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Unformatted text preview: C H A PT E R 2 Differentiation: Basic Concepts PROBLEMS 2.1 In Pro
to its graph for the specier :12
1‘ 3,
\J
I ﬂ 5:; v 5 I
1 /‘
1, 7.1
\/ In Problems 9 through 16, compute the derivative of the given j'(x)=5x—3;x=2 lf(x)=2x23x5;x=0 t_2.t_l
f(x)=\/E;x=9 tangent to its graph for the speciﬁed value x = c. 1
39¢
‘1" “x ' ‘1
LI !’
111 A; ‘\
2731 r“ ‘1. ll}
:13) "‘ \ 1K". 2\1)
l x 23. 25. f(X)=x2;c=1
f(x)=7—2x;c=5 2
f(x)=—;;c= 1 llﬂfﬁ) = 2\/_; c = 4 d y=3;x0=2 y=x(1—x);x0= —1
._ 2 . _
y—x 2x,x0—l Let f(x) = x3. 3. Compute the slope of the secant line joining the
points on the graph of f whose x coordinates are
x = 1 and x = 1.1. b. Use calculus to compute the slope of the line
that is tangent to the graph when x = 1 and
compare with the slope found in part (a). Let f(x) = 2x — x2.
a. Compute the slope of the secant line joining the 1
points where x = 0 and x = 2' b. Use calculus to compute the slope of the tangent
line to the graph of f(x) at x =0 and Compare
with the slope found in part (a). , blems I through 8, compute the derivative of the given
d value of the independent variable. 212 function and ﬁnd the slope of the line that is tangent 2. f(x)=x21;x= 1
4. f(x)=x31;x=2 1 6. f(x)=x—;x=2 1 8. h(u)=—\/—;;u=4 function and ﬁnd the equation of the line that is 10. f(x) = 3; c = 4
12. f(x) = 2 — 3x2; 6 =1 14. f(x)=x3— l;c=1 3
16. f(x) — x2, c — E 18. 20. 22. 24. 26. 1 In Problems 17 through 22, ﬁnd the rate of change i where x = x0. y=6”2X;X()=3 1 .31 2—x Xo
1
y=x—;xO= 1
x
Let f(x) = x2. 3. Compute the slope of the secant line joining the I ‘ points on the graph of f whose x coordinates are
x = ~2 andx = —1.9.
b. Use calculus to compute the slope of the line
that is tangent to the graph when x = 2 and
compare with the slope found in part (a). x
Let f(x) = .
x  1
a. ‘ Compute the slope of the secant line joining the
. 1
points where x = —1 and x = ‘2' b. Use calculus to compute the slope of the tangent
line to the graph of f(x) at x = l and compare
with the slope found in part (a). 29. 225 PROBLEMS 2.2 In Problems 1 through 25, dijj‘erentiate the given function. Simplify your answers.
1 y = x
L3,? y = 2
L5; y = W2 V2?
’y=f+u+3 \y: .'3} f(x) = x9 — 5x8.+ x +12 x27 30. y=2x4—\/;+§:‘;(1,4) s. y=\/?—x2+ (4,—7) 32. f(x)=x4—3x3+2x2—6;x=2
34. f(x)=x3+\/J_c;x=4. 36. f(x) = Jam/2 — 1); x = 4 121 SECTION 2.2 2.
4. 6. 8. 10.
12. 14. 18. 20. 22. 24. In Problems 26 through 31 ﬁnd the equation of the line that is tangent to the graph of the given function at the
speciﬁed point. y=x53x35x+2;(l,—5) Techniques of Differentiation y=x‘
y=r ='n'r‘
y 3 ll 23:2 y = —x3 — 5x2 + 3x —1;<*1,—8> “77:29:: , In Problems 32 through 37 ﬁnd the equation of the line that is tangent to the graph of the given function at the
point (e, f(c)) for the speciﬁed value of x = c. 1——+—— W7 5 3_1,' y = (x2 — x)(3 + 2x); (1, 2) y= 4, x .
