*t* *D(t)*

1994 414.0

1996 469.5

1998 467.3

2000 456.4

2002 442.3

*D(t) - D(1998)*

*t* *t -* 1998

1994 13.3

1996 -1.1

2000 -5.5

2002 -6.3

• A NOTE ON UNITS

Theunits for the*average* rate ofchange *!:iD/!:it*
are theunits for!:iDdivided by the units for*!:it,*
namely, billions of dollars peryear.The instan-
taneous rateofchange isthelimit of the aver-
age rates of change, soit ismeasured in the
same units: billions ofdollars peryear.

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

**EXAMPLE** 7 *Let D(t) 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 1994to 2002. Interpret and estimate the value of D'(l998).

SOLUTION The derivative D'(l998) means the rate of change ofDwith respect to*t*when
*t*= 1998, that is, the rate of increase ofthe national debt in 1998.

According to Equation 5,

D'(l998) = lim

*D(t)* *-*

D(l998)
Hl998 *t -* 1998

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 D'(l998) lies some- where between -1.1 and - 5,5 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 ofchange 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 betoplot the debt function and estimate the slope of the tan-

gent line when *t*= 1998. 0

In Examples 3, 6, and 7 we saw three specific examples of rates of change: the veloci- tyof an object isthe rate of change of displacement with respect totime; marginal cost is the rate of change of production cost with respect to the number of items produced; the rate of change ofthe debt with respect to time isof interest in economics. Here is a small sample ofother rates ofchange: In physics, the rate ofchange of work with respect to time is called power. Chemists who study a chemical reaction are interested in the rate of change in theconcentration of a reactant with respect to time (called the rate of reaction).

A biologist isinterested in the rate ofchange of the population of a colony of bacteria with respect totime. In fact, the computation of rates of change is important in all of the natu- ralsciences, inengineering, and even in the social sciences. Further examples will be given in Section 3.7.

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

### -0

^{EXERCISES}

I. Acurve has equation *y* =*f(x).*

(a) Write an expression for the slope ofthe secant line
through the points *P(3,f(3»* *and Q(x,f(x».*

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

### ffi

^{2.}

^{Graph the}

^{curve y}

^{=}

^{sin}

^{x}^{in the}

^{viewing}rectangles [-2,2]

by[-2,2], [ -I, 1] by[ -1, 1], and [-0.5,0.5] by [-0.5,0.5]'

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
*y* =*4x -* *x**2* atthe point (1, 3)

(i) using Definition 1 (ii) using Equation 2 (b) Find anequation ofthe tangent line in part (a).

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

4. (a) Find the slope of the tangent line tothe 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.

5-8 Find anequation ofthe tangent line to the curve at the given point.

*~;]y=--,*

### x-I

(3,2) 6.*y=2x*

*3*

*-5x,*

*(-1,3)*

*x-2*

*2x*

*8.* *Y*= ( )2' (0, 0)

*x*

### +

1[!](a) Find the slope of the tangent tothecurve
*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).

(c) Graph the curve and both tangents on acommon screen.

10. (a) Find the slope ofthe tangent to the curve *y* ^{=}

### 1/.);

at the point where x = a.(b) Find equations of the tangent lines at the points (1, 1) and (4,

### D-

(c) Graph the curve and both tangents on acommon screen.

**II.** (a) A particle starts bymoving to the right along a horizontal
line; the graph of itsposition function is shown. When is
the particle moving to the right? Moving tothe left?

Standing still?

(b) Draw agraph ofthe velocity function.

ers) 4

2 /

\

### /

\

### /

0 2 4 6 t

12. Shown are graphs ofthe position functions of two runners, A and B, who run a 100-m race and finish inatie.

s (meters)

80 */j*

### ~ /

40

,/

### /

/' /"8

### IL

*V*...-

0 4 8 12 t

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

(c) At what time dothey have the same velocity?

13. If a ball isthrown into the airwith avelocity of 10 mis, its
height (in meters) after t seconds isgiven by y = lOt - *4.9t**2•*

Find the *velocity when t*^{=}2.

14. If a rock isthrown upward on the planet Mars with a velocity
of 10 mis, itsheight (in meters) after t seconds is given by
*H*= lOt - *1.86t**2•*

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

(b) Find the velocity ofthe rock when *t*=*a.*

(c) When willthe rock hit thesurface?

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

15. The displacement (in meters) of aparticle moving ina
straight line is given bythe equation of motion s^{=} *l/t**2,*

where *t*is measured in seconds. Find the velocity ofthe
particle at times *t*=*a,t* =1,*t* =2,*and t*=3.

16. The displacement (in meters) of aparticle moving in a
straight line is given by s^{=}*t**2* *-* *8t*

### +

18,where tismea- sured inseconds.(a) Find the average velocity over each time interval:

(i) [3,4] (ii) [3.5,4]

(iii) [4,5] (iv) [4,4.5]

(b) Find the instantaneous velocity when *t* =4.

(c) Draw the graph ofs asa function of *t*and draw the secant
lines whose slopes are theaverage velocities in part (a)
and the tangent line whose slope isthe instantaneous
velocity in part (b).

### ITLl

^{For}the function

*9*whose graph isgiven, arrange the follow- ing numbers in increasing order and explain your reasoning:

### 1m

(a) Find an equation of the tangent line tothe graph of*y*=

*g(x) atx*=5 if g(5) = -3 and

*g'(5)*=4.

(b)If thetangent line to y =*f(x)* at(4,3) passes through the
point (0, 2), find *f(4)* and1'(4).

### [!!J

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

20. Sketch the graph ofa function *9 for*which g(O)

### =

*g'(O)*

### =

0,*g'(*-1) = -1, g'(1) =3,and

*g'(2)*= 1.

21. If*f(x)* ^{=}*3x**2* *-* *5x,*find*1'(2) and use it to find an equation*
of the tangent line to the parabola y =*3x**2* *-* *5x at the*
point (2,2).

22. If*g(x)*

### =

*1 - x*

*3,*

*find g'(O)*and use it to find an equation of the tangent line to the curve y =1

*- x*

*3*at the point (0, 1).

~ (a) *If F(x)* ^{=}*5x/(t*

### +

*x*

*2),*

*find F'(2)*and use it tofind an equation of the tangent line tothe curve y =

*5x/(t*

### +

*x*

*2)*

atthe point (2, 2).

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

24. (a) If G(x) ^{=}*4x**2* *-* *x**3,* *find G'(a) and* use it to find equa-
tions of the tangent lines to the curve *y* =*4x**2* *-* *x**3* at
the points (2,8)and (3,9).

~ (b) Illustrate part (a) bygraphing the curve and the tangent lines onthe same screen.

25-30 Find *f'(a).*

*25. f(x)* = 3- *2x*

### +

*4x*

*2*

*2t*

### +

1*lmf(t)*

=--
*t*

### +

3*26.* *f(t)* =*t**4* *-* *5t*
*x**2*

### +

1*28.*

*f(x)*=--

*x-2*
1

*29.* *f(x)* = ~
yX

### +

231-36 Each limit represents the derivative ofsome function

*f*

at
some number *a.*State such an

*f*

and *a*in each case.

