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Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

Chapter 4

Calculating the Derivative - all with Video Answers

Educators

+ 3 more educators

Section 1

Techniques for Finding Derivatives

01:47

Problem 1

Find the derivative of each function defined as follows.
$$y=x^{3}-11 x^{2}+7 x+9$$

GD
Gilbert Deleon
Numerade Educator
03:48

Problem 2

Find the derivative of each function defined as follows.
$$y=8 x^{3}-5 x^{2}-\frac{x}{12}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:32

Problem 3

Find the derivative of each function defined as follows.
$$y=x^{3}-\frac{x^{2}}{16}+4 x+9$$

Vishal Parmar
Vishal Parmar
Numerade Educator
03:53

Problem 4

Find the derivative of each function defined as follows.
$$y=5 x^{4}+9 x^{3}+12 x^{2}-7 x$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:34

Problem 5

Find the derivative of each function defined as follows.
$$f(x)=6 x^{3.5}-10 x^{0.5}$$

Phumlani Ngcobo
Phumlani Ngcobo
Numerade Educator
03:20

Problem 6

Find the derivative of each function defined as follows.
$$f(x)=-2 x^{1.5}+12 x^{0.5}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
03:05

Problem 7

Find the derivative of each function defined as follows.
$$y=8 \sqrt{x}+6 x^{3 / 4}$$

GD
Gilbert Deleon
Numerade Educator
03:49

Problem 8

Find the derivative of each function defined as follows.
$$y=-100 \sqrt{x}-11 x^{2 / 3}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:50

Problem 9

Find the derivative of each function defined as follows.
$$y=6 x^{2}-10 x-3 x^{-2}$$

GD
Gilbert Deleon
Numerade Educator
03:45

Problem 10

Find the derivative of each function defined as follows.
$$y=5 x^{-5}-6 x^{-2}+13 x^{-1}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:27

Problem 11

Find the derivative of each function defined as follows.
$$y=\frac{2}{x^{7}}-\frac{8}{x}$$

GD
Gilbert Deleon
Numerade Educator
02:44

Problem 12

Find the derivative of each function defined as follows.
$$y=\frac{8}{x^{2}}-\frac{3}{x}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:58

Problem 13

Find the derivative of each function defined as follows.
$$y=\frac{8}{x^{5}}-\frac{8}{x^{4}}+\frac{6}{x}+\sqrt{7}$$

GD
Gilbert Deleon
Numerade Educator
03:22

Problem 14

Find the derivative of each function defined as follows.
$$y=\frac{6}{x^{4}}-\frac{2}{x^{3}}+\frac{5}{x}+\sqrt{3}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:41

Problem 15

Find the derivative of each function defined as follows.
$$p(x)=-10 x^{-1 / 2}+8 x^{-3 / 2}$$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:09

Problem 16

Find the derivative of each function defined as follows.
$$h(x)=x^{-1 / 2}-14 x^{-3 / 2}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:27

Problem 17

Find the derivative of each function defined as follows.
$$y=\frac{6}{\sqrt[4]{x}}$$

GD
Gilbert Deleon
Numerade Educator
01:37

Problem 18

Find the derivative of each function defined as follows.
$$y=\frac{-2}{\sqrt[3]{x}}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:46

Problem 19

Find the derivative of each function defined as follows.
$$f(x)=\frac{4 x^{3}+5}{x}$$

GD
Gilbert Deleon
Numerade Educator
02:06

Problem 20

Find the derivative of each function defined as follows.
$$g(x)=\frac{x^{3}-4 x}{\sqrt{x}}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:16

Problem 21

Find the derivative of each function defined as follows.
$$g(x)=\left(8 x^{2}-4 x\right)^{2}$$

GD
Gilbert Deleon
Numerade Educator
03:05

Problem 22

Find the derivative of each function defined as follows.
$$h(x)=\left(x^{2}-1\right)^{3}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:17

Problem 23

Which of the following describes the derivative function $f^{\prime}(x)$
of a quadratic function $f(x) ?$
$\begin{array}{ll}{\text { (a) Quadratic }} & {\text { (b) Linear }} \\ {\text { (c) Constant }} & {\text { (d) Cubic (third degree) }}\end{array}$

Melissa Munoz
Melissa Munoz
Numerade Educator
03:29

Problem 24

Which of the following describes the derivative function $f^{\prime}(x)$ of a cubic (third degree) function $f(x) ?$
(a) Quadratic (b) Linear
(d) Cubic (c) Constant

Smita Praharaj
Smita Praharaj
Numerade Educator
01:58

Problem 25

Explain the relationship between the slope and the derivative of $f(x)$ at $x=a$ .