v . 33;"! f(X) = 2x3 + // I 35. _,x’3‘(x) = x — 11x = 1
M; x 37. 21%;): —%x3 + V8“ = 2 'x= —1
x2’ ...\A~\ l“ bnﬂl’ l EN 4 UIIIEICIILIUUUIIZ DUSIC Concepts 5 value x = c. 38. f(x)=x3—3x+5;x=2 40. 42. 44. 45. 46. 47. _f(x)=W+5x;x=4 f(X)=E—x\/E;x=l x NEWSPAPER CIRCULATION It is estimated
that t years from now, the circulation of a local
newspaper will be C(t) = lOOt2 + 400t + 5,000. a. Derive an expression for the rate at which the
circulation will be changing with respect to time
t years from now. b. At what rate will the circulation be changing with
respect to time 5 years from now? Will the circu—
lation be increasing or decreasing at that time? c. By how much will the circulation actually
change during the sixth year? AIR POLLUTION An environmental study of a certain suburban community suggests that t years from now, the average level of carbon monoxide in
the air will be Q(t) = 0.05t2 + 0.1t + 3.4 parts per
million. a. At what rate will the carbon monoxide level be
changing with respect to time 1 year from now? b. By how much will the carbon monoxide level
change this year? c. By how much will the carbon monoxide level
change over the next 2 years? WORKER EFFICIENCY An efﬁciency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 AM. will have assembled f(x) = —x3 + 6x2 + 15x transistor radios x hours later. a. Derive a formula for the rate at which the
worker will be assembling radios after x hours. b. At what rate will the worker be assembling
radios at 9:00 A.M.? c. How many radios will the worker actually
assemble between 9:00 A.M. and 10:00 A.M.? EDUCATIONAL TESTING It is estimated that x years from now, the average SAT mathematics score of the incoming students at an eastern liberal arts college will be f(x) = —6x + 582. a. Derive an expression for the rate at which the
average SAT score will be changing with re
spect to time x years from now. ZZé at”; In Problems 38 through 43 ﬁnd the rate (2f change of the given function f(x) with respect to xfor the prescribed 39. f(x)'=2x4+3x+1,x= —1 41. 43. 48. 49. 50. 51. 52. f(x)=x—\/E+%;x=1 x+\/;
W ;x=l for) = b. What is the signiﬁcance of the fact that the ex
pression in part (a) is a constant? What is the
signiﬁcance of the fact that the constant in part '
(a) is negative? PUBLIC TRANSPORTATION After x weeks, the number of people using a new rapid transit system was approximately N(x) = 6x3 + 500x + 8,000. a. At what rate was the use of the system changing
with respect to time after 8 weeks? b. By how much did the use of the system change
during the eighth week? PROPERTY TAX Records indicate that x years after 2000, the average property tax on a three— bedroom home in a certain community was T(x) = 20x2 + 40): + 600 dollars. » a. At what rate was the property tax increasing
with respect to time in 2000? b. By how much did the tax change between the years 2000 and 2004? GROWTH OF A TUMOR A cancerous tumor
is modeled as a sphere of radius R cm. At what 4
rate is the volume V = g 'lTR3 changing with respect to R when R = 0.75 cm? SPREAD OF AN EPIDEMIC A medical re—
search team determines that t days after an epidemic
begins, N(t) = lOt3 + 5t + Vt people will be in—
fected, for O S t S 20. At what rate is the infected
population increasing on the ninth day? ADVERTISING A manufacturer of motorcycles
estimates that if x thousand dollars are spent on ad—
vertising, then 125 517
M(x) = 2,300 + — — 2
.X X 3SxSlS cycles will be sold. At what rate will sales be
changing when $9,000 is spent on advertising? Are
sales increasing or decreasing for this level of ad
vertising expenditure? 124 CHAPTE R 2 Differentiation: Basic Concepts a. Compute and interpret the derivative T’(t). At what rate is the temperature changing at the beginning of the period (t = 0) and at the end of the period (t = 0.713)? Is the temperature
increasing or decreasing at each of these times? c. At what time is the temperature not changing
(neither increasing nor decreasing)? What is the
bird’s temperature at this time? Interpret your
result. F RECTILINEAR MOTION In Problems 64 through
67, s(t) is the position of a particle moving along a
straight line at time t. 64.
65.
66.
67. 68. 69. 73. 74. (a) Find the velocity and acceleration of the particle. (b) Find all times in the given interval when the
particle is stationary. s(t)= t2—2t+6forOStSZ
s(t)=3t2+2t—5for05tsl s(t) =23 —9t2+15t+25f0r05ts6 s(t) = t4 — 4t3 + 8tfor0 SIS 4 MOTION OF A PROJECTILE A stone is
dropped from a height of 144 feet. a. When will the stone hit the ground?
b. With what velocity does it hit the ground? MOTION OF A PROIECTILE You are stand
ing on the top of a building and throw a ball verti
cally upward. After 2 seconds, the ball passes you 71. 72. Q il’ % 2—28 on the way down, and 2 seconds after that, it hits the ground below. a. What is the initial velocity of the ball? b. How high is the building? c. What is the velocity of the ball when it passes
you on the way down? d. What is the velocity of the ball as it hits the
ground? SPY STORY Our friend, the spy who escaped
from the diamond smugglers in Chapter 1 (Problem
46 of Section 1.4), is on a secret mission in space.
