James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.2 The Product and Quotient Rules 189
45. If f sxd − e x tsxd, where ts0d − 2 and t9s0d − 5, find f 9s0d.
46. If hs2d − 4 and h9s2d − 23, find
d
S hsxd
dx x
DZ
x−2
47. If tsxd − xfsxd, where f s3d − 4 and f 9s3d − 22, find an
equation of the tangent line to the graph of t at the point
where x − 3.
48. If f s2d − 10 and f 9sxd − x 2 fsxd for all x, find f 99s2d.
49. If f and t are the functions whose graphs are shown, let
usxd − f sxdtsxd and vsxd − f sxdytsxd.
(a) Find u9s1d.
(b) Find v9s5d.
y
1
0
1
f
50. Let Psxd − FsxdGsxd and Qsxd − FsxdyGsxd, where F and G
are the functions whose graphs are shown.
(a) Find P9s2d.
(b) Find Q9s7d.
y
1
0 1
51. If t is a differentiable function, find an expression for the
derivative of each of the following functions.
(a) y − xtsxd (b) y − x
tsxd
(c) y − tsxd
x
52. If f is a differentiable function, find an expression for the
derivative of each of the following functions.
(a) y − x 2 f sxd
(b) y − f sxd
x 2
(c) y − x 2
f sxd
F
G
g
(d) y − 1 1 x f sxd
sx
53. How many tangent lines to the curve y − xysx 1 1) pass
through the point s1, 2d? At which points do these tangent lines
touch the curve?
x
x
54. Find equations of the tangent lines to the curve
y − x 2 1
x 1 1
that are parallel to the line x 2 2y − 2.
55. Find R9s0d, where
Rsxd −
x 2 3x 3 1 5x 5
1 1 3x 3 1 6x 6 1 9x 9
Hint: Instead of finding R9sxd first, let f sxd be the numerator
and tsxd the denominator of Rsxd and compute R9s0d from f s0d,
f 9s0d, ts0d, and t9s0d.
56. Use the method of Exercise 55 to compute Q9s0d, where
Qsxd − 1 1 x 1 x 2 1 xe x
1 2 x 1 x 2 2 xe x
57. 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.
58. A manufacturer produces bolts of a fabric with a fixed width.
The quantity q of this fabric (measured in yards) that is sold is
a function of the selling price p (in dollars per yard), so we can
write q − f spd. Then the total revenue earned with selling price
p is Rspd − pf spd.
(a) What does it mean to say that f s20d − 10,000 and
f 9s20d − 2350?
(b) Assuming the values in part (a), find R9s20d and interpret
your answer.
59. The Michaelis-Menten equation for the enzyme chymotrypsin
is
v −
0.14fSg
0.015 1 fSg
where v is the rate of an enzymatic reaction and [S] is the concentration
of a substrate S. Calculate dvydfSg and interpret it.
60. The biomass Bstd of a fish population is the total mass of the
members of the population at time t. It is the product of the
number of individuals Nstd in the population and the average
mass Mstd of a fish at time t. In the case of guppies, breeding
occurs continually. Suppose that at time t − 4 weeks the population
is 820 guppies and is growing at a rate of 50 guppies
per week, while the average mass is 1.2 g and is increasing at
a rate of 0.14 gyweek. At what rate is the biomass increasing
when t − 4?
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