(l

### +

*h)'O -*1 31. Iim---

,,~O *h*

32. lim ---~-2

,,~O *h*

2' - 32
**33.lim---**

,~5 *x* *-* 5

tan*x -* 1
**34.** **Hm ----**

,~",/4 *x* *-* 71/4

I'>Fl *cas(*7T

### +

*h)*

### +

1~lim---

,,~O *h*

*t**4*

### +

*t*

*- 2*

**36.lim----**

1~1 *t* *-* 1

37-38 A particle moves along astraight linewith equation of
motion s^{=}*f(t),* where sismeasured in meters and t inseconds.

Find the velocity and thespeed when t =5.

### m

A warm can ofsoda pop isplaced in a cold refrigerator.Sketch the graph ofthetemperature ofthe soda asa function of time. Isthe initial rate ofchange oftemperature greater or less than the rate of change after an hour?

40. Aroast turkey istaken from anoven when itstemperature hasreached 85°C and isplaced on a table in a room where the temperature is24°C. The graph shows how the tempera- ture ofthe turkey decreases and eventually approaches room

temperature. (In Section 7.5 we will be able to use Newton's
Law of Cooling to find anequation for *T* as a function of
time.) By measuring the slope of the tangent, estimate the
rate of change ofthe temperature after an hour.

(0e) 80

### ~

•••...*P*
40

-...

### ~

r--0 30 60 90 120 150 *t*

41. The table shows the estimated percentage *P*of the population
of Europe that use cell phones. (Midyear estimates are given.)

Year 1998 1999 2000 2001 2002 2003

*P* 28 39 55 68 77 83

(a) Find theaverage rate of cell phone growth

(i) from 2000 to2002 (ii) from 2000 to 200 1 (iii) from 1999 to2000

In each case, include the units.

(b) Estimate the instantaneous rate of growth in 2000 by taking theaverage of two average rates of change. What are itsunits?

(c) Estimate the instantaneous rate of growth in 2000 bymea- suring the slope of a tangent.

42. The number *N of*locations ofapopular coffeehouse chain is
given in the table. (The numbers of locations as of June 30
are given.)

Year 1998 1999 2000 2001 2002

*N* 1886 2135 3501 4709 5886

(a) Find the average rate of growth

(i) from 2000 to 2002 (ii) from 2000 to 2001 (iii) from 1999 to2000

Ineach case, include the units.

(b) Estimate the instantaneous rate of growth in 2000 by taking the average of two average rates of change. What areitsunits?

(c) Estimate the instantaneous rate of growth in 2000 by mea- suring the slope ofa tangent.

### I1!l

The cost (in dollars) ofproducing xunits of a certain com- modity is C(x)^{=}5000

### +

*lOx*

### +

*0.05x*

*2•*

(a) Find the average rate ofchange of C with respect to x when the production level ischanged

(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 iscalled the marginal *cost.*

Its significance will be explained in Section 3.7.)

Ifacylindrical tank holds 100,000 liters of water, which can
be drained from the bottom ofthe tank inan hour, then Torri-
celli's Law gives the volume *V*of water remaining in the tank
after *t*minutes as

*Vet)*= 100,000 ( 1- ;0

*Y*

Find the rate at which the water isflowing out of the tank (the
instantaneous rate of change of *V*with respect to*t)* asafunc-
tion *of t.*What are itsunits? 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 asentence
or two. At what time isthe flowrate thegreatest? The least?

**45. The cost of producing x**kilograms of gold from a new gold
mine isC ^{=}*f(x)* dollars.

(a) What is themeaning of the derivative *1'(x)?*What are its
units?

(b) What does the statement *1'(50)* ^{=}36 mean?

(c) Doyou think the values of*1'(x)* will increase or decrease
in the short term? What about thelong term? Explain.

**46. The** number ofbacteria after *t*hours in a controlled labora-
tory experiment is*n*^{=}*f(t).*

(a) What is the meaning ofthe derivative *1'(5)?* What are its
units?

(b) Suppose there isan unlimited amount ofspace and nutri-
ents for the bacteria. Which doyou think islarger, *1'(5)*
or 1'(IO)? If the supply ofnutrients islimited, would that
affect your conclusion? Explain.

**47.** Let T(t) be the temperature (in °C) in Seoul *t*hours after noon
on August 21,2004. The table shows values ofthis function
recorded every two hours. What is the meaning of*T'(6)?*

Estimate itsvalue.

*t* 0 2 4 6 8 10

*T* 34.4 35.6 38.3 32.8 26.1 22.8

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

(a) What is themeaning of the derivative *1'(8)?* What are its
units?

(b) *Is 1'(8) positive* or negative? Explain.

**49. The quantity of oxygen that can dissolve** in water depends on
thetemperature ofthe water. (So thermal pollution influences

the oxygen content of water.) The graph shows how oxygen
solubility Svaries asa function of the water temperature *T.*

(a) What is the meaning ofthe derivative *S'(T)?* What are its
units?

(b) Estimate the value ofS'(l6) and interpret it.

### s

*(mg/L)*
16

Adaptedfrom*Environmental* *Science: Living Within the System*
*of Nature.* 2d ed..byCharles E.Kupchella. ©1989.Reprinted by
permission of Prentice-Hall,Inc.. Upper SaddleRiver.NJ.

The graph shows the influence of thetemperature *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 ofS'(l5) and S'(25) and interpret them.

### s

*(em/s)*
20

{ . J

*X SIn-*

### [ill

^{f(x)}^{=}

^{0}

^{x}{ . 1

X2SIn-

**52.***f(x)* = 0 *x*

for all values of

*x.*

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

*of x,*

We can interpret the third derivative physically in the case where the function is the position function s =

*set)*

of an object that moves along a straight line. Because
Sill =*(s")'* =*a',* thethird derivative ofthe position function isthe derivative of theaccel-
eration function and is called the**jerk:**

*.* *da* *d**3**s*

### } = *--:it* ^{=}

^{dt}

^{3}Thus thejerkj isthe rate of change of acceleration. It is aptly named because a largejerk means a sudden change in acceleration, which causes an abrupt movement in avehicle.

Wehave seen that one application of second and third derivatives occurs inanalyzing
the motion of objects using acceleration and jerk. We will investigate another applica-
tion of second derivatives in Section 4.3, where we show how knowledge of *f" gives us*
information about the shape of the graph of

*f.*

**In Chapter 12**we will see how second and higher derivatives enable ustorepresent functions as sums of infinite series.

### @

^{EXERCISES}

**1-2** Use the given graph toestimate the value of each derivative.

Then sketch the graph of

### f'. rn

^{Match}

^{the}

^{graph}

^{of}

^{each}

^{function}

^{in}

^{(a)-(d)}with the graph of

itsderivative in I-IV. Give reasons for your choices.