An encounter with an enemy agent leaves him with
a mild concussion and temporary amnesia. Fortu
nately, he has a book that gives the formula for the
motion of a projectile and the values of g for vari
ous heavenly bodies (32 ft/sec2 on earth, 5.5 ft/sec2
on the moon, 12 ft/sec2 on Mars, and 28 ft/sec2 on
Venus). To deduce his whereabouts, he throws a
rock vertically upward (from ground level) and
notes that it reaches a maximum height of 37.5 ft
and hits the ground 5 seconds after it leaves his
hand. Where is he? Find numbers a, b, and c such that the graph of the
function f(x) = ax2 + bx + c will havex intercepts
at (O, 0) and (5, O), and a tangent with slope 1 when
x = 2. Find the equations of all of the tangents to the graph
of the function
f(x) = x2 — 4x + 25 that pass through the origin (0, 0). Prove the sum rule for derivatives. [Hint Note that the difference quotient forf + g can be written as
(f+ gXx + h) — (f+ g)(X) = [for + h) + .206 + M] — lf(X) + 306)] h
f(x + h)  ﬁx)
it a. Iff(x) = x4, show that b. If f(x) = x” for positive integer n, show that f(x + 11) —f(X) _
h —l’l.X h = 4x3 + 6th + 4th + 113 “h + + nxh"_2 + h”“ c. Use the result in part (b) in the deﬁnition of the derivative to prove the power rule: :;[x" = nx n—l 229 75. POL]
that 0
(C02:
apply
the re
bon t:
emiss
a. V at e,
b. L
tl t1
go F F
c L‘JU 1 34 C H A PT E R 2 Differentiation: Basic Concepts d d d
To show that Ex—(fg) = f i + gaix, begin with the appropriate difference quotient
and rewrite the numerator by subtracting and adding the quantity f(x + h)g(x) as
follows:
1 (f1) : 1.1m f(x + 10306 + 11) —.I"(X)g(X)
dx 1’ [1—90 h
= lim[f(x + h)g(x + h) ‘ .f(x + h)g(X) + f(x + h)g(X)  f(X)g(x)
[1—90 h h
) + h — 7 + h _.
2 lim<f(x + MPG ) 506)] + g(x)[f(x ) NOD
h—>() h h Now let h approach zero. Since 1 f(x + 11> —.f(x> _ df
1m — —
h—>() ‘ h dx
1' 301 + h) — gm _ the
1m — ~—
/z—>() h dx
and INF?) f(x + h) =f(x) continuity afﬁx)
c1 1; d
it follows that 67qu :15; + git
P R O B L E M S 2.3
In Problems 1 through 20, défferemime the given function.
1 1.10:) = 12x + 1><3x — 2) 2. f(x) = 1x — 5111 — 2x)
3.‘ y = 10(311 + l)(l  5M) 4. _v = 400(15 — x2)(3x — 2)
”’ 1 1
5.. ﬁx) = 3015 ~ 2x3 + 1)<x — E) 6. f(x) = —3(5x3 — 2x + 5)(\ﬂc + 2x)
7):. _x+l 8 _'2x—3
y x _ 2 ' y 5x + 4
. '1. I l
> 9. 1 ' = 7 l . =
», .__: 1(1) , r _ 2 o for) x _ 2
1 3 [2 + l
7 11.1 = 12. = g
.Jy x+5 'V 1—1“
4 x2—3x+2 [2+2t+l
"3.13: = 7 14. ' = .,
‘ )1 f(x) 2x“ + 5x ‘ 1 ﬂ!) 1 + 3r — 1
‘ _(2x—l)(x+3) _(x2+x+l)(4—x)
415;] foo — x + 1 16' 5‘06) — 2X __ 1
' 171f06) = (2 + 5102 18. f(x) = (x + l)“
. x
I t2 + \ﬂ ' x 4 — x
19' t = . = + I)
‘g0 2t+5 20 W) xz—l x~+1 238 .otient (x) as 2—39 SECTION 2.3 Product and Quotient Rules; HigherOrder Derivatives 135 In. Problems 2] through 25, ﬁnd an equation for the tangent line to the given curve at the point where x = x0. L21. ,.V = (5x — 1)(4 + 3x); x0 = 0 ‘23 I x
_ " Qtmw‘l @ y: (3% + x)(2 — ﬂue, =1 k» 22. y=(x2+3x—1)(2—x);x0=1 x+7 24. 'VZS—Zx;X0=O
2x—l 26. _v=1_x3;Xo=0 InProblems 27 through 3 I , ﬁnd all points on the graph of the given function where the tangent line is horizontal. (/27. f(x) = (x + no? — x — 2) i ) _ X + 1
@ f(x —x2+x+l
‘ . 731, f(x) = x3(x — 5)2 V dv 28. ﬁx): (x — no? — 8x + 7) x2+x—1 30' f(x) : x2 — x +1 In Problems 32 through 35, ﬁnd the rate of change Tfor the prescribed value of x0.