Vishal Parmar
Vishal Parmar
Numerade Educator
03:31

Problem 26

Which of the following do not equal $\frac{d}{d x}\left(4 x^{3}-6 x^{-2}\right) ?$

(a) $\frac{12 x^{2}+12}{x^{3}}$ (b) $\frac{12 x^{5}+12}{x^{3}}$
(b)$12 x^{2}+\frac{12}{x^{3}}$ (d) $12 x^{3}+12 x^{-3}$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:47

Problem 27

Find each derivative.
$$D_{x}\left[9 x^{-1 / 2}+\frac{2}{x^{3 / 2}}\right]$$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:53

Problem 28

Find each derivative..
$$D_{x}\left[\frac{8}{\sqrt[4]{x}}-\frac{3}{\sqrt{x^{3}}}\right]$$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:39

Problem 29

Find each derivative.
$$f^{\prime}(-3)$ if $f(x)=\frac{x^{4}}{6}-9 x$$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:37

Problem 30

Find each derivative.
$$f^{\prime}(3)$ if $f(x)=\frac{x^{3}}{9}-7 x^{2}$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:04

Problem 31

In Exercises $31-34,$ find the slope of the tangent line to the graph of the given function at the given value of $x .$ Find the equation of the tangent line in Exercises 31 and $32 .$

$y=x^{4}-2 x^{2}+1 ; \quad x=-2$

Vishal Parmar
Vishal Parmar
Numerade Educator
04:10

Problem 32

In Exercises $31-34$ , find the slope of the tangent line to the graph of the given function at the given value of $x .$ Find the equation of the tangent line in Exercises 31 and $32 .$
$$y=x^{4}-18 x^{2}+81$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:54

Problem 33

In Exercises $31-34$ , find the slope of the tangent line to the graph of the given function at the given value of $x .$ Find the equation of the tangent line in Exercises 31 and $32 .$
$$y=-5 x^{1 / 2}+x^{3 / 2} ; \quad x=25$$

William Semus
William Semus
Numerade Educator
02:12

Problem 34

In Exercises $31-34$ , find the slope of the tangent line to the graph of the given function at the given value of $x .$ Find the equation of the tangent line in Exercises 31 and $32 .$
$$y=x^{4}-13 x^{2}+36$$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:55

Problem 35

Find all points on the graph of $f(x)=9 x^{2}-8 x+4$ where the slope of the tangent line is $0 .$

Vishal Parmar
Vishal Parmar
Numerade Educator
03:37

Problem 36

Find all points on the graph of $f(x)=x^{3}+9 x^{2}+19 x-10$ where the slope of the tangent line is $-5 .$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:29

Problem 37

In Exercises $37-40,$ for each function find all values of $x$ where the tangent line is horizontal.

$f(x)=2 x^{3}+39 x^{2}+240 x+6$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:27

Problem 38

In Exercises $37-40$ , for each function find all values of $x$ where the tangent line is horizontal.
$$f(x)=2 x^{3}+45 x^{2}+324 x+5$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:48

Problem 39

In Exercises $37-40$ , for each function find all values of $x$ where the tangent line is horizontal.
$$f(x)=x^{3}-4 x^{2}-7 x+8$$

Vishal Parmar
Vishal Parmar
Numerade Educator
04:03

Problem 40

In Exercises $37-40$ , for each function find all values of $x$ where the tangent line is horizontal.
$$f(x)=x^{3}-5 x^{2}+6 x+3$$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:30

Problem 41

At what points on the graph of $f(x)=6 x^{2}+4 x-9$ is the slope of the tangent line $-2 ?$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:45

Problem 42

At what points on the graph of $f(x)=x^{3}+12 x^{2}+55 x+14$ is the slope of the tangent line 7$?$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:55

Problem 43

At what points on the graph of $f(x)=x^{3}+6 x^{2}+17 x+15$ is the slope of the tangent line 5$?$

Vishal Parmar
Vishal Parmar
Numerade Educator
01:46

Problem 44

If g^{\prime}(5)=12$ and $h^{\prime}(5)=-3,$ find $f^{\prime}(5) for
$f(x)=3 g(x)-2 h(x)+3$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:07

Problem 45

$g^{\prime}(2)=7 and h^{\prime}(2)=14,find f^{\prime}(2)$
$f(x)=\frac{1}{2} g(x)+\frac{1}{4} h(x)$

Vishal Parmar
Vishal Parmar
Numerade Educator
01:24

Problem 46

Use the information given in the figure to find the following values.

