51. h(x)= 53. cos x tan x (t)=csc/cot/
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 35E
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190 CHAPTER 3 The Derivative
27. y
sin² x
1+cos.x
28. y = (1+√x)' (1-2√x)*
cos 2x
29. y =
√sin 3x
30. y=√4+x²-√√√4-x²
31. y sin' 2r cos² 2x
32. y = [1 +(2+3x)-³/212/03
33. y sin
34. y=
35. y cos' (√x+1)
=
42. g(t) = 3 sec(t²-1)
1+tan.x
sec x
44. f(x) =
46. f(x)=sin(secx)
48. g(t) tan(sin t)
1
50. h(x)=
cos(tan x)
52. (t) cot(csc)
54. f(x)= (1+sin 3x)/2
56. h(t)=√1+sec² 61
Find the derivatives of the functions given in Problems 36
through 57.
36. f(x) = sec(x¹+x²)
38. f(x) = sec 2x tan 2x
40. f(x) = x'sec 2x
60. y=
1+√ (4.)
62. y secx tan x; (0, 1)
64. lim
1-0 sin x
sin 2x
x-0 sin 5x
3/2
66. lim
68. lim x² sin
1-10
In Problems 58 through 63, write an equation of the line tangent
to the given curve at the indicated point.
(0.-1)
59. y
x + 1
58. y =
x-1
1
Find the limits in Problems 64 through 69.
x-tan x
1
72. h(x) =
2
X
x² +1.
Adasons 250th 3o
37. f(x)= tan² (x¹+x²).
39. f(x) = sec √√xtan √x
sec 21
41. g(t)=
t
43. g(t) = (tan √7)
1- sect
plein not carbure
45. g(t)=
1+ sect
47. f(x)=sin.x sec x
49. g(t) tantsinz
1
51. h(x) =
cos x tan x
53. (f) csc/ cot/
55. g(x)=sin(1+x³)
57. (t)=√(- tant)
sin 3x; (x/6,1)
2x
61. y =
√x+1 (7.7)
63. y = (x²+2x)¹/³; (2.2)
65. lim x cot 3x
x-0
69. lim √xsin
X-0¹
67. lim x² csc 2x cot 2x
x-0
1
X
In Problems 70 through 77, identify two functions f and g such
that h(x) = f(g(x)). Then apply the chain rule to find h'(x).
70. h(x)=√√x+x²
71. h(x) =
1
√x²+25
73. h(x)=√(x-1)³
74. h(x) =
(x+1)10
(x-1) 10
76. h(x)= (1+sin.x)³
75. h(x) = cos(x¹+1)
77. h(x)= sechr
78. What is an equation for the straight line through (1.0
is tangent to the graph of
1
X
h(x)=x+-
at a point in the first quadrant?
79. A rectangle has its base on the x-axis and its upper
the maximum possible area of such a rectangle with
vertices on the graph of y = cos x, -x/2x5x2
places to the right of the decimal point correct or com
rounded. You may, and should, use Newton's meth
help you solve this problem.
80. An oil field containing 20 wells has been producing
barrels of oil daily. For each new well drilled, the daily p
duction of each well decreases by 5 barrels. How many to
wells should be drilled to maximize the total daily prote
tion of the oil field?
81. A triangle is inscribed in a circle of radius R. One
of the triangle coincides with a diameter of the circle
terms of R, what is the maximum possible area of s
triangle?
82. Five rectangular pieces of sheet metal measure 210 cm
336 cm each. Equal squares are to be cut from all their
ners, and the resulting five cross-shaped pieces of metal
to be folded and welded to form five boxes without top
The 20 little squares that remain are to be assembled
groups of four into five larger squares, and these five le
squares are to be assembled into a cubical box with not
What is the maximum possible total volume of the six b
that are constructed in this way?
83. A mass of clay of volume V is formed into two spheres in
what distribution of clay is the total surface area of the
spheres a maximum? A minimum?
84. A right triangle has legs of lengths 3 m and 4 m. What is h
maximum possible area of a rectangle inscribed in the tri
gle in the "obvious" way-with one corner at the triangel
right angle, two adjacent sides of the rectangle lying on th
triangle's legs, and the opposite corner on the hypoteru
85. What is the maximum possible volume of a right circl
cone inscribed in a sphere of radius R?
86. A farmer has 400 ft of fencing with which to build and
angular corral. He will use some or even all of an ex
ing straight wall 100 ft long as part of the perimeter d
corral. What is the maximum area that can be enclosed
87. In one simple model of the spread of a contagious disc
among members of a population of M people, the incide
of the disease, measured as the number of new cases p
day, is given in terms of the number x of individuals alread
infected by
R(x) = kx (M-x) = kMx-kx².
where k is a positive constant. How many individuak
the population are infected when the incidence Ris
greatest?
88. Three sides of a trapezoid have length L, a constant W
should be the length of the fourth side if the trapezoid
have maximal area?
89. A box with no top mu
wide, and the total sur
What is the maximum
90. A small right circular
(Fig. 3.MP.1). The larg
altitude H. What is the
larger cone that the sm
FIGUR
cone in
one (Pr
91. Two vertices of a trap
the other two lie on the
is the maximum possil
area of a trapezoid w
A=h(b + b₂)/2.]
92. Suppose that f is a d
whole real number lin
a point Q(x, y) close
graph. Show that
at Q. Conclude that
the line tangent to the
the square of the dista
93. Use the result of Prob
tance from the point
Ax+By+C =0 is
94. A race track is to be
equal straightaways c
(Fig. 3.MP.2). The leng
4 km. What should its
area within it?