(x . ‘ 32. _v = (x2 + 2)(x + \/)_c); x0 = 4
2x  1. 36. y=x2 + 3x — 5; (0, —5) 38. y = (x + 3)(1— W); (1.0) 40. a. Differentiate the function y = 2x2 — 5x — 3.
b. Now factor the function in part (a) as y =
(2x + l)(x — 3) and differentiate using the
product rule. Show that the two answers are the same. possible before computing the second derivative.) 42. f(x) = 5x‘° ~ 6;? — 27x + 4 The normal line to the curve _v = f(x) at the point Pm),
Problems 36 through 39, ﬁnd an equation for the mutual line to the given curve at the prescribed point. “33. y = (x2 + no — 2x3); x0 = 1 35 "‘x+ 3 'x—O
“V 2—4x’0 f(x())) is the line perpendicular to the tangent line at P. In 2
37. y=;—\f;(1.1) 5x + 7
39. v = ; 1. —12
. 2 _ 3x ( )
41. 3. Use the quotient rule to differentiate the
. 2x — 3
function y = a, .
x‘ b. Rewrite the function as y = x" 30.x — 3) and
differentiate using the product rule. c. Rewrite the function as y = 2x.2 — 3x—3 and
differentiate. d. Show that your answers to parts (a), (b), and
(c) are the same. In Problems 42 through 47, ﬁnd the second derivative of the given function. In each case, use the appropriate nota—
tion for the second derivative and simplifv your answer. (Don’t forget to simp lify the ﬁrst derivative as much as 2 .
43. f(x) = 3x5 — 4x3 + 9x“ — 6x  2 47. _v = (x3 + 2x — 1)(3x + 5) l 136 48. 50. 51. C H A PT E R 2 Differentiation: Basic Concepts DEMAND AND REVENUE The manager of a company that produces graphing calculators
determines that when x thousand calculators are
produced, they will all be sold when the price is ()_ 1,000
1” 0.3x2+8 dollars per calculator. a. At what rate is demand p(x) changing with
respect to the level of production x when
3,000 (x = 3) calculators are produced? b. The revenue derived from the sale of x
thousand calculators is R(x) = xp(x) thousand
dollars. At what rate is revenue changing when
3,000 calculators are produced? ls revenue
increasing or decreasing at this level of
production? SALES The manager of the Many Facets
jewelry store models total sales by the function 2,000:
4 + 0.3t where t is the time (years) since the year 2000 and S is measured in thousands of dollars. a. At what rate are sales changing in the year
2002? ' b. What happens to sales in the “long run”'(that
is, as t——> +00)? PROFIT Bea Johnson, the owner of the Bea Nice boutique, estimates that when a particular kind of perfume is priced at [2 dollars per bottle, she will sell 5(1) = 500 3(1)) = m bottles per month at a total cost of
C(p) = 0.2122 + 3p + 200 dollars a. Express Bea’s proﬁt P(p) as a function of the
price p per bottle.  b. At what rate is the proﬁt changing with
respect to p when the price is $12 per bottle?
Is proﬁt increasing or decreasing at that price? ADVERTISING A company manufactures a
“thin” DVD burner kit that can be plugged into
personal computers. The marketing manager deter
mines that t weeks after an advertising campaign '
begins, P(t) percent of the potential market is
aware of the burners, where t2+5t+5
t2+10t+30 [925 P(t) = 100[ 52. 53. 54. (102.3 é”; , 2—40 3. At what rate is the market percentage P(t)
changing with respect to time after 5 weeks?
