Exercício de Derivadas

Prévia do material em texto

CCET – Unimontes
1st Civil Engineering
Little Homework – Derivative
R. Antonio Gonc¸alves
Montes Claros 7 de junho de 2016
1
1 The Derivative - Problems
1.1 Beginning at the beginning
1. If y =
x− 1
x+ 1
and x changes from x = 2.5 to x = 2. Find ∆x and ∆y.
2. If y = x2 − 3x+ 2 and x changes from x = x to x = x+ ∆x, find ∆y.
3. Let P (x, y) and Q(x + ∆x, y + ∆y) be two points on the curve of Problem 2.
Find the slope of the secant line PQ.
4. Find the derivative of y = x2 + 3x+ 5.
5. Find the derivative of y =
1
x− 2 at x = 1 and x = 3. Show that the derivative
does not exist at x = 2, where the function is discontinuous.
6. Find the derivative of f(x) =
2x− 3
3x+ 4
. Examine the derivative at x = −4
3
, where
the function is discontinuous.
7. Find the derivative of y =
√
2x+ 1.
8. Find the derivative of f(x) = x1/3. Examine f ′(0).
9. The slope of a curve at a point P (x, y) on it is given by the value of derivative
at the point. Find the slope of the curve y = x2 + 3x + 5 at x = −3, x = −3/2
and x = 2.
10. The distance s feet that a body, falling freely from rest, travels in the first t
seconds is given by s = 16t2. (a) Determine the average velocity of the body
during the interval [t, t + ∆t]. (b) Find the average velocity of the body during
the interval t= 10 sec to t = 11sec. (c) Determine the instantaneous velocity of
the body at the end of t sec and at the end of 8 sec.
2
1.2 Application of differentiation formulas
Differentiate each of the following:
11. y = 4 + 2x− 3x2 − 5x3 − 8x4 + 9x5
12. y =
1
x
+
3
x2
+
2
x3
13. y = 2x1/2 + 6x1/3 − 2x3/2
14. y =
2
x1/2
+
6
x1/3
− 2
x3/2
− 4
x3/4
15. y =
3
√
3x2 − 1√
5x
16. s = (t2 − 3)4
17. z =
3
(a2 − y2)2
18. f(x) =
√
x2 + 6x+ 3
19. y = (x2 + 4)2(2x3 − 1)3
20. f(t) = (2t− 1)√3− t2
21. y =
3− 2x
3 + 2x
22. y =
x2√
4− x2
23. y =
√
3− x2
x2 + 1
1.3 Application of inverse functions
24. Find
dy
dx
, given x = y
√
1− y2.
25. Find the slope of the curve x = y2− 4y at the points where it crosses the y-axis.
1.4 Differentiation of a function of a function
26. Find
dy
dx
, given y =
u2 − 1
u2 + 1
and u = 3
√
x2 + 2.
27. A point moves along the curve y = x3 − 3x+ 5 so that x = 1
2
√
t+ 3, where t is
time. At what rate is y changing when t = 4?
3
28. A point moves in the plane according to the law x = t2 + 2t, y = 23 − 6t. Find
dy
dx
when t = 0, 2, 5.
29. If y = x2 − 4x and x = √2t2 + 1, find dy
dx
when t =
√
2.
1.5 Successive differentiation
30. Show that the function f(x) = x3 + 3x2 − 8x+ 2 has derivatives of all orders at
x = a.
31. Investigate the successive derivatives of f(x) = x4/3 at x = 0.
1.6 Implicit differentiation
32. Find y′, given 4x2 + 9y2 = 36.
33. Find y′, given x2y − xy2 + x2 + y2 = 0.
34. Find y′ and y′′, given x2 − xy + y2 = 3
35. Find y′ and y′′, given x3 − 2x2y + 4xy2 − 8xy + 6x = 3 and y = 2.
36. Find y′ and y′′, given x3y + xy3 = 2 and x = 1.
1.7 Differentiation of trigonometric functions
In problems 37-44, find the first derivative.
37. y = sin 3x+ cos 2x
38. y = tanx2
39. y = tan2x
40. y = cotg(1− 2x2)
41. y = sec3
√
x
42. ρ =
√
csc2θ
43. f(x) = x2sinx
44. f(x) =
cosx
x
In problems 45-48, find the indicated derivative.
45. y = xsinx; y′′′
4
46. tan2(3x− 2); y′′
47. y = sin(x+ y); y′′
48. siny + cosx = 1; y′′
49. f ′(pi/3), f ′′(pi/3), f ′′′(pi/3), given f(x) = sin x cos 3x
50. Find the acute angles of intersection of the curves (1) y = 2 sin2x and (2)y =
cos 2x on the interval 0 < x < 2pi.
