**1.** A curve has equation .

(a) Write an expression for the slope of the secant line

through the points and .

*(b) Write an expression for the slope of the tangent line at P.*

; ** _{2.}** Graph the curve in the viewing rectangles

by , by , and by

*y苷 f 共x兲*

*P共3, f 共3兲兲* *Q共x, f 共x兲兲*

*y苷 sin x* 关⫺2, 2兴

关⫺2, 2兴 关⫺1, 1兴 关⫺1, 1兴 关⫺0.5, 0.5兴

. What do you notice about the curve as you zoom in toward the origin?

**3.** (a) Find the slope of the tangent line to the parabola
at the point

( i ) using Definition 1 ( ii ) using Equation 2 (b) Find an equation of the tangent line in part (a).

共1, 3兲
*y苷 4x ⫺ x*^{2}

关⫺0.5, 0.5兴

**2.1** **Exercises**

; Graphing calculator or computer required **1.**Homework Hints available at stewartcalculus.com
**110** **CHAPTER 2 DERIVATIVES**

### In the following example we estimate the rate of change of the national debt with respect to time. Here the function is defined not by a formula but by a table of values.

### Let *be the Canadian gross public debt at time t. The table in the margin* gives approximate values of this function by providing midyear estimates, in billions of dollars, from 1994 to 2002. Interpret and estimate the value of .

SOLUTION

### The derivative *means the rate of change of D with respect to t when* , that is, the rate of increase of the national debt in 1998.

### According to Equation 5,

### So we compute and tabulate values of the difference quotient (the average rates of change) as shown in the table at the left. From this table we see that lies some- where between and billion dollars per year. [Here we are making the reason- able assumption that the debt didn’t fluctuate wildly between 1998 and 2002.] We estimate that the rate of change of the Canadian debt in 1998 was the average of these two numbers, namely

*The minus sign means that the debt was decreasing at that time.*

### Another method would be to plot the debt function and estimate the slope of the tangent line when .

### In Examples 3, 6, and 7 we saw three specific examples of rates of change: the veloc- ity of an object is the rate of change of displacement with respect to time; marginal cost is the rate of change of production cost with respect to the number of items produced; the rate of change of the debt with respect to time is of interest in economics. Here is a small sample of other rates of change: In physics, the rate of change of work with respect to time *is called power. Chemists who study a chemical reaction are interested in the rate of * *change in the concentration of a reactant with respect to time (called the rate of reaction).*

### A biologist is interested in the rate of change of the population of a colony of bacteria with respect to time. In fact, the computation of rates of change is important in all of the natu- ral sciences, in engineering, and even in the social sciences. Further examples will be given in Section 2.7.

### All these rates of change are derivatives and can therefore be interpreted as slopes of tangents. This gives added significance to the solution of the tangent problem. Whenever we solve a problem involving tangent lines, we are not just solving a problem in geome- try. We are also implicitly solving a great variety of problems involving rates of change in science and engineering.

**EXAMPLE 7** *D*

*共t兲*

*D*

### ⬘共1998兲

*D*

### ⬘共1998兲

*t*

### 苷 1998

*D*

### ⬘共1998兲 苷 lim

_{t l1998}*D*

*共t兲 ⫺ D共1998兲*

*t*

### ⫺ 1998

*D*

### ⬘共1998兲

### ⫺1.1 ⫺5.5

*D*

### ⬘共1998兲 ⬇ ⫺3.3 billion dollars per year

*t*

### 苷 1998

**A Note on Units**

The units for the average rate of change are the units for divided by the units for , namely, billions of dollars per year. The instan- taneous rate of change is the limit of the aver- age rates of change, so it is measured in the same units: billions of dollars per year.

*⌬t*

*⌬D* *⌬D兾⌬t*

*t*

1994 414.0

1996 469.5

1998 467.3

2000 456.4

2002 442.3

*D共t兲*

*t*

1994 13.3

1996 2000

2002 ⫺6.3⫺5.5⫺1.1
*D共t兲 ⫺ D共1998兲*

*t*⫺ 1998

98845_02_ch02_p103-111.qk_98845_02_ch02_p103-111 8/16/11 9:52 AM Page 110

**SECTION 2.1 DERIVATIVES AND RATES OF CHANGE** **111**

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

**4.** (a) Find the slope of the tangent line to the curve
at the point

( 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 until the curve and the line appear to coincide.

**5–8** Find an equation of the tangent line to the curve at the
given point.

**5.** , ** _{6.}** ,

**7.** **8.** ,

**9.** (a) Find the slope of the tangent to the curve
at the point where .
(b) Find equations of the tangent lines at the points

and .

; (c) Graph the curve and both tangents on a common screen.

**10.** (a) Find the slope of the tangent to the curve at
the point where .

(b) Find equations of the tangent lines at the points and .

; (c) Graph the curve and both tangents on a common screen.

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

Standing still?

(b) Draw a graph of the velocity function.

**12.** Shown are graphs of the position functions of two runners,
and , who run a 100-m race and finish in a tie.

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

*y苷 1兾sx*
*x苷 a*

共1, 1兲

### (

^{4, }

^{1}2

### )

*s (meters)*

0 2 4 6

4 2

*t (seconds)*

A B

*s (meters)*

0 4 8 12

80 40

*t (seconds)*
A

B

共1, 3兲

*y苷 x ⫺ x*^{3}
共1, 0兲

共1, 0兲

共2, 3兲
*y苷 x*^{3}*⫺ 3x ⫹ 1*
共2, ⫺4兲

*y苷 4x ⫺ 3x*^{2}

共1, 1兲
*y*苷 *2x*⫹ 1

*x*⫹ 2

### (

^{1, 1}兲

*y苷 sx ,*

*x苷 a*
*y苷 3 ⫹ 4x*^{2}*⫺ 2x*^{3}

共1, 5兲 共2, 3兲

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

(c) At what time do they have the same velocity?

**13.** If a ball is thrown into the air with a velocity of 10 m兾s,
its height ( in meters) after seconds is given by

. Find the velocity when .

**14.** If a rock is thrown upward on the planet Mars with a velocity
of , its height ( in meters) after seconds is given by

.

(a) Find the velocity of the rock after one second.

(b) Find the velocity of the rock when . (c) When will the rock hit the surface?

(d) With what velocity will the rock hit the surface?

**15.** The displacement ( in meters) of a particle moving in a
straight line is given by the equation of motion ,
where is measured in seconds. Find the velocity of the par -

ticle at times , and .

**16.** The displacement ( in meters) of a particle moving in a
straight line is given by , where is mea-
sured in seconds.

(a) Find the average velocity over each time interval:

( i ) ( ii )

( iii ) ( iv)

(b) Find the instantaneous velocity when .

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

**17.** For the function t whose graph is given, arrange the follow-
ing numbers in increasing order and explain your reasoning:

**18.** Find an equation of the tangent line to the graph of

at if and .

**19.** If an equation of the tangent line to the curve at the

point where is , find and .

**20.** If the tangent line to at (4, 3) passes through the
point (0, 2), find and .

10 m兾s *t*

*H苷 10t ⫺ 1.86t*^{2}

*t苷 a*

*s苷 1兾t*^{2}
*t*

*t苷 a, t 苷 1, t 苷 2* *t*苷 3

*s苷 t*^{2}*⫺ 8t ⫹ 18* *t*

关3, 4兴 关3.5, 4兴

关4, 5兴 关4, 4.5兴

*t*苷 4

*s* *t*

0 t⬘共⫺2兲 t⬘共0兲 t⬘共2兲 t⬘共4兲

y=©

1 3 4

_1 0 2 x

y

*y苷 t共x兲*
*x*苷 5 t共5兲 苷 ⫺3 t⬘共5兲 苷 4

*y苷 f 共x兲*
*a苷 2 y 苷 4x ⫺ 5* *f*共2兲 *f*⬘共2兲

*y苷 f 共x兲*
*f*共4兲 *f*⬘共4兲

*t*

*y苷 10t ⫺ 4.9t*^{2} *t*苷 2

98845_02_ch02_p103-111.qk_98845_02_ch02_p103-111 8/16/11 9:52 AM Page 111

**112** **CHAPTER 2 DERIVATIVES**

**21.** Sketch the graph of a function for which ,

, and .

