SOLVED:Limits and Derivatives | Calculus: Early Transcendentals

Section 7

Derivatives and Rates of Change

Problem 1

A curve has equation $ y = f(x) $.
(a) Write an expression for the slope of the secant line through the points $ P(3, f(3)) $ and $ Q(x, f(x)) $.
(b) Write an expression for the slope of the tangent line at $ P $.

Ma. Theresa A.

Numerade Educator

View

Problem 2

Graph the curve $ y = e^x $ in the viewing rectangles $ [-1, 1] $ by $ [0, 2] $, $ [-0.5, 0.5] $ by $ [0.5, 1.5] $, and $ [-0.1, 0.1] $ by $ [0.9, 1.1] $. What do you notice about the curve as you zoom in toward the point $ (0, 1) $?

Leon D.

Numerade Educator

View

Problem 3

(a) Find the slope of the tangent line to the parabola $ y = 4x - x^2 $ at the point $ (1, 3) $
(i) using Definition 1 (ii) using Equation 2

(b) Find an equation of the tangent line in part (a).

(c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point
$ (1, 3) $ until the parabola and the tangent line are indistinguishable.

David M.

Numerade Educator

View

Problem 4

(a) Find the slope of the tangent line to the curve $ y = x - x^3 $ at the point $ (1, 0) $
(i) using Definition 1 (ii) using Equation 2

(b) Find an equation of the tangent line in part (a).

(c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at $ (1, 0) $ until the curve and the line appear to coincide.

Leon D.

Numerade Educator

View

Problem 5

Find an equation of the tangent line to the curve at the given point.

$ y = 4x - 3x^2 $, $ (2, -4) $

David M.

Numerade Educator

View

Problem 6

Find an equation of the tangent line to the curve at the given point.

$ y = x^3 - 3x + 1 $, $ (2, 3) $

Anthony H.

Numerade Educator

View

Problem 7

Find an equation of the tangent line to the curve at the given point.

$ y = \sqrt{x} $, $ (1, 1) $

Leon D.

Numerade Educator

View

Problem 8

Find an equation of the tangent line to the curve at the given point.

$ y = \dfrac{2x + 1}{x + 2} $, $ (1, 1) $

Ma. Theresa A.

Numerade Educator

View

Problem 9

(a) Find the slope of the tangent to the curve $ y = 3 + 4x^2 - 2x^3 $ at the point where $ x = a $.

(b) Find equations of the tangent lines at the points $ (1, 5) $ and $ (2, 3) $.

Leon D.

Numerade Educator

View

Problem 10

(a) Find the slope of the tangent to the curve $ y = 1/\sqrt{x} $ at the point where $ x = a $.

(b) Find equations of the tangent lines at the points $ (1, 1) $ and $ (4, \frac{1}{2}) $.

Babita K.

Numerade Educator

05:13

Problem 11

(a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still?

(b) Draw a graph of the velocity function.

Le Anh Thu N.

Numerade Educator

View

Problem 12

Shown are graphs of position functions of two runners, $ A $ and $ B $, who run a 100-meter race and finish in a tie.

(a) Describe and compare how the runners run the race.

(b) At what time is the distance between the runners the greatest?

David M.

Numerade Educator

View

Problem 13

If a ball is thrown into the air with a velocity of $ 40 ft/s $, its height (in feet) after $ t $ seconds is given by $ y = 40t - 16t^2 $. Find the velocity when $ t = 2 $.

Anthony H.

Numerade Educator

View

Problem 14

If a rock is thrown upward on the planet Mars with a velocity of $ 10 m/s $, its height (in meters) after $ t $ seconds is given by $ H = 10t - 1.86t^2 $.

(a) Find the velocity of the rock after one second.
(b) Find the velocity of the rock when $ t = a $.
(c) When will the rock hit the surface?
(d) With what velocity will the rock hit the surface?

Dominique Jan T.

Numerade Educator

View

Problem 15

The displacement (in meters) of a particle moving in a straight line is given by the equation of motion
$ s = 1/t^2 $, where $ t $ is measured in seconds. Find the velocity of the particle at times $ t = a $, $ t = 1 $, $ t = 2 $, and $ t = 3 $.

Ma. Theresa A.

Numerade Educator

View

Problem 16

The displacement (in feet) of a particle moving in a straight line is given by $ s = \frac{1}{2}t^2 - 6t + 23 $, where $ t $ is measured in seconds.