H
ig
h
w
a
y
20
Highway 32
s
256
10
8
lake
P
51. A rectangular plot of ground has two
adjacent sides along highways 20 and
32. In the plot is a small lake, one end
of which is 256 ft from Highway 20 and
108 ft from Highway 32. Find the length
of the shortest straight path which cuts
across the plot from one highway to the
other and passes by the end of the lake.
52. Discuss the curve y = f(x) = 4 sin x− 3 cos x on the interval [0, 2pi]
1pi
2
2pi
2
3pi
2
4pi
2
x
f(x)
−3
5
5
A
b
dc
θ
φ
53. Four bars of lengths a, b, c, d are hinged to-
gether to form a quadrilateral. Show that
the area A is greatest when the opposite
angles are supplementary.
54. A bombardier is sighting on a target on
the ground directly ahead. If the bomber
is flying 2 miles above the ground at 240
mi/hr, how fast must the sighting instru-
ment be turning when the angle between
the path of the bomber and the line of
sight is 30o?
55. A kite, 120 feet above the ground, is mo-
ving horizontally at the rate of 10 ft/sec.
At what rate is the inclination of the string
to the horizontal changing when 240 ft of
string are out?
x
2
240 mi/hr
θ
θ
x
120
y
θ
P
A
Q
B
θ1
θ2
56. A ray of light passes through the air with
velocity v1 form a point P , a units above
the surface of a body of water, to some
point O on the surface and then with ve-
locity v2 to a point Q, b units below the
surface. If OP and OQ make angles θ1
and θ2 with a perpendicular to the sur-
face,, show that the passage from P to Q
is most rapid when
sin θ1
sin θ2
=
v1
v2
.
1.8 Differentiation of inverse trigonometric function
In Problems 57-64, find the first derivative.
6
57. y = arc sin (2x− 3)
58. y = arc cos x2
59. y = arc tan 3x2
60. y = arc cotg
1 + x
1− x
61. y = x
√
a2 − x2 + a2 arc sinx
a
62. y = x arc csc
1
x
+
√
1− x2
63. y =
1
ab
arc tan(
b
a
)tan x
64. y2sinx+ y = arc tan x
65. In a circular arena there is a light at L. A boy starting from B runs at the rate
of 10 ft/sec toward the center O. At what rate will his shadow be moving along
the side when he is half-way from B to O?
L r
B
x
p s
θ
66. The lower edge of a mural, 12 feet high, is 6 feet
above an observer’s eyes. Under the assumption
that the most favorable view is obtained when the
angle subtended by mural at the eye is a maximum,
at what distance from the wall should the observer
stand? x
6
12
θ1
θ2
1.9 Differentiation of exponential and logarithmic functions
In problems 67-77, find the first derivative.
7
67. y = loga 3x
2 − 5
68. y = ln(x+ 3)2
69. y = ln2(x+ 3)
70. y = ln(x3 + 2)(x2 + 3)
71. y = ln
x4
(3x− 4)2
72. y = ln sin 3x
73. y = ln(x+
√
1 + x2)
74. y = e−
1
2
x
75. y = ex
2
76. y = a3x
2
77. y = x23x
78. y =
eax − e−ax
eax + e−ax
79. Find y′′, given y = e−x lnx.
80. Find y′′, given y = e−2x sin3x. Use logarithmic differentiation to find the first
derivative.
81. y = (x2 + 2)3(1− x3)4
82. y =
x(1− x2)2
(1 + x2)1/2
83. Locate (a) the relative maximum and minimum points and (b) the points of
inflection of the curve.
84. Discuss the probability curve y = ae−b2x2 ,a > 0.
85. The equilibrium constant K of a balanced chemical reaction changes with the
absolute temperature T according to the law K = Koe
− 1
2
q
T − To
ToT , where Ko, q
and To are constants. Find the percentage rate of change of K per degree of
change of T .
86. Discuss the damped vibration curve y = f(t) = e−
1
2
t sin2pit.
87. The equation s = ce−btsin(kt + θ), where c, b, k and θ are constants, represents
damped (slowed down) vibratory motion. Show that a = −2bv − (k2 + b2)s.
8
1.10 Tangent, normal, subtangent subnormal
Tangent and normals:
88. Find the points of tangency of horizontal and vertical tangents to the curve
x2 − xy + y2 = 27.
89. Find the equations of the tangent and normal to y = x5 − 2x2 + 4 at (2,4).
90. Find the equations of the tangent and normal to x2 + 3xy + y2 = 5 at (1,1).
91. Find the equationsof the tangents with slope m = −2/9 to the ellipse 4x2+9y2 =
40.
92. Find the equations of the tangent, through the point (2,-2), to the hyperbola
x2 − y2 = 16.
93. Find the equations of the vertical lines which meet the curves (1)y = x3 + 2x2−
4x+ 5 and (2)3y = 2x3 + 9x2 − 3x− 3. Angle of intersection:
94. Find the acute angles of intersection of the curves (1)y2 = 4x and (2)2x2 =
12− 5y.