**22.** Sketch the graph of a function for which

, , ,

, , and .

**23.** If , find and use it to find an equation of
the tangent line to the curve at the point .
**24.** If , find and use it to find an equation of

the tangent line to the curve at the point .
**25.** (a) If , find and use it to find an

equation of the tangent line to the curve at the point .

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

**26.** (a) If , find and use it to find equations
of the tangent lines to the curve at the
points and .

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

**27–32** Find .

**27.** **28.**

**29.** **30.**

**31.** **32.**

**33–38** Each limit represents the derivative of some function at
some number . State such an and in each case.

**33.** **34.**

**35.** **36.**

**37.** **38.**

**39–40** A particle moves along a straight line with equation of
motion , where is measured in meters and in seconds.

Find the velocity and the speed when .

**39.** **40.**

**41.** 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?

**42.** A roast turkey is taken from an oven when its temperature has
reached 85°C and is placed on a table in a room where the
temperature is 24°C. The graph shows how the temperature

*f*共0兲 苷 0
*f*

*f*⬘共2兲 苷 ⫺1
*f*⬘共1兲 苷 0

*f*⬘共0兲 苷 3,

t

t⬘共1兲 苷 t⬘共3兲 苷 0 t共0兲 苷 t共2兲 苷 t共4兲 苷 0

lim*x l 5*⫺t*共x兲 苷 ⬁*
t⬘共2兲 苷 ⫺1

t⬘共0兲 苷 t⬘共4兲 苷 1
lim*x l*⫺1^{⫹}t*共x兲 苷 ⫺⬁*

*f*⬘共1兲
*f共x兲 苷 3x*^{2}*⫺ x*^{3}

共1, 2兲
*y苷 3x*^{2}*⫺ x*^{3}

t⬘共1兲
t*共x兲 苷 x*^{4}⫺ 2

共1, ⫺1兲
*y苷 x*^{4}⫺ 2

*F*⬘共2兲
*F共x兲 苷 5x兾共1 ⫹ x*^{2}兲

*y苷 5x兾共1 ⫹ x*^{2}兲
共2, 2兲

*G⬘共a兲*
*G共x兲 苷 4x*^{2}*⫺ x*^{3}

*y苷 4x*^{2}*⫺ x*^{3}
共3, 9兲

共2, 8兲

*f⬘共a兲*

*f共t兲 苷 2t*^{3}*⫹ t*
*f共x兲 苷 3x*^{2}*⫺ 4x ⫹ 1*

*f共x兲 苷 x*^{⫺2}
*f共t兲 苷* *2t*⫹ 1

*t*⫹ 3

*f共x兲 苷* 4
s1*⫺ x*
*f共x兲 苷 s1 ⫺ 2x*

*f*
*a*

*f*
*a*

lim

*h l0*

s^{4}16*⫹ h ⫺ 2*
lim *h*

*h l0*

*共1 ⫹ h兲*^{10}⫺ 1
*h*

*x l*lim兾4

*tan x*⫺ 1
*x*⫺兾4
lim*x l5*

2* ^{x}*⫺ 32

*x*⫺ 5

lim

*t l1*

*t*^{4}*⫹ t ⫺ 2*
*t*⫺ 1
lim

*h l0*

cos共 ⫹ h兲 ⫹ 1
*h*

*t*
*s*

*s苷 f 共t兲*

*t*苷 5

*f共t兲 苷 100 ⫹ 50t ⫺ 4.9t*^{2} *f共t兲 苷 t*^{⫺1}*⫺ t*

of the turkey decreases and eventually approaches room tem- perature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.

**43.** The table shows the number of passengers that arrived in
Ireland by air, in millions.

(a) Find the average rate of increase of

( i ) from 2001 to 2005 ( ii ) from 2003 to 2005 ( iii ) from 2005 to 2007

In each case, include the units.

(b) Estimate the instantaneous rate of growth in 2005 by taking the average of two average rates of change. What are its units?

**44.** The number 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 2006 to 2007 ( iii ) from 2005 to 2006

In each case, include the units.

(b) Estimate the instantaneous rate of growth in 2006 by taking the average of two average rates of change. What are its units?

(c) Estimate the instantaneous rate of growth in 2006 by mea- suring the slope of a tangent.

(d) Estimate the intantaneous rate of growth in 2007 and com- pare it with the growth rate in 2006. What do you conclude?

**45.** The cost ( in dollars) of producing units of a certain com-

modity is .

(a) Find the average rate of change of with respect to when the production level is changed

( i ) from to ( ii ) from to

(b) Find the instantaneous rate of change of with respect to
when *. (This is called the marginal cost. Its sig-*
nificance will be explained in Section 2.7.)

*T (*°C)

0 30 60 90 120 150

40 80

*t (min)*
P

*N*

*x*
*C共x兲 苷 5000 ⫹ 10x ⫹ 0.05x*^{2}

*x*
*C*

*x*苷 105
*x*苷 100

*x*苷 101
*x*苷 100

*C*
*x*苷 100

*x*

*P*

*P*

Year 2004 2005 2006 2007 2008

*N* 8569 10,241 12,440 15,011 16,680

Year 2001 2003 2005 2007 2009

*P* 8.49 9.65 11.78 14.54 12.84

98845_02_ch02_p112-121.qk_98845_02_ch02_p112-121 8/16/11 9:53 AM Page 112

**SECTION 2.1 DERIVATIVES AND RATES OF CHANGE** **113**
**46.** If a cylindrical tank holds 100,000 liters of water, which can be

drained from the bottom of the tank in an hour, then Torricelli’s Law gives the volume of water remaining in the tank after minutes as

Find the rate at which the water is flowing out of the tank (the
instantaneous rate of change of with respect to ) as a func-
*tion of t. What are its units? For times t*苷 0, 10, 20, 30, 40, 50,
and 60 min, find the flow rate and the amount of water remain-
ing in the tank. Summarize your findings in a sentence or two.

At what time is the flow rate the greatest? The least?

**47.** *The cost of producing x kilograms of gold from a new gold*
mine is dollars.

(a) What is the meaning of the derivative ? What are its units?

(b) What does the statement mean?

(c) Do you think the values of will increase or decrease in the short term? What about the long term? Explain.

**48.** *The number of bacteria after t hours in a controlled laboratory*
experiment is .

(a) What is the meaning of the derivative ? What are its units?

(b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger,

or ? If the supply of nutrients is limited, would that affect your conclusion? Explain.

**49.** Let be the temperature (in ) in Manila hours after noon
on July 19, 2011. The table shows values of this function
recorded every two hours. What is the meaning of ?
Estimate its value.

**50.** The quantity (in kilograms) of a gourmet ground coffee that is
*sold by a coffee company at a price of p dollars per kilogram *
is .

(a) What is the meaning of the derivative ? What are its units?

(b) Is positive or negative? Explain.

**51.** The quantity of oxygen that can dissolve in water depends on
the temperature of the water. (So thermal pollution influences

*n苷 f 共t兲*

*f*⬘共5兲

*f*⬘共10兲
*f*⬘共5兲

*Q苷 f 共 p兲*

*f*⬘共8兲
*f*⬘共8兲

*V* *t*

*C苷 f 共x兲*

*f⬘共x兲*
*f*⬘共50兲 苷 36

*f⬘共x兲*

*V* *t*

0*艋 t 艋 60*
*V共t兲 苷 100,000*

### (

1⫺60^{1}

*t*

### )

^{2}

*T共t兲* ⬚C *t*

*T*⬘共5兲

the oxygen content of water.) The graph shows how oxygen solubility varies as a function of the water temperature . (a) What is the meaning of the derivative ? What are its

units?

(b) Estimate the value of and interpret it.