I. (a) *1'(-3)*
(c)*1'( -1)*
(e)1'(1)
(g) *1'(3)*

2. (a)*1'(0)*
(c)*1'(2)*
(e) *1'(4)*

(b) *1'(-2)*
(d) *1'(0)*
(f) *1'(2)*

(b) *1'(1)*
(d) *1'(3)*
(f) *1'(5)*

**4-11** Trace orcopy thegraph of the given function

*f.*

(Assume
that the axes have equal scales.) Then use the method ofExample
1 tosketch the graph of*l'*

below it.
4.

12. Shown isthe graph of the population function P(t) foryeast cells inalaboratory culture. Use the method ofExample 1 to graph the derivative P'(t). What does the graph of P' tell us about the yeast population?

13. The graph shows how the average age of first marriage of Japanese men has varied inthe last half ofthe 20th century.

Sketch the graph of the derivative function *M'(t).* During
which years was the derivative negative?

## -t

^{l}

^{1960}

^{I}

^{1970}

^{I}

^{1980}

^{I}

^{1990}

^{I}

^{2000}^{I}

^{•}

^{t}14. Make acareful sketch ofthe graph of the sine function and below it sketch the graph of itsderivative in the same manner as in Exercises 4-11.Can you guess what the derivative of the sine function is from itsgraph?

### ffi ~

^{Let f(x)}

^{=}

^{x}

^{2•}(a) Estimate the values off'(O),

*I'm,*

*1'(1),*and

*1'(2)*by using agraphing device to zoom inon the graph

*off*(b) Use symmetry todeduce the values

*off'(*

*-0,1'(*

*-I),*

andf'(-2).

(c) Use the results from parts (a) and (b) toguess a formula
*for f'(x).*

(d) Use the definition of aderivative toprove that your guess in part (c)is correct.

### ffi

^{16.}

^{Letf(x)}^{=}

^{x}

^{3.}(a) Estimate the values *of 1'(0),*

*I'm,*

1'(1), 1'(2), andf'(3)
byusing agraphing device to zoom inon thegraph *off*(b) Use symmetry todeduce the values

*off'(*

### -n

^{1'(}^{-1),}*1'(*-2), and*1'(-3).*

(c) Use thevalues from parts (a) and (b)to graph

### 1'.

(d) Guess aformula for f'(x).

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

**17-27** Find the derivative of the function using the definition of
derivative. State the domain of thefunction and the domain ofits
derivative.

*17. f(x)* =*~x - ~*
*19.* *f(t)* =*5t* *-* *9t**2*

**21.** *f(x)* =*x**3* *-* *3x*

### +

5*18.f(x)* =*mx*

### +

*b*

*20. f(x)*=

*1.5x*

*2*

*-*

*X*

### +

3.7*22.*

*f(x)*=

*x*

### + *j;*

*24. f(x)* =--*3+x*
1 - *3x*

### lID

*G(t) =*--

*4t*

*t*

### +

11
*26.* *g(t)* ^{=}

*Jt*

**28.** (a) Sketch the graph off(x) = ~ by starting with the **35.**

graph of*y*

### = j";

and using the transformations of Sec- tion 1.3.(b) Use the graph from part (a) tosketch the graph of f'.

(c) Use thedefinition of aderivative tofind1'(x). What are the domains of

*f*

and *1'?*

~ (d) Use agraphing device to graph

*l'*

and compare with your
sketch in part (b).
**29.** (a) If f(x) =*x**4*

### +

*2x, find1'(x).*

~ (b) Check tosee that your answer topart (a) isreasonable by comparing the graphs of

*f*

and f'.
**30.** (a) If*f(t)* =*t**2* *-*

*JI,*

find1'(t).
~ (b) Check to see that your answer topart (a) is reasonable by comparing the graphs of

*f*

and f'.
**31.** The unemployment rate *Vet)*varies with time. The table gives
the percentage of unemployed in the Australian labor force
measured at midyear from 1995 to 2004.

*t* *Vet)* *t* *Vet)*

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

(a) What isthe meaning of *V'et)?* What are its units?

(b) Construct a table of values for *V'et).*

**32.** Let*pet)* be the percentage ofthe population ofthe Phillipines
over the age of60at time t. The table gives projections of
values of this function from 1995 to 2020.

*t* *pet)* *t* *pet)*

1995 5.2 2010 6.7

2000 5.5 2015 7.7

2005 6.1 2020 8.9

(a) What isthe meaning of *P'(t)?* What are itsunits?

(b) Construct atable ofestimated values for P'(t).

(c) Graph *P and P'.*

**33-36** The graph of

*f*

is given. State, with reasons, the numbers
at which *f*

isnot differentiable.
**~** **37.** Graph the functionf(x) =*x*

### + *M.*

Zoom in repeatedly,
first toward the point (-1, 0) and then toward the origin.
What isdifferent about the behavior of

*f*

in the vicinity of
these two points? What do you conclude about the differen-
tiabilityoff?
**~** **38. Zoom in**toward the points (1, 0), (0, I), and ( - 1,0) on the
graph of the function g(x) = (x*2* *-* 1?/3.What do you
notice? Account for what you see in terms ofthe differen-
tiability of*g.*

~ The figure shows the graphs of

*f,* *1',*

and ### 1".

Identify each curve, and explain your choices.**40.** The figure shows graphs of f,

*1',*

*f", and*

*1"'.*

Identify each
curve, and explain your choices.
**41. The figure** shows the graphs of three functions. One isthe
position function of a car,one isthe velocity of the car, and
one isitsacceleration. Identify each curve, and explain your
choices.

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

**~** **43-44** Use the definition of aderivative to findf'(x) and

*rCx).*

Then graph *f,*

*1',*

and *r*

^{on}

^{a}common screen and check tosee if your answers arereasonable.

~~ *If f(x)* =*2x**2* *-* *x3,* findf'(x), *r(x),flll(x),* and f(4)(X).

Graph f,

*1',*

*f",*and

*fill on*acommon screen. Are the graphs consistent with the geometric interpretations ofthese derivatives?

**46. (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 accel-
eration at *t*=10seconds?

(b) Use the acceleration curve from part (a)to estimate the jerk at t = 10seconds. What are the units for jerk?

**47. Letf(x)**^{=}

### $.

(a) If*a* ¥- 0, use Equation 3.1.5 to find*f'(a).*

(b) Show thatf'(O) does not exist.

(c) Show that*y* =

### $

has avertical tangent line at(0,0).(Recall the shape ofthe graph of

*f.*

SeeFigure 13inSec-
tion 1.2.)
**48.** (a) If g(x) ^{=}*X**2/**3, show* *that g'(O)* does not exist.

(b) If a¥- 0,find*g'(a).*

(c) Show that *y* ^{=}*X2/3* has a vertical tangent line at (0,0).