$\begin{array}{ll}{\text { (a) } f(1)} & {\text { (b) } f^{\prime}(1)} \\ {\text { (c) The domain of } f} & {\text { (d) The range of } f}\end{array}$.

Smita Praharaj
Smita Praharaj
Numerade Educator
01:28

Problem 47

Explain the concept of marginal cost. How does it relate to cost? How is it found?

Vishal Parmar
Vishal Parmar
Numerade Educator
01:20

Problem 48

In Exercises $43-46$ of Section $2.2,$ the effect of $a$ when graphing $y=a f(x)$ was discussed. Now describe how this relates to the fact that $D_{x}[a f(x)]=a f^{\prime}(x)$

Smita Praharaj
Smita Praharaj
Numerade Educator
00:56

Problem 49

Show that, for any constant $k$

$\frac{d}{d x}\left[\frac{f(x)}{k}\right]=\frac{f^{\prime}(x)}{k}$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:19

Problem 50

Use the differentiation feature on your graphing calculator to solve the problems (to 2 decimal places) below, where $f(x)$ is defined as follows:

$f(x)=1.25 x^{3}+0.01 x^{2}-2.9 x+1$

(a) Find $f^{\prime}(4)$ .
(b) Find all values of $x$ where $f^{\prime}(x)=0$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:59

Problem 51

Revenue Assume that a demand equation is given by $q=5000-100 p .$ Find the marginal revenue for the following production levels (values of $q )$ .

(a) 1000 units (b) 2500 units
(c) 3000 units

Vishal Parmar
Vishal Parmar
Numerade Educator
04:07

Problem 52

Profit Suppose that for the situation in Exercise 51 the cost of producing $q$ units is given by $C(q)=3000-20 q+0.03 q^{2} .$ Find the marginal profit for the following production levels

(a) 500 units (b) 815 units
(c) 1000 units

Smita Praharaj
Smita Praharaj
Numerade Educator
01:54

Problem 53

Revenue If the price in dollars of a stereo system is given by

$p(q)=\frac{1000}{q^{2}}+1000$

where $q$ represents the demand for the product, find the marginal revenue when the demand is $10 .$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:50

Problem 54

Profit Suppose that for the situation in Exercise 53 the cost in dollars of producing $q$ stereo systems is given by $C(q)=0.2 q^{2}+6 q+50 .$ Find the marginal profit when the demand is $10 .$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:56

Problem 55

Marginal Product of Labor The output y of a manufacturing process is a function of the size of the labor force n using the function

$y=k \sqrt{n}$

The marginal product of labor, defined as $d y / d n,$ measures the rate that output increases with the size of the labor force, and is a measure of labor productivity.(a) Show that

$\frac{d y}{d n}=\frac{k}{2 \sqrt{n}}$

(b) How can you tell from your answer to part (a) that as the size of the labor force increases, the marginal product of labor gets smaller? This is a phenomenon known as the law of diminishing returns, discussed more in the next chapter.

Vishal Parmar
Vishal Parmar
Numerade Educator
04:04

Problem 56

Profit An analyst has found that a company's costs and revenues in dollars for a product are given by

$C(x)=\frac{x}{2}$ and $R(x)=2 x-\frac{x^{2}}{5000^{\circ}}$

respectively, where $x$ is the number of items produced.
(a) Find the marginal cost function.
(b) Find the marginal revenue function.
(c) Using the fact that profit is the difference between revenue and costs, find the marginal profit function.
(d) What value of $x$ makes the marginal profit equal 0 ?
(e) Find the profit when the marginal profit is 0 . (As we shall see in the next chapter, this process is used to find maximum profit.)