**52.** The graph shows the influence of the temperature on the
maximum sustainable swimming speed of Coho salmon.

(a) What is the meaning of the derivative ? What are its units?

(b) Estimate the values of and and interpret them.

**53–54** Determine whether exists.

**53.**

**54.**

*T*
*S*

*S⬘共T兲*
*S*⬘共16兲

(mg/ L)

4 8 12 16 S

0 T (°C)

Adapted from Kupchella & Hyland, Environmental Science: Living Within the System of Nature, 2d ed.; © 1989. Printed and electronically reproduced by permission of Pearson Education, Inc., Upper Saddle River, NJ.

8 16 24 32 40

*T*
*S*

*S⬘共T兲*
*S*⬘共25兲
*S*⬘共15兲

0 10 20 T (°C)

S (cm/s) 20

*f*⬘共0兲

*f共x兲 苷*

### 再

^{x sin }^{0}

^{1}

^{x}

^{if x}^{if x}^{苷 0}

^{苷 0}

*f共x兲 苷*

### 再

^{x}^{0}

^{2 }

^{sin }

^{1}

^{x}

^{if x}^{if x}^{苷 0}

^{苷 0}

*t* 1 3 5 7 9 11

*T* 32 32 31 27 26 25

98845_02_ch02_p112-121.qk_98845_02_ch02_p112-121 8/16/11 9:53 AM Page 113

**122** **CHAPTER 2 DERIVATIVES**

**1–2** Use the given graph to estimate the value of each derivative.

Then sketch the graph of .

**1.** (a) (b) (c)

(d) (e) (f)

(g)

**2.** (a) (b) (c)

(d) (e) (f)

(g) (h)

*f*⬘

y

x 1

1

y

0 x

1 1

*f*⬘共⫺3兲 *f*⬘共⫺2兲 *f*⬘共⫺1兲

*f*⬘共0兲 *f*⬘共1兲 *f*⬘共2兲

*f*⬘共3兲

*f*⬘共0兲 *f*⬘共1兲 *f*⬘共2兲

*f*⬘共3兲 *f*⬘共4兲 *f*⬘共5兲

*f*⬘共6兲 *f*⬘共7兲

**3.** Match the graph of each function in (a)–(d) with the graph of
its derivative in I–IV. Give reasons for your choices

### .

y

0 y

0

y

0

y

0

x x

x x

(b) (a)

(c) (d)

II I

III IV

y

0

y

0

y

0

x

x y

0 x

x

**2.2** **Exercises**

### the slope of , we have

### for all values of . So is a constant function and its graph is a horizontal line. There- fore, for all values of ,

### We can also interpret the third derivative physically in the case where the function is the position function of an object that moves along a straight line. Because

### , the third derivative of the position function is the derivative of the accel- **eration function and is called the jerk:**

*Thus the jerk j is the rate of change of acceleration. It is aptly named because a large jerk* means a sudden change in acceleration, which causes an abrupt movement in a vehicle.

### We have seen that one application of second and third derivatives occurs in analyzing the motion of objects using acceleration and jerk. We will investigate another applica- tion of second derivatives in Section 3.3, where we show how knowledge of gives us information about the shape of the graph of . In Chapter 11 we will see how second and higher derivatives enable us to represent functions as sums of infinite series.

*f*

*⬙共x兲*

*f*

*共x兲 苷 6*

*x* *f*

###

*x*

*f*^{共4兲}

*共x兲 苷 0*

*s*

*苷 s共t兲*

*s*

* 苷 共s⬙兲⬘ 苷 a⬘*

*j*

### 苷

*da*

*dt*

### 苷

*d*

^{3}

*s*

*dt*^{3}

*f*

### ⬙

*f*

; Graphing calculator or computer required **1.**Homework Hints available at stewartcalculus.com
98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 122

**SECTION 2.2 THE DERIVATIVE AS A FUNCTION** **123**
**4–11** Trace or copy the graph of the given function . (Assume that

the axes have equal scales.) Then use the method of Example 1 to sketch the graph of below it.

**4.**

**5.** **6.**

**7.** **8.**

**9.** **10.**

**11.**

**12.** Shown is the graph of the population function for yeast
cells in a laboratory culture. Use the method of Example 1 to
graph the derivative . What does the graph of tell us
about the yeast population?

*f*
*f*⬘

0 x

y

*x*
*y*

0 0 x

y

*x*
*y*

0 0 x

y

0 x

y

*x*
*y*

0

0 x

y

*P共t兲*

(yeast cells)

t (hours) P

0 5 10 15

500

*P⬘共t兲* *P*⬘

**13.** A rechargeable battery is plugged into a charger. The graph
shows , the percentage of full capacity that the battery
reaches as a function of time elapsed ( in hours).

(a) What is the meaning of the derivative ?

(b) Sketch the graph of . What does the graph tell you?

**14.** The graph (from the US Department of Energy) shows how
driving speed affects gas mileage. Fuel economy is measured
in miles per gallon and speed is measured in miles per hour.

(a) What is the meaning of the derivative ? (b) Sketch the graph of .

(c) At what speed should you drive if you want to save on gas?

**15.** The graph shows how the average age of first marriage of
Japanese men varied in the last half of the 20th century.

Sketch the graph of the derivative function . During which years was the derivative negative?

**16.** Make a careful sketch of the graph of the sine function and
below it sketch the graph of its derivative in the same manner
as in Exercises 4 –11. Can you guess what the derivative of the
sine function is from its graph?

*C共t兲*

*t*

*C⬘共t兲*
*C⬘共t兲*

*t (hours)*

2 4 6 8 10 12

20 40 60 80 100

percentage of full charge

*C*

*F*
*v*

*F*⬘共* ^{v}*兲

*F*⬘共

*兲*

^{v}√ (mi/ h) 0

10 30 20

70 60 40

20 30 50

10
*F (mi/ gal)*

*M⬘共t兲*

1990 2000 25

M

1960 1970 1980 27

t 98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 123

**124** **CHAPTER 2 DERIVATIVES**

;** _{17.}** Let .

(a) Estimate the values of , , , and by using a graphing device to zoom in on the graph of . (b) Use symmetry to deduce the values of , ,

and .

(c) Use the results from parts (a) and (b) to guess a formula for .

(d) Use the definition of derivative to prove that your guess in part (c) is correct.

;** _{18.}** Let .

(a) Estimate the values of , , , , and
*by using a graphing device to zoom in on the graph of f.*

(b) Use symmetry to deduce the values of , , , and .

(c) Use the values from parts (a) and (b) to graph . (d) Guess a formula for .

(e) Use the definition of derivative to prove that your guess in part (d) is correct.

**19–29** Find the derivative of the function using the definition of
derivative. State the domain of the function and the domain of its
derivative.

**19.** **20.**

**21.** **22.**

**23.** **24.**

**25.** **26.**

**27.** **28.**

**29.**

**30.** (a) Sketch the graph of by starting with the
graph of and using the transformations of Sec -
tion 1.3.

(b) Use the graph from part (a) to sketch the graph of . (c) Use the definition of a derivative to find . What are

*the domains of f and ?*

; (d) Use a graphing device to graph and compare with your sketch in part (b).

**31.** (a) If , find .

; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of and .

**32.** (a) If , find .

; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of and .