~ (d) Illustrate part (c) bygraphing *y* =*X2/3•*

### lID

Show that the function*f(x)*=

### I

*x*

*-*

### 61

isnot differentiable at 6.Find a formula for*I'*

and sketch itsgraph.
**50. Where** isthegreatest integer function *f(x)* =*[x ~*not differ-
entiable? Find a formula for

*I'*

and sketch its graph.
### [!.J

^{(a)}Sketch the graph ofthe function

*f(x)*=

*x*

### I

*x*

*I·*

(b) Forwhat values of*x* is

*f*

differentiable?
(c) Find a formula for

*1'.*

**52. The** **left-hand and right-hand** derivatives of

*f*

at*a are*defined by

*f'-(a)* = Jim *f(a*

### +

*h)-*

*f(a)*

*h~O-* *h*

*f'* *( )* *-* I' *f(a*

### +

*h)*

*-*

*f(a)*

*+* *a* *-* 1m

*h~O+* *h*

ifthese limits exist. Then *f'(a)* exists ifand only ifthese one-
sided derivatives exist and are equal.

(a) *Find f'-(4)* andf~(4) for the function

### 1

^{0}

*5-x*

*f(x)* = _I_

*s-x*

if *x* *~* 0

if 0

### <

*x*

### <

4(b) Sketch the graph of

*f.*

(c) Where is

*f*

discontinuous?
(d) Where is

*f*

not differentiable?
**53. Recall that a function f is called** *even if f( -x)* =*f(x)* for all
*x* in its domain and *odd if f(* *-x)* =*-f(x)* for all such *x.*

Prove each of the following.

(a) The derivative ofan even function isan odd function.

(b) The derivative ofan odd function is aneven function.

**54.** When you turn on ahot-water faucet, the temperature *T*of
the water depends on how long the water has been running.

(a) Sketch apossible graph *of T as a function* ofthe time *t*
that haselapsed since the faucet wasturned on.

(b) Describe how the rate of change *of T with respect tot*
*varies as t increases.*

(c) Sketch a graph of the derivative of*T.*

**55. Let**

*e*

^{be}the tangent line to the parabola

*y*=

*x*

*2*at the point (I, I).The

*angle of inclination*of

*e*

^{is}

^{the}

^{angle}

^{4>}

^{that}

*e*

makes with the positive direction ofthe x-axis. Calculate 4>

correct tothe nearest degree.

**TABLE OF**

**DIFFERENTIATION** **FORMULAS**

### §

^{EXERCISES}

**1-20** Differentiate the function.

I. *f(x)* ^{=} 186.5
*3.* *f(x)* =*5x -* 1
*5.* *f(x)* ^{=}*x**3* *-* *4x*

### +

61-3'12 1

### 20

^{(l}

^{+}

^{12)2}= --

^{4}

We use the point-slope form to write an equation ofthe tangent line at (1,

### D:

The slope of the normal line at(1,

### D

^{is}

^{the}negative reciprocal of

### -i,

namely 4, so an*x* equation is

**EXAMPLE** **13** At what points on the hyperbola *xy* = 12 isthe tangent line parallel tothe
line 3x

### +

*y*=O?

SOLUTION Since *xy* =12 can be written as*y* = l2/x, we have

*dy* *d* 12

- =

### 12-

(X-I) =*12(-x-*

*= --*

^{2)}*dx* *dx* *x**2*

Let the x-coordinate of one ofthe points inquestion be

*a.*

Then the slope of the tangent
line atthat point is -12/ a2. This tangent line will be parallel tothe line 3x ### +

^{y}^{=}

^{0,}

^{or}

*y* = *-3x,* if ithas the same slope, that is, -3. Equating slopes, we get

--=12 -3
*a**2*

Therefore the required points are (2,6) and (-2, -6). The hyperbola and the tangents

are shown inFigure 6.

### 0

*-d* *(c)* =0

*dx*

*-d* *(x")* =*nx,,-I*
*dx*

*gl'* *-* *fg'*
*g2*

*8. f(t)* =*it**6* *-* *3t**4*

### +

*t*

*10.* *R(t)* =*5t**-3/5*

*4. F(x)* ^{=}*-4x* 10

*6. h(x)* =*(x* *- 2)(2x*

### +

3)

_{12.}

_{R(x)}*JW*

=-7-
*X*

1
*14.**f(t)* =

*Jt* *-* *Jt*

*16. R(x)* =-7-

### JiG

*X*

*18. g(u) =*

*J2*

^{u}### + J3;;

*20.* *v* ^{= (}

*JX* + ~ *y*

21. Find the derivative of y =*(x**2* + 1)(x3 + 1) in two ways: by
using the Product Rule and by performing the multiplication
first. Do your answers agree?

*()* *x* *-* *3xJX*

*Fx=*

*JX*

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

23-42 Differentiate.

23. *Vex) = (2x**3*

### +

*3)(x*

*4*

*-*

*2x)*

*24.* *feu)* ^{=}*(u-**2*

### +

*U-3)(U5*

*-*

*2u*

*2)*

### [ill

*F(y)*=

### (;2 - :4

^{)(y}^{+}

^{5y3)}*26. y* ^{=}

*JX*

^{(x}*1)*

^{-}*3x -*1

*27.*

*g(x)*=--

*2x*

### +

1*28. f(t)* ^{=}--2*2t*
4

### +

*t*

*x*

### +

1**30.***Y=-3---*

*x* *+x-2*
*x**3*

*29.* *y* =---2

*I-x*

*v*

^{3}

^{-}*2vJi;*

31. *y=----*
*v*

*t**3*

### +

*t*

*32. Y*=

*t*

*4*

*-*2

*t* *-*

*Jt*

*34. get) =* --1/-3 -

*t*

*B*

### C

*36. y = A*

### +- +-

*X* *x**2*

*37 Y**=*

• 1

### + v'7

*38. y=---ex*
1

### +

*ex*

*I1IJf(x)* =-- *x*
*x*

### +-

e*x*

*ax*

### +

*b*

*42. f(x)*=--

*ex*

### +

*d*

43. The general polynomial *of degree n has* the form
*P(x)* =*a"x"*

### +

*a,,-Ix,,-I*

### + ... +

*a2x*

*2*

### +

*alx*

### +

*ao*

### ffi

^{44-46}

*Find f'(x).*Compare the graphs of

*f and f'*and use them to explain why your answer is reasonable.

*44. f(x)* =*x/(x2* *-* 1)

*45. f(x)* =3Xl5 - *5x**3*

### +

3 1*46. f(x)* =*x*

### + -

*x*

### ffi

47. (a) Use a graphing calculator orcomputer to graph the func-*tion f(x)*=

*x*

*4*

*-*

*3x*

*3*

*-*

*6x*

*2*

### +

*7x*

### +

30 in the viewing rectangle [-3, 5Jby [-10,50].(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of

### f'.

(See Example 1 in Section 3.2.)*(c) Calculate f'(x)* and use this expression, with a graphing
device, to graph

### f'.

Compare with your sketch in part (b).### ffi

48. (a) Use a graphing calculator orcomputer to graph the func-*tion g(x)*=

*X2/(X2*

### +

1) in the viewing rectangle [-4, 4J by [ -1, 1.5].(b) Using the graph in part (a) to estimate slopes, make a
rough sketch, by hand, of the graph *of g'.*(See Example 1
in Section 3.2.)