Smita Praharaj
Smita Praharaj
Numerade Educator
01:12

Problem 57

Postal Rates U.S. postal rates have steadily increased since $1932 .$ Using data depicted in the table in the next column for the years $1932-2014,$ the cost in cents to mail a single letter can be modeled using a quadratic formula as follows:

$C(t)=0.007714 t^{2}-0.03969 t+0.726$

where $t$ is the number of years since $1932 .$ Source: U.S. Postal
Service.
(a) Find the predicted cost of mailing a letter in 1982 and 2002 and compare these estimates with the actual rates.
(b) Find the rate of change of the postage cost for the years 1982 and 2002 and interpret your results.
(c) Using the regression feature on a graphing calculator, find a cubic function that models these data, letting $t=0$ correspond to the year $1932 .$ Then use your answer to find the rate of change of the postage cost for the years 1982 and 2002 .
(d) Discuss whether the quadratic or cubic function best describes the data. Do the answers from part (b) or from part (c) best describe the rate that postage was going up in the years 1982 and 2002$?$
(e) Explore other functions that could be used to model the data, using the various regression features on a graphing calculator, and discuss to what extent any of them are useful descriptions of the data.

Carson Merrill
Carson Merrill
Numerade Educator
06:36

Problem 58

Money The total amount of money in circulation for the years $1955-2014$ can be closely approximated by
$M(t)=0.006934 t^{3}-0.07837 t^{2}+1.480 t+27.78$
where $t$ represents the number of years since 1955 and $M(t)$ is in billions of dollars. Find the derivative of $M(t)$ and $M(t)$ is find the rate of change of money in circulation in the following
years. Source: U.S. Treasury.
$\begin{array}{ll}{\text { (a) } 1960} & {\text { (b) } 1980} \\ {\text { (c) } 2000} & {\text { (d) } 2010}\end{array}$
(e) What do your answers to parts (a)-(d) tell you about the amount of money in circulation in those years?

Smita Praharaj
Smita Praharaj
Numerade Educator
02:06

Problem 59

Cancer Insulation workers who were exposed to asbestos and employed before 1960 experienced an increased likelihood of lung cancer. If a group of insulation workers has a cumulative
total of 100,000 years of work experience with their first date of employment $t$ years ago, then the number of lung cancer cases occurring within the group can be modeled using the function

$N(t)=0.00437 t^{3.2}$

Find the rate of growth of the number of workers with lung cancer in a group as described by the following first dates of employment. Source: Observation and Inference: An Introduction to the Methods of Epidemiology.

(a) 5 years ago (b)10 years ago

Vishal Parmar
Vishal Parmar
Numerade Educator
View

Problem 60

Blood Sugar Level Insulin affects the glucose, or blood sugar, level of some diabetics according to the function

$G(x)=-0.2 x^{2}+450$,

where $G(x)$ is the blood sugar level 1 hour after $x$ units of insulin are injected. (This mathematical model is only approximate, and it is valid only for values of $x$ less than about $40 .$ Find the
blood sugar level after the following numbers of units of insulin are iniected.

(a) 0 (b) 25

Find the rate of change of blood sugar level after injection of the following numbers of units of insulin.
$\begin{array}{ll}{\text {

(c) 10 (d) 25

Danielle Fairburn
Danielle Fairburn
Numerade Educator
02:42

Problem 61

Bighorn Sheep The cumulative horn volume for certain types of bighorn rams, found in the Rocky Mountains, can be described by the quadratic function

$V(t)=-2159+1313 t-60.82 t^{2}$

where $V(t)$ is the horn volume $\left($ in $\mathrm{cm}^{3}\right)$ and $t$ is the year of growth, $2 \leq t \leq 9 .$ Source: Conservation Biology.
(a) Find the horn volume for a 3 -year-old ram.
(b) Find the rate at which the hom volume of a 3 -year-old ram is changing.

William Semus
William Semus
Numerade Educator
04:25

Problem 62

Brain Mass The brain mass of a human fetus during the last rimester can be accurately estimated from the circumference of the head by

$m(c)=\frac{c^{3}}{100}-\frac{1500}{c}$

where $m(c)$ is the mass of the brain (in grams) and $c$ is the circumference (in centimeters) of the head. Source: Early Human Development.
(a) Estimate the brain mass of a fetus that has a head circumference of 30 $\mathrm{cm} .$
(b) Find the rate of change of the brain mass for a fetus that has a head circumference of 30 $\mathrm{cm}$ and interpret your results.

Smita Praharaj
Smita Praharaj
Numerade Educator
01:34

Problem 63

Velocity of Marine Organism The typical velocity (in centimeters per second) of a marine organism of length $I$ (in centimeters) is given by $v=2.691^{186} .$ Find the rate of change of the velocity with respect to the length of the organism. Source: Mathematical Topics in Population Biology Morphogenesis and Neurosciences.