*f共x兲 苷 x*^{2}

*f*⬘共2兲
*f*⬘共1兲

*f*⬘

### (

^{1}2

### )

*f*⬘共0兲

*f*
*f*⬘共⫺1兲
*f*⬘

### (

⫺^{1}2

### )

*f*⬘共⫺2兲
*f⬘共x兲*

*f共x兲 苷 x*^{3}

*f*⬘共3兲
*f*⬘共2兲

*f*⬘共1兲
*f*⬘

### (

^{1}2

### )

*f*⬘共0兲

*f*⬘共⫺1兲
*f*⬘

### (

⫺^{1}2

### )

*f*⬘共⫺3兲
*f*⬘共⫺2兲

*f*⬘
*f⬘共x兲*

*f共x兲 苷 mx ⫹ b*
*f共x兲 苷*^{1}^{2}*x*⫺^{1}^{3}

*f共x兲 苷 1.5x*^{2}*⫺ x ⫹ 3.7*
*f共t兲 苷 5t ⫺ 9t*^{2}

*f共x兲 苷* *x*^{2}⫺ 1
*2x*⫺ 3
t*共x兲 苷 s9 ⫺ x*

*f共x兲 苷 x*^{3}^{兾2}
*G共t兲 苷* 1*⫺ 2t*

3*⫹ t*
*f共x兲 苷 x*^{4}

*f共x兲 苷 s6 ⫺ x*
*y苷 sx*

*f*⬘
*f⬘共x兲*
*f*⬘

*f*⬘

*f⬘共x兲*
*f共x兲 苷 x*^{4}*⫹ 2x*

*f*⬘
*f*

*f⬘共x兲*
*f共x兲 苷 x ⫹ 1兾x*

*f*⬘
*f*

*f共x兲 苷 x*^{3}*⫺ 3x ⫹ 5* *f共x兲 苷 x ⫹ sx*

**33.** The unemployment rate varies with time. The table
gives the percentage of unemployed in the Australian labor
force measured at midyear from 1995 to 2004.

(a) What is the meaning of ? What are its units?

(b) Construct a table of estimated values for .

**34.** Let be the percentage of the population of the Philippines
over the age of 60 at time . The table gives projections of
values of this function from 1995 to 2020.

(a) What is the meaning of ? What are its units?

(b) Construct a table of estimated values for . (c) Graph and .

**35–38** The graph of is given. State, with reasons, the numbers
at which is not differentiable.

**35.** **36.**

**37.** **38.**

;** _{39.}** Graph the function . Zoom in repeatedly,
first toward the point (⫺1, 0) and then toward the origin.

What is different about the behavior of in the vicinity of
these two points? What do you conclude about the differen-
*tiability of f ?*

;** _{40.}** Zoom in toward the points (1, 0), (0, 1), and (⫺1, 0) on
the graph of the function . What do you
notice? Account for what you see in terms of the differen-
tiability of t.

*U⬘共t兲*

*U⬘共t兲*

*P⬘共t兲*

*P⬘共t兲*
*P*⬘

*P*
*f*
*f*

2 4 *x*

*y*

0

_2 2 *x*

*y*

0

_2 2 *x*

*y*

_2 *4 x* 0

*y*

0

*f共x兲 苷 x ⫹ s*

### ⱍ

^{x}### ⱍ

*f*

t*共x兲 苷 共x*^{2}⫺ 1兲^{2}^{兾3}
*U共t兲*

*P共t兲*

*t*

*t* *t*

1995 8.1 2000 6.2

1996 8.0 2001 6.9

1997 8.2 2002 6.5

1998 7.9 2003 6.2

1999 6.7 2004 5.6

*U共t兲*
*U共t兲*

*t* *t*

1995 5.2 2010 6.7

2000 5.5 2015 7.7

2005 6.1 2020 8.9

*P共t兲*
*P共t兲*

98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 124

**SECTION 2.2 THE DERIVATIVE AS A FUNCTION 125**
**41.** The figure shows the graphs of , , and . Identify each

curve, and explain your choices.

**42.** The figure shows graphs of , *, and * . Identify each
curve, and explain your choices.

**43.** The figure shows the graphs of three functions. One is the posi-
tion function of a car, one is the velocity of the car, and one is
its acceleration. Identify each curve, and explain your choices.

**44.** The figure shows the graphs of four functions. One is the
position function of a car, one is the velocity of the car, one is
its acceleration, and one is its jerk. Identify each curve, and
explain your choices.

*f*⬙
*f*⬘
*f*

x

y a

b

c

*f*
*f*⬙
*f*⬘
*f, *

x

y a b c d

t

y a

b c

0

0 t

y a

b c

d

;** _{45–46}** Use the definition of a derivative to find and .
Then graph , , and on a common screen and check to see if
your answers are reasonable.

**45.**

**46.**

;** _{47.}** If , find , , , and .

Graph , , , and on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?

**48.** *(a) The graph of a position function of a car is shown, where s*
*is measured in meters and t in seconds. Use it to graph the*
velocity and acceleration of the car. What is the acceleration
at seconds?

(b) Use the acceleration curve from part (a) to estimate the jerk at seconds. What are the units for jerk?

**49.** Let .

(a) If , use Equation 2.1.5 to find . (b) Show that does not exist.

(c) Show that has a vertical tangent line at . (Recall the shape of the graph of . See Figure 13 in Section 1.2.)

**50.** (a) If , show that does not exist.

(b) If , find .

(c) Show that has a vertical tangent line at .

; (d) Illustrate part (c) by graphing .

**51.** Show that the function is not differentiable
at 6. Find a formula for and sketch its graph.

**52.** Where is the greatest integer function not differen-
tiable? Find a formula for and sketch its graph.

**53.** (a) Sketch the graph of the function .
(b) For what values of is differentiable?

(c) Find a formula for .

*f⬙共x兲*
*f⬘共x兲*

*f*⬙
*f*⬘
*f*

*f共x兲 苷 3x*^{2}*⫹ 2x ⫹ 1*
*f共x兲 苷 x*^{3}*⫺ 3x*

*f*^{共4兲}*共x兲*
*f共x兲*

*f⬙共x兲*
*f⬘共x兲*
*f共x兲 苷 2x*^{2}*⫺ x*^{3}

*f*
*f*⬙
*f*⬘
*f*

0 10 t

s

100

20

*t*苷 10

*f共x兲 苷 s*^{3}*x*

*f⬘共a兲*
*a*苷 0

*f*⬘共0兲

共0, 0兲
*y*苷 s^{3}*x*

*f*

t⬘共0兲
t*共x兲 苷 x*^{2}^{兾3}

t*⬘共a兲*
*a*苷 0

共0, 0兲
*y苷 x*^{2}^{兾3}

*y苷 x*^{2}^{兾3}

*f共x兲 苷*

### ⱍ

^{x}^{⫺ 6}

### ⱍ

*f*⬘

*f共x兲 苷 冀 x冁*
*f*⬘

*f共x兲 苷 x*

### ⱍ

^{x}### ⱍ

*f*
*x*
*f*⬘
*t*苷 10

98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 125

**126** **CHAPTER 2 DERIVATIVES**

**54.** **The left-hand and right-hand derivatives of at are **defined
by

and

if these limits exist. Then exists if and only if these one- sided derivatives exist and are equal.

(a) Find and for the function

(b) Sketch the graph of .
*f共x兲 苷*

0
5*⫺ x*

*if x*艋 0
if 0*⬍ x ⬍ 4*
1

5*⫺ x* *if x*艌 4

*f* *a*

*f*⬘⫺*共a兲 苷 lim*_{h l0}_{⫺} *f共a ⫹ h兲 ⫺ f 共a兲*
*h*
*f*⬘⫹*共a兲 苷 lim*_{h l0}_{⫹} *f共a ⫹ h兲 ⫺ f 共a兲*

*h*
*f⬘共a兲*
*f*⬘⫹共4兲
*f*⬘⫺共4兲

*f*

(c) Where is discontinuous?

(d) Where is not differentiable?

**55.** *Recall that a function is called even if for *all
*in its domain and odd if * for all such . Prove

each of the following.

(a) The derivative of an even function is an odd function.

(b) The derivative of an odd function is an even function.

**56.** When you turn on a hot-water faucet, the temperature of the
water depends on how long the water has been running.

(a) Sketch a possible graph of as a function of the time that has elapsed since the faucet was turned on.

(b) Describe how the rate of change of with respect to varies as increases.

(c) Sketch a graph of the derivative of .

**57.** Let be the tangent line to the parabola at the point
*. The angle of inclination of is the angle * that makes
with the positive direction of the -axis. Calculate correct to
the nearest degree.