(a) Find the average velocity over each time interval:
(i) $ [4, 8] $ (ii) $ [6, 8] $
(iii) $ [8, 10] $ (iv) $ [8, 12] $

(b) Find the instantaneous velocity when $ t = 8 $.

(c) Draw the graph of $ s $ as a function of $ t $ and draw the secant lines whose slopes are the average velocities in part (a). Then draw the tangent line whose slope is the instantaneous velocity in part (b).

Michael C.

Numerade Educator

07:20

Problem 17

For the function $ g $ whose graph is given, arrange the following numbers in increasing order and explain your reasoning:

$$ 0 \hspace{5mm} g'(-2) \hspace{5mm} g'(0) \hspace{5mm} g'(2) \hspace{5mm} g'(4) $$

Korrina M.

Numerade Educator

06:18

Problem 18

The graph of a function $ f $ is shown.

(a) Find the average rate of change of $ f $ on the interval $ [20, 60] $.
(b) Identify an interval on which the average rate of change of $ f $ is 0.
(c) Which interval gives a larger average rate of change, $ [40, 60] $ or $ [40, 70] $?
(d) Compute $ \dfrac{f(40) - f(10)}{40 - 10} $; what does this value represent geometrically?

Heather B.

Numerade Educator

View

Problem 19

For the function $ f $ graphed in Exercise 18:

(a) Estimate the value of $ f'(50) $.
(b) Is $ f'(10) > f'(30) $?
(c) Is $ f'(60) > \dfrac{f(80) - f(40)}{80 - 40} $? Explain.

Carolyn B.

Numerade Educator

View

Problem 20

Find an equation of the tangent line to the graph of $ y = g(x) $ at $ x = 5 $ if $ g(5) = -3 $ and $ g'(5) = 4 $.

Leon D.

Numerade Educator

View

Problem 21

If an equation of the tangent line to the curve $ y = f(x) $ at the point where $ a = 2 $ is $ y = 4x - 5 $, find $ f(2) $ and $ f'(2) $.

Leon D.

Numerade Educator

View

Problem 22

If the tangent line to $ y = f(x) $ at $ (4, 3) $ passes through the point $ (0, 2) $, find $ f(4) $ and $ f'(4) $.

Babita K.

Numerade Educator

View

Problem 23

Sketch the graph of a function $ f $ for which $ f(0) = 0 $, $ f'(0) = 3 $, $ f'(1) = 0 $, and $ f'(2) = -1 $.

Linda H.

Numerade Educator

View

Problem 24

Sketch the graph of a function $ g $ for which
$ g(0) = g(2) = g(4) = 0 $, $ g'(1) = g'(3) = 0 $, $ g'(0) = g'(4) = 1 $, $ g'(2) = -1 $, $ \displaystyle \lim_{x \to \infty} g(x) = \infty $, and $ \displaystyle \lim_{x \to -\infty} g(x) = -\infty $.

Anjali K.

Numerade Educator

View

Problem 25

Sketch the graph of a function $ g $ that is continuous on its domain $ (-5, 5) $ and where $ g(0) = 1 $, $ g'(0) = 1 $, $ g'(-2) = 0 $, $ \displaystyle \lim_{x \to -5^+} g(x) = \infty $, and $ \displaystyle \lim_{x \to 5^-} g(x) = 3 $.

Linda H.

Numerade Educator

View

Problem 26

Sketch the graph of a function $ f $ where the domain is $ (-2, 2) $, $ f'(0) = -2 $, $ \displaystyle \lim_{x \to 2^-} f(x) = \infty $, $ f $ is continuous at all numbers in its domain except $ \pm 1 $, and $ f $ is odd.

Anjali K.

Numerade Educator

View

Problem 27

If $ f(x) = 3x^2 - x^3 $, find $ f'(1) $ and use it to find an equation of the tangent line to the curve
$ y = 3x^2 - x^3 $ at the point $ (1, 2) $.

Ashley B.

Numerade Educator

View

Problem 28

If $ g(x) = x^4 - 2 $, find $ g'(1) $ and use it to find an equation of the tangent line to the curve
$ y = x^4 - 2 $ at the point $ (1, -1) $.

Leon D.

Numerade Educator

04:57

Problem 29

(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Daniel J.