Vishal Parmar
Vishal Parmar
Numerade Educator
02:00

Problem 64

Heart The left ventricular length (viewed from the front of the heart) of a human fetus that is at least 18 weeks old can be estimated by

$l(x)=-2.318+0.2356 x-0.002674 x^{2}$

where $l(x)$ is the ventricular length (in centimeters) and $x$ is the age (in weeks) of the fetus. Source: American Journal of Cardiology.

(a) Determine a meaningful domain for this function.
(b) Find $I^{\prime}(x)$ .
(c) Find $I^{\prime}(25)$ .

Smita Praharaj
Smita Praharaj
Numerade Educator
03:19

Problem 65

Track and Field In 1906 Kennelly developed a simple formula for predicting an upper limit on the fastest time that humans could ever run distances from 100 yards to 10 miles. His formula is given by

$t=0.0588 s^{1.125}$

where $s$ is the distance in meters and $t$ is the time to run that distance in seconds. Source: Proceedings of the American Academy of Arts and Sciences.

(a) Find Kennelly's estimate for the fastest mile. (Hint: 1 mile $\approx 1609$ meters.)
(b) Find $d t / d s$ when $s=100$ and interpret your answer.
(c) Compare this and other estimates to the current world records. Have these estimates been surpassed?

Vishal Parmar
Vishal Parmar
Numerade Educator
03:48

Problem 66

Human Cough To increase the velocity of the air flowing through the trachea when a human coughs, the body contracts the windpipe, producing a more effective cough. Tuchinsky formulated that the velocity of air that is flowing through the trachea during a cough is

$V=C\left(R_{0}-R\right) R^{2}$

where $C$ is a constant based on individual body characteristics, $R_{0}$ is the radius of the windpipe before the cough, and $R$ is the radius of the windpipe during the cough. It can be shown that the maximum velocity of the cough occurs when $d V / d R=0$ . Find the value of $R$ that maximizes the velocity.* Source: COMAP, Inc.

Smita Praharaj
Smita Praharaj
Numerade Educator
07:01

Problem 67

Body Mass Index The body mass index (BMI) is a number that can be calculated for any individual as follows: Multiply a person's weight by 703 and divide by the person's height squared. That is,

$B M I=\frac{703 w}{h^{2}}$

where $w$ is in pounds and $h$ is in inches. The National Heart, Lung, and Blood Institute uses the BMI to determine whether
a person is "overweight" $(25 \leq B M I<30)$ or "obese"

$(B M I \geq 30) .$ Source: The National Institutes of Health.
(a) Calculate the BMI for Lebron James, basketball player for the Cleveland Cavaliers, who is 250 Ib. and $6^{\prime} 8^{\prime \prime}$ tall.
(b) How much weight would Lebron James have to lose until he reaches a BMI of of 24.9 and is no
longer "overweight"? Comment on whether BMI cutoffs are appropriate for athletes with considerable muscle mass.
(c) For a $125-$ lb female, what is the rate of change of BMI with respect to height? (Hint: Take the derivative of the function: $f(h)=703(125) / h^{2} . )$
(d) Calculate and interpret the meaning of $f^{\prime}(65)$ .
(e) Use the TABLE feature on your graphing calculator to construct a table for BMI for various weights and heights.
(f) Using the fact that 1 in. $=0.0254 \mathrm{m}$ and $1 \mathrm{lb}=0.4536 \mathrm{kg}$ ,transform this formula to handle metric units.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
01:38

Problem 68

Velocity We saw in the previous chapter that if a function $s(t)$ gives the position of an object at time $t,$ the derivative gives the velocity, that is, $v(t)=s^{\prime}(t) .$ For each position function in Exercises $68-71,$ find (a) $v(t)$ and (b) the velocity when $t=0, t=5,$ and $t=10$ .
$s(t)=11 t^{2}+4 t+2$

Smita Praharaj
Smita Praharaj
Numerade Educator
01:49

Problem 69

Velocity We saw in the previous chapter that if a function $s(t)$ gives the position of an object at time $t,$ the derivative gives the velocity, that is, $v(t)=s^{\prime}(t) .$ For each position function in Exercises $68-71,$ find (a) $v(t)$ and (b) the velocity when $t=0, t=5,$ and $t=10$ .
$$s(t)=18 t^{2}-13 t+8$$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:30

Problem 70

Velocity We saw in the previous chapter that if a function $s(t)$ gives the position of an object at time $t,$ the derivative gives the velocity, that is, $v(t)=s^{\prime}(t) .$ For each position function in Exercises $68-71,$ find (a) $v(t)$ and (b) the velocity when $t=0, t=5,$ and $t=10$ .
$$s(t)=4 t^{3}+8 t^{2}+t$$

