MATH 220: Worksheet 13
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October 8, 2013
101 Derivatives!
Find y 0 or f 0 .
1. y = ln(2x2 +3) cos(5x−
1)
t4 − 1
19. y = 4
t +1
q
√
38. y = sin x
2. y = x ln x − x
20. xey = y sin x
39. y = log5 (1 + 2x)
3. y = ln(sin x)
21. y = ln(x ln x)
1
4. y =
ln x
22. y = emx cos nx
√
√
23. y = x cos x
2
x
5. y = log5 (xe )
x
1 + ln x
√
7. y = ln(x + x2 − 1)
cos x
9. y = (ln x)
10. y = (tan x)
11. y =
1/x
e−x cos2 x
x2 + x + 1
12. y = (x2 + x3 )4
13. y = tan[ln(ax + b)]
1
1
14. y = √ − √
5
x
x3
x2 − x + 2
√
15. y =
x
tan x
1 + cos x
42. y =
25. y =
1 2
sin x
2
(x2 + 1)4
(2x + 1)3 (3x − 1)5
43. y = x tan−1 (4x)
1/x
e
x2
44. y = ecos x + cos(ex )
26. y = ln sec x
45. y = ln | sec 5x + tan 5x|
27. y + x cos y = x2 y
4
x−1
28. y =
x2 + x + 1
√
29. y = arctan x
46. y = 10tan πx
30. y = cot(csc x)
x
31. y = tan
1 + x2
x sec x
32. y = e
33. y = 3x ln x
34. y = sec(1 + x2 )
17. y = x2 sin πx
35. y = (1 − x−1 )−1
q
√
3
36. y = 1/ x + x
18. y = x cos−1 x
37. sin(xy) = x2 − y
16. y =
41. y = ln sin x −
24. y = (arcsin 2x)2
6. y =
8. y = (sin x)ln x
40. y = (cos x)x
1
47. y = cot(3x2 + 5)
p
48. y = x ln(x4 )
√
49. y = sin(tan 1 + x3 )
√
50. y = arctan(arcsin x)
51. y = tan2 (sin x)
52. xey = y − 1
√
x + 1(2 − x)5
53. y =
(x + 3)7
54. y =
(x + λ)4
x4 + λ4
sin mx
x
2
x − 4
56. y = ln 2x + 5 55. y =
√
57. y = cos e tan 3x
2
√
58. y = sin (cos sin πx)
59. y = (3x2 + 7)(x2 − 2x +
3)
60. y = (x3 − 3x)(x + 2)
√
61. y = x sin x
3
62. y = x cos x
63. y =
x2 + x − 1
x2 − 1
6x − 5
64. y = 2
x +1
65. y =
1
4 − 3x2
66. y =
9
2
3x − 2x
2
67. y =
x
cos x
68. y =
sin x
x2
69. y = 3x2 sec x
72. y = 3x sin x + x2 cos x
2
x−3
73. y =
x2 + 1
5
1
2
74. y = x +
x
75. y = (x2 − 1)5/2 (x3 + 5)
76. y =
77. y = 3 cos(3x + 1)
78. y = 1 − cos 2x + 2 cos2 x
79. y =
x sin 2x
−
2
4
sec7 x sec5 x
80. y =
−
7
5
2 3/2
81. y
=
sin x −
3
2 7/2
sin x
7
82. y = √
3x
x2 + 1
83. y =
sin πx
x+2
84. y =
cos(x − 1)
x−1
70. y = 2x − x2 tan x
71. y = x cos x − sin x
x
(1 − x)3
85. x2 + 3xy + y 3 = 10
86. x2 + 9y 2 − 4x + 3y = 0
√
√
87. y x − x y = 16
88. y 2 = (x − y)(x2 + y)
89. x sin y = y cos x
90. cos(x + y) = x
91. f (x) = x2 g(x)
92. f (x) = g(x2 )
93. f (x) = [g(x)]2
94. f (x) = g(g(x))
95. f (x) = g(ex )
96. f (x) = eg(x)
97. f (x) = ln |g(x)|
98. f (x) = g(ln x)
g(x)h(x)
g(x) + h(x)
s
g(x)
100. f (x) =
h(x)
99. f (x) =
101. f (x) = g(h(sin 4x))
An Ending Thought: Practice doesn’t make perfect. Perfect practice makes
perfect.
– Rick Brockway (my middle school band director)
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Rates of Change
1. When you give some a drug, the rate at which the drug is absorbed is proportional to
the amount left in the bloodstream. For example, it might be that
dv
= −2v
dt
where v(t) represents the amount of the drug left in the bloodstream (in ml) at time t
(in hours).
(a) Suppose that someone gets a dose of 10 ml at time t = 0. How much is left after
time t?