Is the percentage increasing or decreasing at
this time? b. What happens to the percentage P(t) in the
“long run”; that is, as t——)+00? What
happens to the rate of change of P(t) as
1—) +00? BACTERIAL POPULATION A bacterial
colony is estimated to have a population of 242‘ + 10
t2 + 1 million 1‘ hours after the introduction of a toxin. a. At what rate is the population changing 1 hour
after the toxin is introduced (1 = 1)? Is the
population increasing or decreasing at this
time? b. At what time does the population begin to
decline? POLLUTION CONTROL A large city commissions a study that indicates that spending money on pollution control is effective up to a point but eventually becomes wasteful. Suppose it is known that when x million dollars is spent on controlling pollution, the percentage of pollution removed is given by 1’0) = a. At what rate is the percentage of pollution
removal P(x) changing when 16 million dollars
are spent? Is the percentage increasing or
decreasing at this level of expenditure? b. For what values of x is P(x) increasing? For
what values of x is P(x) decreasing? PHARMACOLOGY An oral painkiller is
administered to a patient, and t hours later, the
concentration of drug in the patient’s bloodstream
is given by
2t
3t2 + 16
a. At what rate R(t) is the concentration of drug
in the patient’s bloodstream changing 1 hours
after being administered? At what rate is R(t)
changing at time t? ‘
b. At what rate is the concentration of drug
changing after 1 hour? Is the concentration changing at an increasing or decreasing rate at
this time? ' C(t) = 240 1ding
o a
)ose it
it on
ition )n
dollars
r ‘ For the
stream f drug
hours is R(t) lg
tion
; rate at c. When does the concentration of the drug begin
to decline? d. Over what time period is the concentration
changing at a declining rate? 55. WORKER EFFICIENCY An efﬁciency study of
the morning shift at a certain factory indicates that
an average worker arriving on the job at 8:00 AM.
will have produced Q(t) = t3 + 8t2 + 15t units
t hours later. :1. Compute the worker’s rate of production
Rtt) = Q’tt). b. At what rate is the worker’s rate of production
changing with respect to time at 9:00 A.M.? 56. POPULATION GROWTH It is estimated that
t years from now, the population of a certain suburban community will be P(t) = 20 — :1 thousand. a. Derive a formula for the rate at which the
population will be changing with respect to
time t years from now. b. At what rate will the population be growing
1 year from now? 0. By how much will the population actually
increase during the second year? d. At what rate will the population be growing
9 years from now? e. What will happen to the rate of population
growth in the long run? ‘ In Problems 57 through 60, the position s(t) of an
object moving along a straight line is given. In each
case: (a) Find the object’s velocity v( t) and acceleration a( t).
(17) Find all times t when the acceleration is 0. 57. s(t) = 3:5 — 52:3 — 7 58. s(t) = 2t4 — 5t3 + t — 3
59. s(t) = —r3 + 7t2 + t + 2
60. s(t) = 425/2 — 15:2 + t 3 ELOCITY An object moves along a straight
line so that after t minutes, its distance from a. At what velocity is the object moving at the
end of 4 minutes? b. How far does the object actually travel during
the ﬁfth minute? . . . . 5
its starting pomt is D(t) = 10t + TF—l — 5 meters. 63. 64. 2—41 ‘ ECTGION 2.3 Product and Quotient Rules; HigherOrder Derivatives 137 62. ACCELERATION After t hours of an 8hour 10 2
trip, a car has gone D(t) = 64t + $12 — 613 kilometers.
a. Derive a formula expressing the acceleration of the car as a function of time. b. At what rate is the velocity of the car chang
ing with respect to time at the end of 6
hours? Is the velocity increasing or
decreasing at this time? c. By how much does the velocity of the car
actually change during the seventh hour? DRUG DOSAGE One biological model* sug—
gests that the human body’s reaction to a dose of
medicine can be measured by a function of the form
F = gum2 — M) where K is a positive constant and M is the
amount of medicine absorbed in the blood. The dF
derivative S = EAT can be thought of as a measure of the sensitivity of the body to the medicine. a. Find the sensitivity S. dS dZF
b. Fi d W = W and give an interpretation of the second derivative. BLOOD CELL PRODUCTION A biological
modeli measures the production of a certain
type of white blood cell (granulocytes) by the
function p(x) = where A and B are positive constants, the exponent m is positive, and x is the number of cells present. a. Find the rate of produCtion p’(x). . b. Find p”(x) and determine all values of x
for which p"(x) = 0 (your answer will
involve m). *Thrall et al., Some Mathematical Models in Biology, US. Dept. of Commerce, 1967.
7“M. C. Mackey and L. Glass, “Oscillations and Chaos in Physiological Control Systems,” Science, Vol. 197, pp. 287289. 251 ‘ SECTION 2.4 The Chain Rule 147 ' The manager estimates that t months from now, the unit price of the blenders will be
p(t) = 0.0613/2 + 22.5 dollars. At what rate will the monthly demand for blenders D( p)
be changing 25 months from now? Will it be increasing or decreasing at this time? Solution dD
We want to ﬁnd ; when t = 25. We have £12 _ _d_[8,000] _ ~8,000
dp dp p p2 and d]7 d 2/2 3 1/2 1/2
——~ 2 — . ‘ + _ = ....
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