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

49-50 Find an equation of the tangent line to the curve at the given point.

*49. y* =--,*2x* (1, 1)
*x*

### +

151. (a) The curve *y* ^{=}1/(1

### +

*x*

*2)*iscalled awitch of Maria Agnesi. Find an equation of the tangent line to this curve at the point

### (-1, n

### ffi

(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.~ *(a) The curve y = x/(l*

### +

*x*

*2)*is called a serpentine.

Find an equation of the tangent line to this curve at the point (3, 0.3).

### ffi

(b) Illustrate part (a) by graphing thecurve and thetangent line on the same screen.53-56 Find equations of the tangent line and normal line to the curve at the given point.

*54.* *Y*^{=} (1

### +

*2X)2,*(1,9)

*56.*

*y*= --,

*JX*

(4,0.4)
*x*

### +

1*3x*

### +

1*55. y =*-2--' (1,2)
*x*

### +

1*x**2*

*59.* *f(x)* = I

### +

*2x*

I
*60.* *f(x)* = 3 - *x*

### [MJ

The equation ofmotion of aparticle is s = t*3*

*-*

*3t,*where s isin meters and

*t*is inseconds. Find

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

(c) the acceleration when the velocity isO.

62. The equation of motion ofaparticle is

s=*2t**3* *-* *7t**2*

### +

*4t*

### +

I, where sisinmeters and t isin seconds.(a) Find the velocity and acceleration as*functions of t.*

(b) Find the acceleration after Is.

~ (c) Graph the position, velocity, and acceleration functions onthe same screen.

~ Suppose that f(5) = 1,1'(5) =*6, g(5)* = -3, *and g'(5)* =2.

Find the following values.

*(a) (fg)'(5)* (b) (f/g)'(5)

*(c) (g/f)'(5)*

64. Find h'(2), given thatf(2) = -3, *g(2)* =*4,1'(2)* = -2,
and *g'(2)* =7.

*(a) hex)* =*5f(x)* *- 4g(x)*

(c) *hex)* =

*~i:~*

^{(b) hex)}

^{(d)}

^{hex) =}^{=}

*I*

^{f(x)}

^{g(x)}### +

^{g(x)}f(x)67. If

*f*

and *9*are the functions whose graphs areshown, let

*u(x)*=

*f(x)g(x)*and vex) =

*f(x)/g(x).*

(a) *Find u'(l).* (b) Find *v'(5).*

68. Let P(x) ^{=}*F(x)G(x)* and Q(x) ^{=}*F(x)/G(x),* where *F and* G
are the functions whose graphs are shown.

(a) *Find P'(2).* (b) Find Q'(7).

*y*

### \ ..-

### \

^{'\.}

_{/}

^{F_}### -

./*V*

### "-...

./

./ GI'---.

f-l '-...

0 1

69. If*9*isa differentiable function, find an expression for the
derivative of each of the following functions.

*x* *g(x)*

*(a) y* =*xg(x)* (b) y =-(-) (c) *y* =-

*9* *x* *x*

### I7QJ

^{If}

*f*

is a differentiable function, find an expression for the
derivative of each of the following functions.
(b) *y* = *f(x)*
*x**2*

*x**2*

*(c)y=-*

*f(x)*

I

### +

*xf(x)*

(d) y =

### j;

### ITIJ

Find the points on the curve*y*=

*2x*

*3*

### +

*3x*

*2*

*-*l2x

### +

I where the tangent ishorizontal.72. Forwhat values of*x* does the graph of

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

### +

*3x*

*2*

### +

*X*

### +

3have ahorizontal tangent?73. Show that the curve *y* =*6x**3*

### +

*5x*

*-*3 has no tangent line with slope 4.

74. Find anequation of the tangent line to the curve *y* ^{=}*x ~*
that isparallel to the line*y* =I

### +

*3x.*

75. Find equations of both lines that are tangent to the curve
*y* = I

### +

*x*

*3*and areparallel to the line 12x -

*y*= I.

76. Find equations of the tangent lines to the curve x-I

*y=~*

77. Find an equation of the normal line to the parabola
*y* =*x**2* *-* *5x*

### +

4 that isparallel to the line x -*3y*= 5.

78. Where does the normal line tothe parabola *y* ^{=}*x* *-* *x**2* atthe
point (I, 0)intersect the parabola asecond time? Illustrate
with a sketch.

~ Draw adiagram to show that there are two tangent lines to
the parabola *y* =*x**2* that pass through the point (0, -4). Find
the coordinates of the points where these tangent lines inter-
sect the parabola.

*80.* (a) Find equations of both lines through the point (2, - 3) that
are tangent to the parabola *y* =*x**2*

### +

*x.*

(b) Show that there isno line through the point (2,7)that is tangent to the parabola. Then draw a diagram to seewhy.

### lID:J

(a) Use the Product Rule twice to prove that if*f,*

*g,*and

*hare*

differentiable, then *(fgh)'* =*f'gh*

### +

*fg'*

*h*

### +

*f9h'.*

(b) Taking

*f*

=*9*=

*h in part*(a), show that

*~* *[J(X)J3* *= 3[J(x)J**2**1'(x)*

(c) Use part (b) to differentiate y = *(x**4*

### +

*3x3*

### +

*l7x*

### +

82)3.82. *Find the nth derivative of each* function bycalculating the first
few derivatives and observing the pattern that occurs.

(a) f(x)

### =

*x"*(b)

*f(x)*

### =

*llx*

83. Find a second-degree polynomial *P*such that *P(2)* =5,
*P'(2)* =3,and *P"(2)* =2.

84. The equation *y"*

### +

*y' -*

*2y*=

*x*

*2*is called adifferential equation because it involves an unknown function

*y and its*derivatives y' and

*y". Find constants A,B,*and Csuch that the function

*y*=

*Ax*

*2*

### +

*Bx*

### +

Csatisfies this equation. (Differen- tial equations will bestudied in detail inChapter 10.)85. Find a cubic function *y* ^{=}*ax**3*

### +

*bx*

*2*

### +

*ex*

### +

*d whose*graph has horizontal tangents at the points (-2,6) and (2,0).

86. Find a parabola with equation *y* =*ax**2* + *bx* + ethat has
slope 4*at x* = I, slope - 8 at*x* = -I, and passes through the
point (2, 15).

87. 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 thepopulation wasincreasing at roughly 9200 people per year.The average annual income was $30,593 per capita, and this average was increasing at about $1400 peryear (a little above the national average ofabout $1225 yearly). Use the Product Rule and these figures toestimate therate at which total personal income was rising inthe Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.

88. Amanufacturer produces bolts ofa fabric with a fixedwidth.

The quantity *q*of this fabric (measured in meters) that issold is
afunction of the selling price*p* (in dollars per meter), sowe
can write *q* =*f(p).* Then the total revenue earned with selling
price *pis* *R(p)*

### =

*pf(p).*

(a) What does it mean tosay that f(20) = 10,000 and
*1'(20)* = -350?