95. The cable of a certain suspension bridge is attached to supporting pillars 250 feet
apart. If it hangs in the form of a parabola with the lowest point 50 feet below
the point of suspension, find the angle between the cable and pillar.
θ1
θ2
(125,50)
Length of tangent, normal, subtangent, subnormal:
96. Find the length of the subtangent, subnormal, tangent, and normal of xy+ 2x−
y = 5 at the point (2,1).
9
1.11 Functions and their graphs
Increasing and decreasing functions, maxima and minima.
97. Given y =
1
3
x3 +
1
2
x2 − 6x + 8, find (a) the critical points, (b) the intervals on
which y is increasing and decreasing, and (c) the maximum and minimum values
of y.
98. Given y = x4 + 2x3− 3x2− 4x+ 4, find (a) the intervals on which y is increasing
and decreasing, and (b) the maximum and minimum values of y.
99. Show that the curve y = x3 − 8 has no maximum or minimum value.
100. Locate the maximum and minimum values of f(x) = 2 + x2/3 and the intervals
on which the function is increasing and decreasing.
101. Examine y = f(x) =
1
x− 2 for maxima and minima, and locate the intervals on
which the function is increasing and decreasing.
102. Examine f(x) =
√
x− 1 for maximum and minimum values.
103. Examine f(x) = x(12−2x)2 for maxima and minima using the second derivative
method.
104. Examine y = x2 +
250
x
for maxima and minima using the second derivative
method.
105. Examine y = (x−2)2/3 for maximum and minimum values. Directions of bending,
points of inflection.
106. Examine y = 3x4 − 10x3 − 12x2 + 12x − 7 for directions of bending and points
of inflection.
107. Examine y = x4 − 6x+ 2 for directions of bending and points of inflection.
X
Y
(0,2)
(Problem 107)
X
Y
(-2,-6)
(Problem 108)
10
108. Examine y = 3x+ (x+ 2)3/5 for directions of bending and points of inflection.
109. Find the equation of the tangents at the points of inflection of
y = f(x) = x4 − 6x3 + 12x2 − 8x.
1.12 Related rates
110. Gas is escaping from a spherical balloon at the rate of 2 cubic feet per minute
(2ft3/min). How fast is the surface area shrinking when the radius is 12 ft?
8
4
r
h–
111. Water is running out of a conical funnel at the
rate of 1 cubic inch per sec(1in.3/sec). If the
radius of the base of the funnel is 4 in. and the
altitude is 8 in., find the rate at which the water
level is dropping when it is 2 in. from the top.
112. Sand falling from a chute forms a conical pile whose altitude is always equal to
4/3 the radius of the base.
A. How fast is the volume increasing when the radius of the base is 3 feet and is
increasing at the rate of 3in./min?
B. How fast is the radius increasing when it is 6 ft and the volume is increasing
at the rate of 24ft3/min?
113. One ship is sailing due south at 16 mi/hr and a
second ship B, 32 miles south of A, is sailing due
east at 12 mi/hr.
A. At what rate are they approaching or separating
at the end of 1 hr?
B. At the end of 2 hr?
C. When do they cease to approach each other and
how far apart are they at that time?
A0
1
6t
12t
32
-1
6
t
B0
114. Two parallel sides of a rectangle are being lengthened at the rate of 2 in./sec,
while the other two sides are shortened in such a way that figure remains a
11
rectangle with constant are A of 50 square inches. What is the rate of change of
the perimeter P when the length if an increasing side is
A. 5 in.?
B. 10 in.?
C. What are the dimensions when the perimeter ceases to decrease?
115. The radius of a sphere is rin. at the time t sec. Find the radius when the rates
of increase of the surface area and the radius are numerically equal.
y
P
W
x
2
A
30 +
x2
0
−
x
116. A weight W is attached to a rope 50 ft long which
passes over a pulley at P, 20 ft above the ground.
The other end of the rope is attached to a truck
at a point A, 2 ft above the ground. If the truck
moves off at the rate of 9 ft/sec, how fast is the
weight rising when it is 6 ft above the ground?
y
h
L
H
T
vto
117. A light L hangs H ft above a street and an object h
ft tall at O, directly under the light, is moved in a
straight line along the street at v ft/sec. Investigate
the velocity V of the tip of the shadow on the street
after t sec.
12
20
5
h
r
118. Water is running into a conical reservoir, 20 ft deep
and 10 ft in diameter, at 1ft3/minute.
a) At what rate is the water level rising when the
water is 6 ft deep?
b) At what rate is the area of the water surface
increasing when the water is 8 ft deep?
c) At what rate is the wetted surface of the reservoir
increasing when the water is 10 ft deep?
13