Numerade Educator

View

Problem 30

(a) If $ G(x) = 4x^2 - x^3 $, find $ G'(a) $ and use it to find equations of the tangent lines to the curve
$ y = 4x^2 - x^3 $ at the points $ (2, 8) $ and $ (3, 9) $.

(b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

Leon D.

Numerade Educator

View

Problem 31

Find $ f'(a) $.

$ f(x) = 3x^2 - 4x + 1 $

Leon D.

Numerade Educator

View

Problem 32

Find $ f'(a) $.

$ f(t) = 2t^3 + t $

David M.

Numerade Educator

View

Problem 33

Find $ f'(a) $.

$ f(t) = \dfrac{2t + 1}{t + 3} $

Anthony H.

Numerade Educator

View

Problem 34

Find $ f'(a) $.

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

David M.

Numerade Educator

View

Problem 35

Find $ f'(a) $.

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

Leon D.

Numerade Educator

View

Problem 36

Find $ f'(a) $.

$ f(x) = \dfrac{4}{\sqrt{1 - x}} $

Leon D.

Numerade Educator

View

Problem 37

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h} $

Leon D.

Numerade Educator

View

Problem 38

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{h \to 0} \frac{e^{-2 + h} - e^{-2}}{h} $

Leon D.

Numerade Educator

View

Problem 39

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{x \to 2} \frac{x^6 - 64}{x - 2} $

Babita K.

Numerade Educator

View

Problem 40

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{x \to 1/4} \frac{\frac{1}{x} - 4}{x - \frac{1}{4}} $

Suman Saurav T.

Numerade Educator

View

Problem 41

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{h \to 0} \frac{\cos (\pi + h) + 1}{h} $

Babita K.

Numerade Educator

View

Problem 42

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{\theta \to \pi/6} \frac{\sin \theta - \frac{1}{2}}{\theta - \pi/6} $

Darshan M.

Numerade Educator

View

Problem 43

A particle moves along a straight line with equation of motion $ s = f(t) $, where $ s $ is measured in meters and $ t $ in seconds. Find the velocity and the speed when $ t = 4 $.

$ f(t) = 80t - 6t^2 $

Ma. Theresa A.

Numerade Educator

View

Problem 44

A particle moves along a straight line with equation of motion $ s = f(t) $, where $ s $ is measured in meters and $ t $ in seconds. Find the velocity and the speed when $ t = 4 $.

$ f(t) = 10 + \frac{45}{t + 1} $

Cindy R.

Numerade Educator

View

Problem 45

A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?

Aparna S.

Numerade Educator

04:54

Problem 46

A roast turkey is taken from an oven when its temperature has reached $ 185^{\circ}F $ and is placed on a table in a room where the temperature is $ 75^{\circ}F $. The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.

Albert T.

Numerade Educator

View

Problem 47

Researchers measured the average blood alcohol concentration $ C(t) $ of eight men starting one hour after consumption of 30 mL of ethanol (corresponding to two alcoholic drinks).

(a) Find the average rate of change of $ C $ with respect to $ t $ over each time interval:
(i) $ [1.0, 2.0] $ (ii) $ [1.5, 2.0] $
(iii) $ [2.0, 2.5] $ (iv) $ [2.0, 3.0] $
In each case, include the units.

(b) Estimate the instantaneous rate of change at $ t = 2 $ and interpret your result. What are the units?

David M.

Numerade Educator

01:10

Problem 48

The number $ N $ of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.)

(a) Find the average rate of growth
(i) from 2006 to 2008
(ii) from 2008 to 2010
In each case, include the units. What can you conclude?

(b) Estimate the instantaneous rate of growth in 2010 by taking the average of two average rates of change. What are its units?

Carson M.

Numerade Educator

View

Problem 49

The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day.
(a) Compute and interpret the average rate of change from 1990 to 2005. What are the units?
(b) Estimate the instantaneous rate of change in 2000 by taking the average of two average rates of change. What are its units?

David M.

Numerade Educator

View

Problem 50

The table shows values of the viral load $ V(t) $ in HIV patient 303, measured in RNA copies/mL, $ t $ days after ABT-538 treatment was begun.

(a) Find the average rate of change of $ V $ with respect to $ t $ over each time interval:
(i) $ [4, 11] $ (ii) $ [8, 11] $
(iii) $ [11, 15] $ (iv) $ [11, 22] $
What are the units?