(b) Assuming the values in part (a), find R'(20) and interpret your answer.

{2 - *x* if *x* *~* 1

*f(x)* = *x**2* *-* *2x*

### +

2 if*x*

### >

1{-1-2X ifx<-I

*g(x)* = *x**2* if -1~*x ~* I

*x* ifx>1

Give a formula for*g'* and sketch the graphs of 9 and *g'.*

91. (a) For what values of*x* isthe functionf(x) ^{=}1*x**2* *-* 91differ-
entiable? Find a formula forf'.

(b) Sketch the graphs of

*f*

and*1'.*

92. Where is*the function hex)* =

### I

x-II### +

1*x*

### +

21 differenti- able? Give aformula for h' and sketch thegraphs of*hand*

*h'.*

~ For what values of a*and b is*the line *2x*

### +

*y*

^{=}

*b*tangent to the parabola

*y*

^{=}

*ax*

*2*when

*x*

^{=}2?

94. (a) *If F(x)* =*f(x)* *g(x),* where f and *9*have derivatives of all
orders, show that F" ^{=}*f"g* + *21'g'* +*fg".*

(b) Find similar formulas for *Fin* and *F(4).*

(c) Guess a formula forFC").

95. Find the value of c such that theline y =*~x*

### +

6is tangent to the curve*y*=

*e£.*

*()* *{X2* if x *~* 2

*f*

*x*

*=*

*mx*

### +

*b*if

*x*>2

Find the values *of m and* *b*that make

*f*

differentiable
everywhere.
97. An easy proof of theQuotient Rule can begiven ifwemake
the prior assumption that F'(x) exists, where F ^{=}

*fig.*

Write
*f*

=*Fg;*then differentiate using theProduct Rule and solve the resulting equation for F'.

98. Atangent line isdrawn tothe hyperbola *xy* ^{=}c at a point P.

(a) Show that the midpoint of the line segment cut from this
tangent line by the coordinate axes is*P.*

(b) Show that the triangle formed by the tangent line and the
coordinate axes always has the same area, no matter where
*P*islocated on the hyperbola.

*X*1000 - 1

~ Evaluate lim ----.

x~1 *x* *-* I

100. Draw a diagram showing two perpendicular lines that intersect
on the y-axis and are both tangent tothe parabola *y* =*x**2•*

Where dothese lines intersect?

10I. Ife >

### L

how many lines through the point (0,c) are normal lines to the parabola y =*x*

*2*

*?*What ife ~ ~?

102. Sketch the parabolas y ^{=}*x**2* and *y* ^{=}*x**2* *-* *2x*

### +

2.Do you think there is aline that istangent to both curves? Ifso,find its equation. If not, why not?SOLUTION Here wedivide numerator and denominator by x:

1imcos*x*

.\"----1>0

*x cos x* cos x

Jim*x*cot*x*

### =

tim -.--### =

Jim-. -### =

*x**•...•*0 *x**•...•*0 sIn*x* *x**•...•**O* sIn*x* l' sin*x*

IID--

*x----').o* *X*

### G

^{EXERCISES}

I. *f(x)* =*x* *- 3 sin x*
*3.* *y* =sin*x* *+ 10 tan x*
*5. g(')* =

### ,3

^{cos}

^{t}*7. h(e)* = *ecsc e-* cot *e*

*[1;]y=*

*x*

2 - tan*x*

*2.* *f(x)* =*x*sin x
*4.* *y* = 2csc x + 5 cos x
*6.* *g(t)* =4 sec , + tan'
*8.* *y* =*u(a cos* *u* + *b* cot*u)*

I + sin*x*
*10.* *y* *=*

*x* *+ cos x*
sec

*e*

**II.****f(****fJ)**=_I_+_s_ec_e I - sec x

*12.* *y* *=*

tan*x*
*13. y* =-2-sinx

*X*

17. Prove that -*d* *(csc x)* ^{=} -csc *x* cot*x.*

*dx*

18. Prove that -*d* *(sec x)* =*sec x* tan*x.*

*dx*

19. Prove that -*d* *(cot x)* = -csc2x.

*dx*

20. Prove, using the definition of derivative, that if f(x) =cos x,

then *.f'(x)* =-sin *x.*

**21-24** Find an equation of the tangent line tothe curve at the
given point.

*24.* *y* = . I , (0, 1)

sm*x +* cos *x*

25. (a) Find an equation of the tangent line to the curve
*y* =*2x* *sin x at the point* (71/2, 7T).

~ (b) Illustrate part (a) bygraphing thecurve and the tangent line on the same screen.

26. (a) Find anequation of the tangent line to thecurve
*y* =sec x *-* 2 cos*x* at the point (7T/3, 1).

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

27. (a) If f(x) =sec x *-* *x,* find*f'(x).*

~ (b) Check to see that your answer topart (a) isreasonable by graphing both

*f*

and *f'*

for ### I

*x*

### I <

7T/2.28. (a) If*f(x)* =~ sin*x,* find*f'(x).*

### ffi

^{(b)}

^{Check}

^{to see}

^{that}your answer topart (a) isreasonable by graphing both

*f*and

*f'*for 0:S X :S 27T.

~ If*H(e)* = *e*sin e,find H'(e) and W(e).

30. If f(x) ^{=}secx, find f"(7T/4).

() tan*x -* I
*fx=---*

secx

(b) Simplify the expression *for f(x)* by writing it interms of
sin*x*and cos*x,* and then find f'(x).

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

*h(x)* = cos*x*
*f(x)*

Find (a) g'(7T/3) and (b) h'(7T/3).

### InJ

^{For}what values of

*x*does thegraph of f(x) =

*x +*2sin

*x*have ahorizontal tangent?

34. Find the points on the curve *y* =*(cos x)/(2* + sin*x) at*whjch
the tangent ishorizontal.

35. Amass ona spring vibrates horizontally ona smooth
level surface (see the figure). Its equation ofmotion is
*x(t) = 8 sint,*where *t is*in seconds and*x* in centimeters.

(a) Find the velocity and acceleration at time t.

(b) Find the position, velocity, and acceleration ofthe mass
at time *t*=271/3. Inwhat direction isit moving atthat
time?

equilibrium position

### 1

### ffi

^{36. An}elastic band ishung onahook and amass ishung onthe lower end of the band. When the mass is pulled downward and then released, itvibrates vertically. The equation of motion is s =2 cos

*t*

### +

3 sin*t,t*

*;.*0, where s ismeasured incentimeters and

*t*inseconds. (Take the positive direction to be downward.)

(a) Find the velocity and acceleration attime *t.*

(b) Graph the velocity and acceleration functions.

(c) When does the masspass through the equilibrium position for the firsttime?

(d) How farfrom itsequilibrium position does themass travel?

(e) When is the speed the greatest?