FIGUR
race tr
rectan
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190 CHAPTER 3 The Derivative
27. y
sin² x
1+cos.x
28. y = (1+√x)' (1-2√x)*
cos 2x
29. y =
√sin 3x
30. y=√4+x²-√√√4-x²
31. y sin' 2r cos² 2x
32. y = [1 +(2+3x)-³/212/03
33. y sin
34. y=
35. y cos' (√x+1)
=
42. g(t) = 3 sec(t²-1)
1+tan.x
sec x
44. f(x) =
46. f(x)=sin(secx)
48. g(t) tan(sin t)
1
50. h(x)=
cos(tan x)
52. (t) cot(csc)
54. f(x)= (1+sin 3x)/2
56. h(t)=√1+sec² 61
Find the derivatives of the functions given in Problems 36
through 57.
36. f(x) = sec(x¹+x²)
38. f(x) = sec 2x tan 2x
40. f(x) = x'sec 2x
60. y=
1+√ (4.)
62. y secx tan x; (0, 1)
64. lim
1-0 sin x
sin 2x
x-0 sin 5x
3/2
66. lim
68. lim x² sin
1-10
In Problems 58 through 63, write an equation of the line tangent
to the given curve at the indicated point.
(0.-1)
59. y
x + 1
58. y =
x-1
1
Find the limits in Problems 64 through 69.
x-tan x
1
72. h(x) =
2
X
x² +1.
Adasons 250th 3o
37. f(x)= tan² (x¹+x²).
39. f(x) = sec √√xtan √x
sec 21
41. g(t)=
t
43. g(t) = (tan √7)
1- sect
plein not carbure
45. g(t)=
1+ sect
47. f(x)=sin.x sec x
49. g(t) tantsinz
1
51. h(x) =
cos x tan x
53. (f) csc/ cot/
55. g(x)=sin(1+x³)
57. (t)=√(- tant)
sin 3x; (x/6,1)
2x
61. y =
√x+1 (7.7)
63. y = (x²+2x)¹/³; (2.2)
65. lim x cot 3x
x-0
69. lim √xsin
X-0¹
67. lim x² csc 2x cot 2x
x-0
1
X
In Problems 70 through 77, identify two functions f and g such
that h(x) = f(g(x)). Then apply the chain rule to find h'(x).
70. h(x)=√√x+x²
71. h(x) =
1
√x²+25
73. h(x)=√(x-1)³
74. h(x) =
(x+1)10
(x-1) 10
76. h(x)= (1+sin.x)³
75. h(x) = cos(x¹+1)
77. h(x)= sechr
78. What is an equation for the straight line through (1.0
is tangent to the graph of
1
X
h(x)=x+-
at a point in the first quadrant?
79. A rectangle has its base on the x-axis and its upper
the maximum possible area of such a rectangle with
vertices on the graph of y = cos x, -x/2x5x2
places to the right of the decimal point correct or com
rounded. You may, and should, use Newton's meth
help you solve this problem.
80. An oil field containing 20 wells has been producing
barrels of oil daily. For each new well drilled, the daily p
duction of each well decreases by 5 barrels. How many to
wells should be drilled to maximize the total daily prote
tion of the oil field?
81. A triangle is inscribed in a circle of radius R. One
of the triangle coincides with a diameter of the circle
terms of R, what is the maximum possible area of s
triangle?
82. Five rectangular pieces of sheet metal measure 210 cm
336 cm each. Equal squares are to be cut from all their
ners, and the resulting five cross-shaped pieces of metal
to be folded and welded to form five boxes without top
The 20 little squares that remain are to be assembled
groups of four into five larger squares, and these five le
squares are to be assembled into a cubical box with not
What is the maximum possible total volume of the six b
that are constructed in this way?
83. A mass of clay of volume V is formed into two spheres in
what distribution of clay is the total surface area of the
spheres a maximum? A minimum?
84. A right triangle has legs of lengths 3 m and 4 m. What is h
maximum possible area of a rectangle inscribed in the tri
gle in the "obvious" way-with one corner at the triangel
right angle, two adjacent sides of the rectangle lying on th
triangle's legs, and the opposite corner on the hypoteru
85. What is the maximum possible volume of a right circl
cone inscribed in a sphere of radius R?
86. A farmer has 400 ft of fencing with which to build and
angular corral. He will use some or even all of an ex
ing straight wall 100 ft long as part of the perimeter d
corral. What is the maximum area that can be enclosed
87. In one simple model of the spread of a contagious disc
among members of a population of M people, the incide
of the disease, measured as the number of new cases p
day, is given in terms of the number x of individuals alread
infected by
R(x) = kx (M-x) = kMx-kx².
where k is a positive constant. How many individuak
the population are infected when the incidence Ris
greatest?
88. Three sides of a trapezoid have length L, a constant W
should be the length of the fourth side if the trapezoid
have maximal area?
89. A box with no top mu
wide, and the total sur
What is the maximum
90. A small right circular
(Fig. 3.MP.1). The larg
altitude H. What is the
larger cone that the sm
FIGUR
cone in
one (Pr
91. Two vertices of a trap
the other two lie on the
is the maximum possil
area of a trapezoid w
A=h(b + b₂)/2.]
92. Suppose that f is a d
whole real number lin
a point Q(x, y) close
graph. Show that
at Q. Conclude that
the line tangent to the
the square of the dista
93. Use the result of Prob
tance from the point
Ax+By+C =0 is
94. A race track is to be
equal straightaways c
(Fig. 3.MP.2). The leng
4 km. What should its
area within it?
FIGUR
race tr
rectan
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