(b) Estimate and interpret the value of the derivative $ V'(11) $.

David M.

Numerade Educator

View

Problem 51

The cost (in dollars) of producing $ x $ units of a certain commodity is $ C(x) = 5000 + 10x + 0.05x^2 $.

(a) Find the average rate of change of $ C $ with respect to $ x $ when the production level is changed
(i) from $ x = 100 $ to $ x = 105 $
(ii) from $ x = 100 $ to $ x = 101 $

(b) Find the instantaneous rate of change of $ C $ with respect to $ x $ when $ x = 100 $. ( This is called the \textit{marginal cost}. Its significance will be explained in Section 3.7.)

Leon D.

Numerade Educator

View

Problem 52

If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $ V $ of water remaining in the tank after $ t $ minutes as
$$ V(t) = 100,000 (1 - \frac{1}{60}t)^2 \hspace{5mm} 0 \le t \le 60 $$

Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $ V $ with respect to $ t $) as a function of $ t $. What are its units? For times $ t $ = 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? the least?

Linda H.

Numerade Educator

View

Problem 53

The cost of producing $ x $ ounces of gold from a new gold mine is $ C = f(x) $ dollars.
(a) What is the meaning of the derivative $ f'(x) $? What are its units?
(b) What does the statement $ f'(800) = 17 $ mean?
(c) Do you think the values of $ f'(x) $ will increase or decrease in the short term? What about the long term? Explain.

Linda H.

Numerade Educator

View

Problem 54

The number of bacteria after $ t $ hours in a controlled laboratory experiment is $ n = f(t) $.

(a) What is the meaning of the derivative $ f'(5) $? What are its units?

(b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, $ f'(5) $ or $ f'(10) $? If the supply of the nutrients is limited, would that affect your conclusion? Explain.

Leon D.

Numerade Educator

View

Problem 55

Let $ H(t) $ be the daily cost (in dollars) to heat an office building when the outside temperature is $ t $ degrees Fahrenheit.

(a) What is the meaning of $ H'(58) $? What are its units?

(b) Would you expect $ H'(58) $ to be positive or negative? Explain.

Leon D.

Numerade Educator

View

Problem 56

The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of $ p $ dollars per pound is $ Q = f(p) $.

(a) What is the meaning of the derivative $ f'(8) $? What are its units?

(b) Is $ f'(8) $ positive or negative? Explain.

Leon D.

Numerade Educator

View

Problem 57

The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how oxygen solubility $ S $ varies as a function of the water temperature $ T $.
(a) What is the meaning of the derivative $ S'(T) $? What are its units?

(b) Estimate the value of $ S'(16) $ and interpret it.

David M.

Numerade Educator

View

Problem 58

The graph shows the influence of the temperature $ T $ on the maximum sustainable swimming speed
$ S $ of Coho salmon.

(a) What is the meaning of the derivative $ S'(T) $? What are its units?

(b) Estimate the values of $ S'(15) $ and $ S'(25) $ and interpret them.

David M.

Numerade Educator

View

Problem 59

Determine whether $ f'(0) $ exists.

$ f(x) = \left\{
\begin{array}{ll}
x \sin \frac{1}{x} & \mbox{if $ x \neq 0 $}\\
0 & \mbox{if $ x = 0 $}
\end{array} \right.$

David M.

Numerade Educator

View

Problem 60

Determine whether $ f'(0) $ exists.

$ f(x) = \left\{
\begin{array}{ll}
x^2 \sin \frac{1}{x} & \mbox{if $ x \neq 0 $}\\
0 & \mbox{if $ x = 0 $}
\end{array} \right.$

Suman Saurav T.

Numerade Educator

View

Problem 61

(a) Graph the function $ f(x) = \sin x - \frac{1}{1000} \sin (1000x) $ in the viewing rectangle
$ [-2 \pi, 2 \pi] $ by $ [-4, 4] $. What slope does the graph appear to have at the origin?

(b) Zoom in to the viewing window $ [-0.4, 0.4] $ by $ [-0.25, 0.25] $ and estimate the value of $ f'(0) $. Does this agree with your answer from part (a)?

(c) Now zoom in to the viewing window $ [-0.008, 0.008] $ by $ [-0.005, 0.005] $. Do you wish to revise your estimate for $ f'(0) $?

Patrick D.

Numerade Educator