37. Aladder 610 long rests against a vertical wall. Let *0*be the
angle between the top ofthe ladder and the wall and let*x* be
the distance from the bottom ofthe ladder tothe wall. If the
bottom ofthe ladder slides away from the wall, how fast does
*x*change with respect to 0 when *0 = 71/3?*

38. An object with mass m isdragged along a horizontal plane by aforce acting along arope attached to the object. If the rope makes anangle 0 with the plane, then the magnitude of the force is

*F=* *f.Lmg*

*f.L*sin 0

### +

cos*0*

where *f.L*is aconstant called the *coefficient* *oijriction.*

(a) Find the rate of change of F with respect to O.

(b) When isthis rate of change equal to O?

(c) If*m*=20 kg,*9* = 9.8m/s2, and*f.L* =0.6, draw thegraph
*of F as*a function of *0 and use* itto locate the value of 0
for which *dF/dO* =O.Isthe value consistent with your
answer topart (b)?

sin *3x*
39.1im--

*x-a* *x*

sin*4x*
40.lim--

*x~o* sin 6x

### IiQ

lim tan*6t*42. lim cos

*0 -*1

HO sin *2t* * ^{e~o}* sin

*0*

43. lim sin(cos 0) *sin**2**3t*

44. lim---

*e~o* sec*0* ^{(_0}*t**2*

~ lim sinO 46. . sin(x2)

1110---

*e~o* *0*

### +

tan*0*

*x_a*

*x*

47. lim I- tan*x* *sin(x* *-* I)

48. lim

*x--"'7T/4* sinx *- cosx* ^{x~1}*x**2*

### +

^{X}*2*

^{-}49. Differentiate each trigonometric identity to obtain a new (or familiar) identity.

sin*x* I

(a) tan*x* =-- (b) sec*x* =--

cos x cos x

1

### +

*cot x*(c) sin

*x*

### +

cos*x*=----

csc*x*

50. Asemicircle with diameter *PQ* sitson an isosceles triangle
*PQR to*form aregion shaped like atwo-dimensional ice-
cream cone, as shown inthe figure. If*A(O)* isthe area of the
semicircle and B(O) is the area of the triangle, find

lim *A(O)*

*e~o+* *B(O)*

### [Q

The figure shows a circular arc of length s and a chord of length*d,*both subtended by a central angle

*O.*Find

lim !...-

*e~o+* *d*

### =================~

TH E CHAI N RULESuppose you are asked todifferentiate the function

*F(x)* =

### R+1

The differentiation formulas you learned in the previous sections of this chapter do not
enable you to calculate *F'(x).*

If we define £ to be 0 when *Llx* =0, then £ becomes acontinuous function of *Llx.* Thus,
for a differentiable functionj, we can write

and £ is a continuous function of *Llx. This* property of differentiable functions is what
enables us to prove the Chain Rule.

PROOF OF THE CHAIN RULE *Suppose u*

### =

*g(x) is differentiable*

*at a*

*and y*

### =

*f(u)*isdifferen- tiable at

*b*=

*g(a).*If

*Llx*is an increment in

*x*and

*Llu and*

*Lly*are the corresponding incre- ments

*in u and*

*y,*then we can use Equation 5 towrite

where £2 ----0>0as *Llu* ----0>O.If we now substitute the expression for*Llu from Equation 6*
into Equation 7, we get

*Lly* = *[f'(b)*

### +

^{£2]}

^{[g'(a)}### +

^{£J]}

^{Llx}*Lly*

*Llx* =*[f'(b)*

### +

*£2][g'(a)*

### + £J]

As *Llx* ----0>0, Equation 6shows that *Llu* ----0>O.So both £, ----0>0 and £2 ----0>0as *Llx* ----0>O.

Therefore

*dy*

### =

*lim ~y*

### =

*lim [f'(b)*

### +

^{£2][g'(a)}### + £1]

*dx* *ilx-->O* uX *ilx-->O*

### §

^{EXERCISES}

**1-6** Write the composite function in the form *f(g(x».* [Identify the

inner function u =*g(x)* and theouter function *y* =*f(u).]* Then find

the derivative dy*/dx.*

I.*y* ^{=}sin*4x*

*3.* *y* =(l - X2)IO

### [I]

*Y*

^{=}Jsinx

*2.* *y* =*J4*

### +

*3x*

*4.*

*y*= tan(sin

*x)*

*6.*

*y*=sin

*JX*

*15.Y*=*x sec kx* 16. *y* ^{=}*3 cot(nfJ)*

*17.* *g(x)* =(l

### +

*4X)5(3*

### +

*x - X2)8*

*18.*

*h(t)*

^{=}

*(t*

*4*

*-*

*1)3(t*

*3*

### +

1)4*~* *Y*=*(2x -* *W(8x**2* *-* 5)-3 20. *y* ^{=}*(x**2*

### +

*1).ijx*

*2*

### +

2( 2

### + 1)3

**21.** *Y* = *_x__*

*x**2* *-* 1

*7.F(x)* =*(x**3*

### +

*4xr*

*9. F(x)*= ~l

### +

*2x*

### +

*x*

*3*

1

**II.** *get)* = (4 )l

*t*

### +

1*13.*

*Y*=

*cos(a*

*3*

### +

Xl)*8.* *F(x)* =*(x**2* *-* *X*

### +

1)3*10.f(x)*=(I

### +

*X4)2/3*

*x*
*24.* *f(x)* = ~3

'017 *-.JX*

~-I
*25.* *F(z)* = --

*z*

### +

1*(y* *_ 1)4*
*26.* *G(y)* =

*(l* +

*2y)5*

*r*

*27.y=* *~*

*yr**2*

### +

1COS*7TX*
*28.* *Y*^{=}

sin TTX

### +

cos*7TX*

*30.* *G(y)* =

### (~)5

_{y+1}*32.* *Y*=tan2(38)
I
*34.* *Y* =xsin-

*x*
(1 - COS

*2X)4*

*35.* *y=* *----*

I

### +

cos 2x*36.f(t)*=

### -V '

^{t}*t-*

### +

4*38.*

*Y*=

*(ax* + *ji2+jii)'2*

*39.* *Y*^{=}*[x**2*

### +

(I -*3xn*

*3*40.

*y*

^{=}sin(sin(sin

*x))*

**41.***y*

### = *.jx +* *jX*

42. *Y*

### = *Jx +* *.jx* + *JX*

*43.* *g(x)*

### =

*sin*

^{(2r}*rx*

### +

*n)P*44.

*y*

### =

cos4(sin3x)*45.* *y* ^{=}cos.)sin(tan *7TX)* 46. *y* ^{=}*[x*

### +

*(x*

### +

sin2xn4*48.* *Y*=sin2( *7Tt)*
*50.* *y=* *~4x*

*yX*

### +

1**51-54** Find an equation ofthe tangent line to the curve atthe
given point.

**51.** *Y* = ~,8 (4,2) 52. *Y*=sin*x*

### +

*cos 2x,*(7T/6,1) y4

### +

*3x*

*53.* *y* = sin(sin x), (7T, 0) 54. *Y*=.j5+X2, (2,3)

*I*

55. (a) Find anequation of the tangent line tothe curve
*y* ^{=}tan(^{7Tx2}*/4)* at thepoint (1, 1).