ᐍ *y苷 x*^{2}

共1, 1兲 ᐍ ᐍ

*x*

*T*
*t*
*T*

*t*
*T*

*t*

*T*
*f*

*f共⫺x兲 苷 f 共x兲*
*f*

*x*
*f共⫺x兲 苷 ⫺f 共x兲*

*x*

*f*

### If it were always necessary to compute derivatives directly from the definition, as we did in the preceding section, such computations would be tedious and the evaluation of some limits would require ingenuity. Fortunately, several rules have been developed for finding derivatives without having to use the definition directly. These formulas greatly simplify the task of differentiation.

### Let’s start with the simplest of all functions, the constant function . The graph *of this function is the horizontal line y * *苷 c, which has slope 0, so we must have * . (See Figure 1.) A formal proof, from the definition of a derivative, is also easy:

### In Leibniz notation, we write this rule as follows.

**Derivative of a Constant Function**

**Power Functions**

### We next look at the functions *, where n is a positive integer. If * , the graph of *is the line y * *苷 x, which has slope 1. (See Figure 2.) So*

*f*

*共x兲 苷 c*

*f*

*⬘共x兲 苷 0*

*f*

*⬘共x兲 苷 lim*

*h l 0*

*f*

*共x ⫹ h兲 ⫺ f 共x兲*

*h*

### 苷 lim

*h l 0*

*c*

*⫺ c*

*h*

### 苷 lim

*h l 0*

### 0 苷 0

*d*

*dx*

*共c兲 苷 0*

*f*

*共x兲 苷 x*

^{n}*n*

### 苷 1

*f*

*共x兲 苷 x*

*d*

*dx*

*共x兲 苷 1*

**1**

**2.3** **Differentiation Formulas**

**FIGURE 1**

The graph of ƒ=c is the line y=c, so fª(x)=0.

*y*

*c*

0 x

y=c slope=0

*y*

0

x y=x

slope=1

**FIGURE 2 **

The graph of ƒ=x is the line y=x, so fª(x)=1.

98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 126

### Let the -coordinate of one of the points in question be . Then the slope of the tangent line at that point is . This tangent line will be parallel to the line , or

### , if it has the same slope, that is, . Equating slopes, we get

### Therefore the required points are and . The hyperbola and the tangents are shown in Figure 6.

### We summarize the differentiation formulas we have learned so far as follows.

**Table of Differentiation Formulas**

### 共2, 6兲 共⫺2, ⫺6兲

*x* *a*

*⫺12兾a*

^{2}

*3x* *⫹ y 苷 0*

*y*

*苷 ⫺3x* ⫺3

### ⫺ 12

*a*^{2}

### 苷 ⫺3 or

*a*

^{2}

### 苷 4 or

*a*

### 苷 ⫾2

*d*

*dx*

*共c兲 苷 0*

*d*

*dx*

*共x*

^{n}*兲 苷 nx*

^{n}^{⫺1}

*共cf 兲⬘ 苷 cf⬘* *共 f ⫹ t兲⬘ 苷 f⬘⫹ t⬘* *共 f ⫺ t兲⬘ 苷 f⬘⫺ t⬘*

*共 ft兲⬘ 苷 ft⬘ ⫹ tf⬘* 冉 ^{t}

^{f}### 冊 ^{⬘} ^{苷} ^{t}

^{f}^{⬘ ⫺ ft⬘} ^{t}

^{⬘ ⫺ ft⬘}

^{2}

**136** **CHAPTER 2 DERIVATIVES**

**FIGURE 6**

3x+y=0 xy=12

0 x

y

(_2, _6)

(2, 6)

**1–22** Differentiate the function.

**1.** **2.**

**3.** **4.**

**5.** **6.**

**7.** **8.**

**9.** **10.**

**11.** **12.**

**13.** **14.**

**15.** **16.**

**17.** **18.**

**19.** **20.**

**21.** **22.**

*f共x兲 苷 x*^{3}*⫺ 4x ⫹ 6* *f共t兲 苷*^{1}2*t*^{6}*⫺ 3t*^{4}*⫹ t*
t*共x兲 苷 x*^{2}*共1 ⫺ 2x兲* *h共x兲 苷 共x ⫺ 2兲共2x ⫹ 3兲*

*B共y兲 苷 cy*^{⫺6}
*A共s兲 苷 ⫺*12

*s*^{5} *y苷 x*^{5}^{兾3}*⫺ x*^{2}^{兾3}

*S共 p兲 苷 sp ⫺ p* *y苷 sx 共x ⫺ 1兲*

*R共a兲 苷 共3a ⫹ 1兲*^{2} *S共R兲 苷 4R*^{2}

*y*苷 *x*^{2}*⫹ 4x ⫹ 3*

s*x* *y*苷 sx *⫹ x*

*x*^{2}

*H共x兲 苷 共x ⫹ x*^{⫺1}兲^{3} t*共u兲 苷 s2 u ⫹ s3u*

*u*苷 s^{5}*t* *⫹ 4st*^{5 } * ^{v}*苷冉

^{s}

^{x}^{⫹}

^{s}

^{3}

^{1}

*冊*

^{x}^{2}

*f共x兲 苷 186.5* *f共x兲 苷 s30*

*f共x兲 苷 5x ⫺ 1* *F共x兲 苷 ⫺4x*^{10}

*y苷 x*^{⫺2兾5}

**23.** Find the derivative of in two ways:

by using the Product Rule and by performing the multiplication first. Do your answers agree?

**24.** Find the derivative of the function

in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?

**25–44** Differentiate.

**25.**

**26.**

**27.**

**28.**

**29.** **30.**

*V共x兲 苷 共2x*^{3}*⫹ 3兲共x*^{4}*⫺ 2x兲*
*L共x兲 苷 共1 ⫹ x ⫹ x*^{2}*兲共2 ⫺ x*^{4}兲

*F共y兲 苷*冉^{y}^{1}^{2} ^{⫺} ^{y}^{3}^{4}冊^{共y ⫹ 5y}^{3}^{兲}

*J*共* ^{v}*兲 苷 共

^{v}^{3}⫺ 2

*兲共*

^{v}

^{v}^{⫺4}⫹

^{v}^{⫺2}兲

*F共x兲 苷* *x*^{4}*⫺ 5x*^{3}*⫹ sx*
*x*^{2}

*f共x兲 苷 共1 ⫹ 2x*^{2}*兲共x ⫺ x*^{2}兲

t*共x兲 苷* *3x*⫺ 1

*2x*⫹ 1 *f共t兲 苷* *2t*

4*⫹ t*^{2}

**2.3** **Exercises**

; Graphing calculator or computer required **1.**Homework Hints available at stewartcalculus.com
98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 136

**SECTION 2.3 DIFFERENTIATION FORMULAS** **137**

**31.** **32.**

**33.** **34.**

**35.** **36.**

**37.** **38.**

**39.** **40.**

**41.** **42.**

**43.** **44.**

**45.** The general polynomial of degree has the form

where . Find the derivative of

;** _{46–48}** Find . Compare the graphs of and and use them
to explain why your answer is reasonable.

**46.**

**47.** **48.**

;** _{49.}** (a) Use a graphing calculator or computer to graph the func-

tion in the viewing

rectangle by .

(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.2.)

(c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b).

;** _{50.}** (a) Use a graphing calculator or computer to graph the func-
tion in the viewing rectangle
by .

(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.2.)

(c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b).

