\infty

Checkpoint 4.40 §4.8 source ↗

Evaluate

lim_{x \to 0} x cot x .

Answer

1

Checkpoint 4.41 §4.8 source ↗

Evaluate

lim_{x \to 0^{+}} (\frac{1}{x} - \frac{1}{sin x}) .

Answer

0

Checkpoint 4.42 §4.8 source ↗

Evaluate

lim_{x \to \infty} x^{1 / ln (x)} .

Answer

e

Checkpoint 4.43 §4.8 source ↗

Evaluate

lim_{x \to 0^{+}} x^{x} .

Answer

1

Checkpoint 4.44 §4.8 source ↗

Compare the growth rates of

x^{100}

and

2^{x} .

Answer

The function

2^{x}

grows faster than

x^{100} .

4-357 §4.8 source ↗

For the following exercises, evaluate the limit.

Evaluate the limit $lim_{x \to \infty} \frac{e^{x}}{x^{k}} .$

Answer

\infty

4-359 §4.8 source ↗

For the following exercises, evaluate the limit.

Evaluate the limit $lim_{x \to a} \frac{x - a}{x^{2} - a^{2}},$ $a \neq 0$ .

Answer

\frac{1}{2 a}

4-361 §4.8 source ↗

For the following exercises, evaluate the limit.

Evaluate the limit $lim_{x \to a} \frac{x - a}{x^{n} - a^{n}},$ $a \neq 0$ .

Answer

\frac{1}{n a^{n - 1}}

4-363 §4.8 source ↗

For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.

$lim_{x \to \infty} x^{1 / x}$

Answer

Cannot apply directly; use logarithms

4-365 §4.8 source ↗

$lim_{x \to 0} \frac{x^{2}}{1 / x}$

Answer

Cannot apply directly; rewrite as

lim_{x \to 0} x^{3}

4-367 §4.8 source ↗