Smita Praharaj
Smita Praharaj
Numerade Educator
02:03

Problem 71

Velocity We saw in the previous chapter that if a function $s(t)$ gives the position of an object at time $t,$ the derivative gives the velocity, that is, $v(t)=s^{\prime}(t) .$ For each position function in Exercises $68-71,$ find (a) $v(t)$ and (b) the velocity when $t=0, t=5,$ and $t=10$ .
$$s(t)=-3 t^{3}+4 t^{2}-10 t+5$$

Vishal Parmar
Vishal Parmar
Numerade Educator
03:24

Problem 72

Velocity If a rock is dropped from a 200 ft building, its posi-
tion (in feet above the ground) is given by $s(t)=-16 t^{2}+200$ ,
where $t$ is the time in seconds since it was dropped.
(a) What is the velocity 1 second after being dropped? 2 seconds
after being dropped?
(b) When will it hit the ground?
(c) What is its velocity upon impact?

Smita Praharaj
Smita Praharaj
Numerade Educator
02:35

Problem 73

Velocity A ball is thrown vertically upward from the ground at a velocity of 64 ft per second. Its distance from the ground at $t$ seconds is given by $s(t)=-16 t^{2}+64 t$
(a) How fast is the ball moving 2 seconds after being thrown? 3 seconds after being thrown?
(b) How long after the ball is thrown does it reach its maximum height?
(c) How high will it go?

Vishal Parmar
Vishal Parmar
Numerade Educator
02:51

Problem 74

Dead Sea Researchers who have been studying the alarming rate in which the level of the Dead Sea has been dropping have shown that the density $d(x)$ (in g per cm's) of the Dead Sea brine during evaporation can be estimated by the function

$d(x)=1.66-0.90 x+0.47 x^{2}$

where $x$ is the fraction of the remaining brine, $0 \leq x \leq 1$ Source: Geology.
(a) Estimate the density of the brine when 50$\%$ of the brine remains.
(b) Find and interpret the instantaneous rate of change of the density when 50$\%$ of the brine remains.

Smita Praharaj
Smita Praharaj
Numerade Educator
01:53

Problem 75

Echoes Acoustical experts have found that clapping one's hands near the staircases of the Pyramid of Kukulkan at Chichen Itza results in an echo that sounds like a chirp. The initial frequency of the chirp $f$ depends on the tread length of the stairs $T$ according to the formula

$f=\frac{c}{2 T}$

where $c$ is the speed of sound, 340 $\mathrm{m} / \mathrm{sec} .$ Find the initial frequency and the rate of change of the initial frequency with respect to tread length when the tread is 0.5 $\mathrm{m} .$ Source: Acoustical Society of America.

Vishal Parmar
Vishal Parmar
Numerade Educator
04:49

Problem 76

AP Examinations The probability (as a percent) of scoring 3 or more on the Calculus AB Advanced Placement Examination can be very closely predicted as a function of a student's PSATI NMSOT Score $x$ by the function

$P(x)=-0.00209 x^{3}+0.3387 x^{2}-15.15 x+208.6$

Source: The College Board.
(a) Find the probability of scoring 3 or more for each of the following PSATNMSQT scores, as well as the rate that this probability is increasing with respect to the PSATI NMSQT score.
(i) 45 (ii)75

(b) Based on your results from part (a), what recommendations would you make about who should take the Calculus AB examination?

Smita Praharaj
Smita Praharaj
Numerade Educator
03:01

Problem 77

Dog's Human Age From the data printed in the following table from the Minneapolis Star Tribune a dog's age when compared to a human's age can be modeled using either a linear formula or a quadratic formula as follows:

$y_{1}=4.13 x+14.63$
$y_{2}=-0.033 x^{2}+4.647 x+13.347$

where $y_{1}$ and $y_{2}$ represent a dog's human age for each formula and $x$ represents a dog's actual age. Source: The Mathematics Teacher.

(a) Find $y_{1}$ and $y_{2}$ when $x=5 .$
(b) Find $d y_{1} / d x$ and $d y_{2} / d x$ when $x=5$ and interpret your answers.
(c) If the first two points are eliminated from the table, find the equation of a line that perfectly fits the reduced set of data. Interpret your findings.
(d) Of the three formulas, which do you prefer?

Carson Merrill
Carson Merrill
Numerade Educator