*y*苷 ^{v}

3⫺ 2* ^{v}*s

*v*

*v* *y*苷 *t*

*共t ⫺ 1兲*^{2}
*y*苷 *x*^{3}

1*⫺ x*^{2} *y*苷 *x*⫹ 1

*x*^{3}*⫹ x ⫺ 2*

t*共t兲 苷* *t⫺ st*
*t*^{1}^{兾3}
*y*苷 *t*^{2}⫹ 2

*t*^{4}*⫺ 3t*^{2}⫹ 1

*y苷 A ⫹* *B*
*x* ⫹ *C*

*x*^{2}
*y苷 ax*^{2}*⫹ bx ⫹ c*

*y*苷 *cx*
1*⫹ cx*
*f共t兲 苷* *2t*

2*⫹ st*

*y*苷 *u*^{6}*⫺ 2u*^{3}⫹ 5
*u*^{2}
*y*苷 s^{3}*t共t*^{2}*⫹ t ⫹ t*^{⫺1}兲

*f共x兲 苷* *ax⫹ b*
*cx⫹ d*
*f共x兲 苷* *x*

*x*⫹ *c*
*x*

*n*

*P共x兲 苷 a*^{n}*x*^{n}*⫹ a** ^{n}*⫺1

*x*

^{n}^{⫺1}

*⫹ ⭈ ⭈ ⭈ ⫹ a*

^{2}

*x*

^{2}

*⫹ a*

^{1}

*x⫹ a*

^{0}

*P.*

*a**n*苷 0

*f*⬘
*f*
*f⬘共x兲*

*f共x兲 苷 x兾共x*^{2}⫺ 1兲

*f共x兲 苷 x ⫹* 1
*f共x兲 苷 3x*^{15}*⫺ 5x*^{3}⫹ 3 *x*

*f共x兲 苷 x*^{4}*⫺ 3x*^{3}*⫺ 6x*^{2}*⫹ 7x ⫹ 30*
关⫺10, 50兴

关⫺3, 5兴

*f*⬘
*f⬘共x兲*

*f*⬘

关⫺4, 4兴
t*共x兲 苷 x*^{2}*兾共x*^{2}⫹ 1兲

关⫺1, 1.5兴

t⬘
t*⬘共x兲*

t⬘

**51–52** Find an equation of the tangent line to the curve at the
given point.

**51.** ,

**52.** ,

**53.** (a) The curve **is called a witch of Maria**
**Agnesi. Find an equation of the tangent line to this curve**
at the point .

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

**54.** (a) The curve **is called a serpentine. **

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

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

**55–58** Find equations of the tangent line and normal line to the
curve at the given point.

**55.** , ** _{56.}** ,

**57.** , ** _{58.}** ,

**59–62** Find the first and second derivatives of the function.

**59.** **60.**

**61.** **62.**

**63.** The equation of motion of a particle is , where
is in meters and is in seconds. Find

(a) the velocity and acceleration as functions of , (b) the acceleration after 2 s, and

(c) the acceleration when the velocity is 0.

**64.** The equation of motion of a particle is

where is in meters and is in seconds.

(a) Find the velocity and acceleration as functions of . (b) Find the acceleration after 1 s.

; (c) Graph the position, velocity, and acceleration functions on the same screen.

共1, 2兲
*y苷 x*^{4}*⫹ 2x*^{2}*⫺ x*

共1, 1兲
*y*苷 *2x*

*x*⫹ 1

*y苷 1兾共1 ⫹ x*^{2}兲

### (

_{⫺1, }

^{1}2

### )

*y苷 x兾共1 ⫹ x*^{2}兲
共3, 0.3兲

共1, 9兲
*y苷 共1 ⫹ 2x兲*^{2}
共1, 2兲

*y苷 x ⫹ sx*

共4, 0.4兲
*y*苷 s*x*

*x*⫹ 1
共1, 2兲

*y*苷 *3x*⫹ 1
*x*^{2}⫹ 1

*G共r兲 苷 sr ⫹ s*^{3}*r*
*f共x兲 苷 x*^{4}*⫺ 3x*^{3}*⫹ 16x*

*f共x兲 苷* 1
3*⫺ x*
*f共x兲 苷* *x*^{2}

1*⫹ 2x*

*s*
*s苷 t*^{3}*⫺ 3t*
*t*

*t*

*t*
*s*

*s苷 t*^{4}*⫺ 2t*^{3}*⫹ t*^{2}*⫺ t*

*t*
98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 137

**138** **CHAPTER 2 DERIVATIVES**

**65.** Boyle’s Law states that when a sample of gas is compressed
at a constant pressure, the pressure of the gas is inversely
proportional to the volume of the gas.

(a) Suppose that the pressure of a sample of air that occupies at is . Write as a function of . (b) Calculate when . What is the meaning

of the derivative? What are its units?

;** _{66.}** Car tires need to be inflated properly because overinflation or
underinflation can cause premature treadware. The data in the
table show tire life ( in thousands of kilometers) for a cer-
tain type of tire at various pressures ( in ).

(a) Use a graphing calculator or computer to model tire life with a quadratic function of the pressure.

(b) Use the model to estimate when and when . What is the meaning of the derivative?

What are the units? What is the significance of the signs of the derivatives?

**67.** Suppose that , , , and .

Find the following values.

(a) (b)

(c)

**68.** Find , given that , , ,

and .

(a) (b)

(c) (d)

**69.** If , where and , find .

**70.** If and , find

**71.** If and are the functions whose graphs are shown, let
and .

(a) Find (b) Find
*P*

*V*

*P*
*V*

50 kPa 25⬚C

0.106 m^{3}

*P*苷 50 kPa
*dV兾dP*

t⬘共5兲 苷 2 t共5兲 苷 ⫺3

*f*⬘共5兲 苷 6
*f*共5兲 苷 1

*共 f兾t兲⬘共5兲*
*共 ft兲⬘共5兲*

*共t兾f 兲⬘共5兲*

*f*⬘共2兲 苷 ⫺2
t共2兲 苷 4

*f*共2兲 苷 ⫺3
*h*⬘共2兲

t⬘共2兲 苷 7

*h共x兲 苷 f 共x兲 t共x兲*
*h共x兲 苷 5f 共x兲 ⫺ 4t共x兲*

*h共x兲 苷* t*共x兲*
1*⫹ f 共x兲*
*h共x兲 苷* *f共x兲*

t*共x兲*

*f*⬘共4兲
t⬘共4兲 苷 7

t共4兲 苷 8
*f共x兲 苷 sx t共x兲*

*h*⬘共2兲 苷 ⫺3
*h*共2兲 苷 4

*d*

*dx* 冉* ^{h}^{共x兲}^{x}* 冊冟

^{x}^{苷2}

*f* t

*v共x兲 苷 f 共x兲兾t共x兲*
*u共x兲 苷 f 共x兲t共x兲*

*v*⬘共5兲.

*u*⬘共1兲.

f

g

*x*
*y*

0 1

1
*L*

*P* kPa

*dL兾dP* *P*苷 200
*P*苷 300

**72.** Let and , where and

are the functions whose graphs are shown.

(a) Find . (b) Find .

**73.** If is a differentiable function, find an expression for the
derivative of each of the following functions.

(a) (b) (c)

**74.** If is a differentiable function, find an expression for the
derivative of each of the following functions.

(a) (b)

(c) (d)

**75.** Find the points on the curve
where the tangent is horizontal.

**76.** For what values of does the graph of

have a horizontal tangent?

**77.** Show that the curve has no tangent line
with slope 4.

**78.** Find an equation of the tangent line to the curve
that is parallel to the line .

**79.** Find equations of both lines that are tangent to the curve
and are parallel to the line .
**80.** Find equations of the tangent lines to the curve

that are parallel to the line .

**81.** Find an equation of the normal line to the parabola
that is parallel to the line .
**82.** Where does the normal line to the parabola at the

point (1, 0) intersect the parabola a second time? Illustrate with a sketch.

**83.** Draw a diagram to show that there are two tangent lines to
the parabola that pass through the point . Find
the coordinates of the points where these tangent lines inter-
sect the parabola.