(b) Let’s say this was poison, and we find the dead body with 1 ml of poison in his
bloodstream. How long ago was he poisoned?
2. Scientists use a technique called carbon dating to determine how old fossils are. Every living thing has a fixed proportion of a particular radioactive isotope of carbon
(called C-14) while it’s alive. Once it dies, that radioactive carbon starts decaying
exponentially with a half-life of about 5500 (so after 5500 years, half of the carbon has
decayed).
(a) After 11000 years, how much of the carbon has decayed? How much is left?
(Answer will be as a fraction of the original amount of carbon)
(b) Find an equation for the amount of carbon left since the thing died.
(c) Suppose an organism started with 1 gram of C-14 when it died, and it’s measured
to have .1 grams now. How long ago did it die?
3. Newton’s Law of Cooling states the rate of change of the temperature of an object
is proportional to the difference between its own temperature and the temperature of
its surroundings. In a murder investigation, the temperature of the corpse was 32.5◦ C
at 1:30 PM and 30.3◦ C an hour later. Normal body temperature is 37.0◦ C and the
temperature of the surroundings was 20.0◦ C. When did the murder take place? (Hint:
perform a change of variables y(t) = T (t) − 22)
4. The rate of change of atmospheric pressure P with respect to altitude h is proportional
to P , provided the termperature is constant. At 15◦ C, the pressure is 101.3 kPa at sea
level and 87.14 kPa at h = 1000m.
(a) What is the pressure at an altitude of 3000 m?
(b) What is the pressure at the top of Mount MicKinley, at an altitude of 6187 m?
5. The height of a ball at time t is given (in feet) by s(t) = −16t2 + 64t + 32. At what
speed was the ball initially thrown into the air? From where? What is the greatest
height it reaches? When does it hit the ground? With what velocity does it hit the
ground?
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Related Rates
Related rates involve computing the change of one quantity (like volume of a cube) with
respect to the rate of change of another quantity (like the side of the cube). The big picture
idea is to find an equation that relates the two quantities and then use the Chain Rule to
differentiate both sides with respect to time.
Tips for Success with Related Rates
1. Draw a picture
2. Give variables to all of your quantities.
3. Before working on the problem, make one list of everything you know. Make another
list of what you are being asked to find.
4. Before taking the derivative, make sure you have one equation with just the variables
you know something about.
5. Remember that you will always need the Chain Rule. If you don’t use the chain rule,
you probably made a mistake.
6. Pay attention to units! They are important in your answer and also give you clues
as to what information you already have. A unit like m3 is volume, anything that is
unit/s is going to some derivative, etc.
Some examples
1. Imagine we are blowing up a spherical balloon. We are blowing air into it at a rate of
100 cm3 /s. How fast is the radius of the balloon increasing when the diameter is 50
cm? The steps below will walk you through the problem.
(a) What quantity is changing at a rate of 100 cm3 /s? What derivative is equal to
this value? (Hint: look at the units)
(b) What quantity do we want to find? (Hint: this is a derivative)
(c) If the diameter is 50 cm, what do we know about the radius?
(d) What is an equation that relates our two quantities? Take the derivative with
respect to t. (This will involve the chain rule)
(e) Solve the equation you got above for the quantity we want to find. Use the values
we have to compute it. What are the units on this quantity?
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2. A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away
from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall
when the bottom of the ladder is 6 ft from the wall?
(a) Draw a picture of the situtation described. What side of the triangle do you
know? Label it. The other two sides are changing. Give them variables.
(b) One of the sides, you know how fast it is changing. Which one is it? Write this
=? (fill in the ?s).
as d?
dt
(c) You know one of the value of the sides. Which one is it? Write this as ? =?
(d) What quantity are you being asked to find in terms of the variables you have?
)
(This will be like d?
dt
(e) Consider the two unknown sides. How are they related? Write an equation for
this relationship. You don’t need to solve for y
(f) Take the derivative of this equation. (Use the Chain Rule and Implicit Differentiation)
(g) Plug in the values you have to find the quantity you want. Remember your units.
3. A water tank has the shape of an inverted circular cone with base radius 2 m and
height 4 m. If water is being pumped into the tank at a rate of 2 m3 /min, find the
rate at which the water level is rising when the water is 3 m deep.
(a) Draw a picture. Label the quantities you are interested in (either you know
something about them or you want to know something about them).
(b) What quantities do we already know? Write these with their variables. What
quantity are we looking for?
(c) We know the volume of a cone is V = 31 πr2 h, but we don’t really know anything
about the radius. Use similar triangles to find the radius in terms of the height
of the triangle. Substitute this into your equation for volume.
(d) Take the derivative (what rule do you need?).
(e) Use the information you have to find the answer.
An Ending Thought: I am not a product of my circumstances. I am a product
of my decisions.
– Stephen Covey
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