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

56. (a) The curve *y* =

### I

*x*

### I/~

iscalled a*bullet-nose*

*curve.*

Find anequation ofthe tangent line to this curve atthe point (I, 1).

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

57. (a) Iff(x) =*x~,* find*f'(x).*

~ (b) Check to seethat your answer topart (a) isreasonable by comparing the graphs of

*f*

and### f'.

~ 58. The function f(x) =sin(x

### +

sin*2x),*0

*~ x*

*~*7T, arises in applications to frequency modulation (FM) synthesis.

(a) Use a graph of

*f*

produced by a graphing device tomake
arough sketch ofthe graph of### f'.

(b) Calculate *f'(x)* and use this expression, with a graphing
device, tograph f'. Compare with your sketch inpart (a).

### m

^{Find all}points on the graph ofthe function

*f(x)* = 2 sin x

### +

sin2x atwhich the tangent line ishorizontal.60. Find thex-coordinates of all points onthe curve

*y* =sin*2x* *-* 2sin *x*atwhich the tangent line ishorizontal.

### m

^{If}

^{F(x)}^{=}

^{f(g(x»,}^{where}

^{f(}^{-2)}

^{=}

^{8,}

^{1'( -2)}^{=}

^{4,1'(5)}^{=}

^{3,}

*g(5)* =-2,and *g'(5)* =6, find F'(5).

*62. If hex)* = .)4

### +

*3f(x),*where f(l) = 7and

*I'*

(l) = 4,
*find 17'(1).*

*x* *f(x)* *g(x)* *f'(x)* *g'(x)*

I 3 2 4 6

2 I 8 5 7

3 7 2 7 9

(a) *If hex)* =*f(g(x)),* *find 17'(1).*

*(b) If H(x)* =*g(f(x),* *find H'(l).*

64. Let

*f*

and 9 bethe functions in Exercise 63.
*(a) If F(x)* =*f(f(x»,* find *F'(2).*

(b) If G(x) =*g(g(x»),* find G'(3).

~ If

*f*

and *9*are the functions whose graphs are shown, let

*u(x)*=

*f(g(x»,*

*vex)*=

*g(j(x»,*and

*w(x)*=

*g(g(x».*Find each derivative, ifit exists. Ifit does not exist, explain why.

(a) *u'(1)* (b) v'(l) (c) *w'(l)*

66. If*f* isthe function whose graph isshown, *let hex)* ^{=}*f(f(x»)*
and *g(x)* =*f(x**2).* Use the graph of*f* to estimate the value
of each derivative.

(a) *17'(2)* (b) *g'(2)*

~ Suppose *f* is differentiable on IR.Let *F(x)* =*f(cos* *x)*
and G(x) =*cos(f(x)).* Find expressions for (a)*F'(x)* and
(b)*G'(x).*

68. Suppose

*f*

isdifferentiable on IRand *a*is areal number.

Let *F(x)* =*f(x** ^{U)}* and G(x) =

*[f(x)Y*Find expressions for(a)

*F'(x) and*(b)

*G'(x).*

69. Let *rex)* =*f(g(h(x))),* where h(l) = *2, g(2)* = 3,*h'(l)* = 4,
*g'(2)*

### =

5,and*1'(3)*

### =

6. Find r'(l).70. *If 9*is atwice differentiable function and*f(x)* =*xg(x**2),* find
*f"* in terms of g, g', and *g".*

71. If*F(x)* =*f(3f(4f(x))),* wheref(O) = 0 andl'(O) = 2,
find *F'(O).*

72. If*F(x)* =*f(xf(xf(x))),* wheref(l) = *2,f(2)* = *3,f'(l)* =4,
*1'(2)* =5,andl'(3) ^{=}6,find F'(l).

**73-74** Find the given derivative by finding the firstfew derivatives
and observing thepattern that occurs.

~ The displacement of aparticle on a vibrating string is given by theequation

where s ismeasured incentimeters and tin seconds. Find the
velocity of the particle after *t seconds.*

76. Ifthe equation of motion of a particle is given by
s=*Acos(wt*

### +

*a),*the particle is said toundergo simple

*harmonic*

*motion.*

(a) Find the velocity ofthe particle at time t.

(b) When is the velocity O?

77. ACepheid variable star isa star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maxi- mum brightness is 5.4 days. The average brightness of this star is4.0 and its brightness changes by :!::0.35. In view of these data, the brightness of Delta Cephei at time t,where tismea- sured in days, has been modeled bythe function

*B(t)* =4.0

### +

0.35 sin(*27Tt)*5.4

(a) Find the rate of change ofthe brightness after t days.

(b) Find, correct totwo decimal places, the rate of increase after one day.

78. In Example 4inSection 1.3 we arrived at amodel forthe length of daylight (in hours) inAnkara, Turkey, on the tth day of the year:

*L(t)* = 12

### +

2.8 sin [ ;6~ (t - 80)]Use this model to compare howthe number ofhours of day- light isincreasing inAnkara on March 21and May 21.

79. A particle moves along astraight line with displacement *set),*
velocity vet), and acceleration *aCt).* Show that

*aCt)* =*v(t)-dv*
*ds*

Explain the difference between the meanings of thederivatives
*dv/dt* and *dv/ds.*

~ Air isbeing pumped into aspherical weather balloon. At any time t, the volume of theballoon is Vet) and its radius is r(t).

(a) What do thederivatives *dV/dr* and*dV/dt* represent?

(b) Express dV/dt interms of dr/dt.

### [ill]

81. Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and sofurther commands may benecessary tosimplify the answer.(a) Use aCAS to find the derivative inExample 5and com- pare with the answer inthat example. Then use the sim- plify command and compare again.

(b) Use a CAS to find thederivative inExample 6. What hap- pens if you use the simplify command? What happens if you use the factor command? Which form of the answer would be best for locating horizontal tangents?

and tosimplify the result.

(b) Where does the graph of

*f*

have horizontal tangents?
(c) Graph

*f*

and*I'*

on the same screen. Are the graphs con-
sistent with your answer topart (b)?
83. Use the Chain Rule toprove the following.

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

(b) The derivative ofan odd function isan even function.

84. Use the Chain Rule and the Product Rule togive an alternative proof of the Quotient Rule.

*[Hint:* Write f(x)/g(x) ^{=}*f(x)[g(xW'.]*

-*d* (sin"x cos nx) =*n sin"-Ix cos(n*

### +

Ox*dx*

(b) Find aformula for the derivative of y =*cos"x cosnx*
that is similar tothe one inpart (a).

86. Suppose *y* =*f(x)* is a curve that always lies above the x-axis
and never has ahorizontal tangent, where

*f*

isdifferentiable
everywhere. For what value of y isthe rate of change of y5
with respect to*x*eighty times the rate of change of

*y*with respect to

*x?*

~ Use the Chain Rule to show that if

*e*

^{is}measured indegrees, then

*d* 7T

- (sin

*e)* = --

cos*e*

*de* *180*

(This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The differentiation formulas would notbe as simple if weused degree measure.)