**84.** (a) Find equations of both lines through the point
that are tangent to the parabola .

F

G
*x*
*y*

0 1

1

t

*y*苷 t*共x兲*
*y*苷 *x* *x*

t*共x兲*
*y苷 xt共x兲*

*f*

*y*苷 *f共x兲*
*x*^{2}
*y苷 x*^{2}*f共x兲*

*y*苷 1*⫹ xf 共x兲*
*y*苷 *x*^{2} sx

*f共x兲*

*y苷 2x*^{3}*⫹ 3x*^{2}*⫺ 12x ⫹ 1*

*x*
*f共x兲 苷 x*^{3}*⫹ 3x*^{2}*⫹ x ⫹ 3*

*y苷 6x*^{3}*⫹ 5x ⫺ 3*

*y苷 xsx*
*y苷 1 ⫹ 3x*

*12x⫺ y 苷 1*
*y苷 1 ⫹ x*^{3}

*y*苷 *x*⫺ 1
*x*⫹ 1
*x⫺ 2y 苷 2*

*x⫺ 3y 苷 5*
*y苷 x*^{2}*⫺ 5x ⫹ 4*

*y苷 x ⫺ x*^{2}

共0, ⫺4兲
*y苷 x*^{2}

*G*
*F*
*Q共x兲 苷 F共x兲兾G共x兲*

*P共x兲 苷 F共x兲 G共x兲*

*Q*⬘共7兲
*P*⬘共2兲

共2, ⫺3兲
*y苷 x*^{2}*⫹ x*

*P* 179 193 214 242 262 290 311

*L* 80 106 126 130 119 113 95

98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 138

**SECTION 2.3 DIFFERENTIATION FORMULAS** **139**
(b) Show that there is no line through the point that is

tangent to the parabola. Then draw a diagram to see why.

**85.** (a) Use the Product Rule twice to prove that if , , and are

differentiable, then .

(b) Taking in part (a), show that

(c) Use part (b) to differentiate .

**86.** Find the derivative of each function by calculating the first
few derivatives and observing the pattern that occurs.

(a) (b)

**87.** Find a second-degree polynomial such that ,

, and .

**88.** The equation **is called a differential **
**equation because it involves an unknown function and its**
derivatives and . Find constants such that the
function satisfies this equation. (Differen-
tial equations will be studied in detail in Chapter 9.)

**89.** Find a cubic function whose graph

has horizontal tangents at the points and .
**90.** Find a parabola with equation that has

slope 4 at , slope at , and passes through the point .

**91.** In this exercise we estimate the rate at which the total personal
income is rising in the Richmond-Petersburg, Virginia, metro-
politan area. In 1999, the population of this area was 961,400,
and the population was increasing at roughly 9200 people per
year. The average annual income was $30,593 per capita, and
this average was increasing at about $1400 per year (a little
above the national average of about $1225 yearly). Use the
Product Rule and these figures to estimate the rate at which
total personal income was rising in the Richmond-Petersburg
area in 1999. Explain the meaning of each term in the Product
Rule.

**92.** A manufacturer produces bolts of a fabric with a fixed width.

*The quantity q of this fabric (measured in meters) that is sold is*
*a function of the selling price p ( in dollars per meter), so we*
can write . Then the total revenue earned with selling
*price p is .*

(a) What does it mean to say that and

?

(b) Assuming the values in part (a), find and interpret your answer.

**93.** Let

Is differentiable at 1? Sketch the graphs of and .
*h*
*f* t
*共 fth兲⬘ 苷 f ⬘th ⫹ ft⬘h ⫹ fth⬘*

*f苷 t 苷 h*
*d*

*dx* *关 f 共x兲兴*^{3}*苷 3关 f 共x兲兴*^{2}*f⬘共x兲*

*y苷 共x*^{4}*⫹ 3x*^{3}*⫹ 17x ⫹ 82兲*^{3}
*nth*

*f共x兲 苷 1兾x*
*f共x兲 苷 x*^{n}

*P*共2兲 苷 5
*P*

*P*⬙共2兲 苷 2
*P*⬘共2兲 苷 3

*y⬙ ⫹ y⬘ ⫺ 2y 苷 x*^{2}

*y*
*A, B, and C*
*y*⬙

*y*⬘

*y苷 Ax*^{2}*⫹ Bx ⫹ C*

*y苷 ax*^{3}*⫹ bx*^{2}*⫹ cx ⫹ d*
共2, 0兲
共⫺2, 6兲

*y苷 ax*^{2}*⫹ bx ⫹ c*
*x*苷 ⫺1

⫺8
*x*苷 1

共2, 15兲

*f*共20兲 苷 10,000
*f*⬘共20兲 苷 ⫺350

*R*⬘共20兲

*f共x兲 苷*再^{x}^{x}^{2}^{⫹ 1}^{⫹ 1} ^{if x}^{if x}^{⬍ 1}^{艌 1}

*f*⬘
*f*
*f*

共2, 7兲

*q苷 f 共p兲*
*R共p兲 苷 pf 共p兲*

**94.** At what numbers is the following function differentiable?

Give a formula for and sketch the graphs of and .
**95.** (a) For what values of is the function

differentiable? Find a formula for . (b) Sketch the graphs of and .

**96.** Where is the function differenti -

able? Give a formula for and sketch the graphs of and .
**97.** For what values of and is the line tangent to

the parabola when ?

**98.** (a) If , where and have derivatives of all

orders, show that .

(b) Find similar formulas for and . (c) Guess a formula for .

**99.** Find the value of such that the line is tangent to
the curve .

**100.** Let

Find the values of and that make differentiable everywhere.

**101.** An easy proof of the Quotient Rule can be given if we make
the prior assumption that exists, where . Write

; then differentiate using the Product Rule and solve the resulting equation for .

**102.** A tangent line is drawn to the hyperbola at a point .
(a) Show that the midpoint of the line segment cut from this

tangent line by the coordinate axes is .

(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where

is located on the hyperbola.

**103.** Evaluate .

**104.** Draw a diagram showing two perpendicular lines that intersect
on the -axis and are both tangent to the parabola .
Where do these lines intersect?

**105.** If , how many lines through the point are normal
lines to the parabola ? What if ?

**106.** Sketch the parabolas and . Do you

think there is a line that is tangent to both curves? If so, find its equation. If not, why not?

*P*

lim

*x l 1*

*x*^{1000}⫺ 1
*x*⫺ 1

*y* *y苷 x*^{2}

*c*⬎^{1}2 *共0, c兲*

*y苷 x*^{2} *c*艋^{1}2

*y苷 x*^{2} *y苷 x*^{2}*⫺ 2x ⫹ 2*
*P*

*F苷 f兾t*
*F⬘共x兲*

*f苷 Ft*

*F*⬘

*P*
*xy苷 c*

*f*
*b*

*m*

*f共x兲 苷*再^{x}^{mx}^{2} ^{⫹ b}^{if x}^{if x}^{艋 2}^{⬎ 2}

*y苷 csx*

*F⬙ 苷 f ⬙t ⫹ 2f ⬘t⬘ ⫹ ft⬙*

*F*^{共4兲}
*F*
*F*^{共n兲}

*y*苷^{3}2*x*⫹ 6
*c*

*f* t
*F共x兲 苷 f 共x兲t共x兲*

*x*苷 2
*y苷 ax*^{2}

*2x⫹ y 苷 b*
*b*

*a*

*h共x兲 苷*

### ⱍ

^{x}^{⫺ 1}

### ⱍ

^{⫹}

### ⱍ

^{x}^{⫹ 2}

### ⱍ

*h*⬘
*h*
*h*⬘

*f*⬘
*f*

*f*⬘

*f共x兲 苷*

### ⱍ

^{x}^{2}

^{⫺ 9}

### ⱍ

*x*

t*共x兲 苷*

### 再

^{2x}^{2x}^{2}

^{⫺ x}^{⫺ x}^{2}

^{if x}^{if 0}

^{if x}^{艋 0}

^{艌 2}

^{⬍ x ⬍ 2}t⬘ t t⬘

t 98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 139

**146** **CHAPTER 2 DERIVATIVES**

### If we let , then as , so by Equation 2 we have

### Calculate .

SOLUTION

*Here we divide numerator and denominator by x :*

(by the continuity of cosine and Equation 2)

### lim

*x l 0*

*sin 7x* *4x* 苷 7

### 4 lim

*x l 0*

### 冉 ^{sin 7x} ^{7x} 冊

^{sin 7x}

^{7x}

### 苷 7 4 lim

l 0

### sin

### 苷 7

### 4 ⴢ 1 苷 7 4

**v**

**EXAMPLE 6**

### lim

*x l 0**x cot x*

### lim

*x l 0*

*x cot x*

### 苷 lim

*x l 0*

*x cos x*

*sin x* 苷 lim

*x l 0*

*cos x* *sin x*

*x*

### 苷 lim

*x l 0*

*cos x* lim

*x l 0*

*sin x*

*x*

### 苷 cos 0

### 1 苷 1

*x l 0*

### l 0

* 苷 7x*

**1–16** Differentiate.

**1.** **2.**

**3.** **4.**

**5.** **6.**

**7.** **8.**

**9.** **10.**

**11.** **12.**

**13.** **14.**

**15.** **16.**

**17.** Prove that .

**18.** Prove that .

**19.** Prove that .

*f共x兲 苷 sx sin x*
*f共x兲 苷 3x*^{2}* 2 cos x*

*y苷 2 sec x csc x*
*f共x兲 苷 sin x *^{1}2*cot x*

*y苷 u共a cos u b cot u兲*
*y苷 c cos t t*^{2}*sin t*

*y*苷 sin cos

*y*苷 *x*

2* tan x*

*y*苷 *cos x*
1* sin x*
*f*共兲 苷 sec

1 sec

*y*苷 1* sec x*
*tan x*
*y*苷 *t sin t*

1* t*

*y苷 x*^{2 }*sin x tan x*
*h*共兲 苷 csc cot

*d*

*dx* *共csc x兲 苷 csc x cot x*
*d*

*dx* *共sec x兲 苷 sec x tan x*
*d*

*dx* *共cot x兲 苷 csc*^{2}*x*

t*共t兲 苷 t*^{3}*cos t* t*共t兲 苷 4 sec t tan t*

**20.** Prove, using the definition of derivative, that if ,
then .

**21–24** Find an equation of the tangent line to the curve at the given
point.

**21.** **22.** ,

**23.** , ** _{24.}** ,

**25.** (a) Find an equation of the tangent line to the curve
at the point .

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

**26.** (a) Find an equation of the tangent line to the curve

at the point .

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

**27.** (a) If , find .

; (b) Check to see that your answer to part (a) is reasonable by graphing both and for .

**28.** (a) If , find .

; (b) Check to see that your answer to part (a) is reasonable by graphing both and for .

*f共x兲 苷 cos x*
*f共x兲 苷 sin x*

*y苷 sec x, 共兾3, 2兲* *y苷 共1 x兲 cos x 共0, 1兲*
*y苷 cos x sin x 共, 1兲* *y苷 x tan x 共, 兲*

*y苷 2x sin x* 共兾2, 兲

*y苷 3x 6 cos x* 共兾3, 3兲

*f共x兲 苷 sec x x* *f共x兲*
*f* *f*

### ⱍ

^{x}### ⱍ

^{ 兾2}

*f共x兲 苷 sx sin x* *f共x兲*

*f* *f* 0* x 2*

**2.4** **Exercises**

; Graphing calculator or computer required **1.**Homework Hints available at stewartcalculus.com
98845_02_ch02_p142-151.qk_98845_02_ch02_p142-151 8/16/11 9:55 AM Page 146

**SECTION 2.4 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 147**

**29.** If , find .

**30.** If , find .

**31.** (a) Use the Quotient Rule to differentiate the function

(b) Simplify the expression for by writing it in terms of and , and then find .

(c) Show that your answers to parts (a) and (b) are equivalent.

**32.** Suppose and , and let

and . Find

(a) (b)
**33.** For what values of does the graph of

have a horizontal tangent?

**34.** Find the points on the curve at which
the tangent is horizontal.

**35.** A mass on a spring vibrates horizontally on a smooth
level surface (see the figure). Its equation of motion is

, where is in seconds and in centimeters.

(a) Find the velocity and acceleration at time .

(b) Find the position, velocity, and acceleration of the mass at time . In what direction is it moving at that time?

;** _{36.}** An elastic band is hung on a hook and a mass is hung on the
lower end of the band. When the mass is pulled downward
and then released, it vibrates vertically. The equation of

motion is , , where is measured

in centi meters and in seconds. (Take the positive direction to be downward.)

(a) Find the velocity and acceleration at time . (b) Graph the velocity and acceleration functions.

(c) When does the mass pass through the equilibrium position for the first time?

(d) How far from its equilibrium position does the mass travel?

(e) When is the speed the greatest?

**37.** A ladder 6 m long rests against a vertical wall. Let be the
angle between the top of the ladder and the wall and let be
the distance from the bottom of the ladder to the wall. If the
bottom of the ladder slides away from the wall, how fast does

change with respect to when ?
*H*共兲 苷 sin *H*共兲 and H 共兲
*f共t兲 苷 csc t* *f* 共兾6兲

*f共x兲 苷* *tan x* 1
*sec x*

*f共x兲*

*sin x* *cos x* *f共x兲*

*f*共兾3兲 苷 4 *f*共兾3兲 苷 2
t*共x兲 苷 f 共x兲 sin x* *h共x兲 苷 共cos x兲兾f 共x兲*

t共兾3兲 *h*共兾3兲

*x* *f共x兲 苷 x 2 sin x*

*y苷 共cos x兲兾共2 sin x兲*

*x共t兲 苷 8 sin t* *t* *x*

*t*
*t*苷 2兾3

x x

0

equilibrium position

*s苷 2 cos t 3 sin t t 0* *s*
*t*

*t*

*x*

*x* 苷 兾3

**38.** An object with mass is dragged along a horizontal plane
by a force acting along a rope attached to the object. If the
rope makes an angle with the plane, then the magnitude of
the force is

where *is a constant called the coefficient of friction.*

(a) Find the rate of change of with respect to . (b) When is this rate of change equal to 0?

; (c) If , , and , draw the graph

of as a function of and use it to locate the value of for which . Is the value consistent with your answer to part (b)?

**39–48** Find the limit.

**39.** **40.**

**41.** **42.**

**43.** **44.**

**45.** **46.**

**47.** **48.**

**49–50** Find the given derivative by finding the first few deriva-
tives and observing the pattern that occurs.

**49.** **50.**

**51.** Find constants such that the function

satisfies the differential equation .

**52.** (a) Evaluate .

(b) Evaluate .

; (c) Illustrate parts (a) and (b) by graphing .
**53.** Differentiate each trigonometric identity to obtain a new

(or familiar) identity.

(a) (b)

(c) lim

*x l 0*

*sin 3x*

*x* lim

*x l 0*

*sin 4x*
*sin 6x*
lim

*t l 0*

*tan 6t*

*sin 2t* lim

l 0

cos 1 sin lim

*x l 0*

*sin 3x*

*5x*^{3}* 4x* lim

*x l 0*

*sin 3x sin 5x*
*x*^{2}

lim l 0

sin

tan lim

*x l 0*

sin*共x*^{2}兲
*x*

lim

*x l*兾4

1* tan x*

*sin x cos x* lim

*x l 1*

sin*共x 1兲*
*x*^{2}* x 2*

*d*^{99}

*dx*^{99}*共sin x兲* *d*^{35}

*dx*^{35}*共x sin x兲*

*A and B*
*y苷 A sin x B cos x*
*y y 2y 苷 sin x*

lim

*x l**x sin *1
*x*
lim

*x l 0**x sin *1
*x*

*y苷 x sin共1兾x兲*

*tan x*苷 *sin x*

*cos x* *sec x*苷 1

*cos x*
*sin x cos x 苷* 1* cot x*

*csc x*
*m*

*F*苷 *
mt*

sin cos

*F*

*m*苷 20 kg t 苷 9.8 m兾s^{2}
苷 0.6

*F*

*dF兾d 苷 0*
98845_02_ch02_p142-151.qk_98845_02_ch02_p142-151 8/16/11 9:55 AM Page 147