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Calculus Problems for Cutting and Pasting
By: Patrick Bourque
Chapter The First: Limits and Continuity
1.
Find a suitable δ for the following limit.
limx→5
(2x− 6) = 4
2.
Find a suitable δ for the following limit.
limx→−2
(2− 3x) = 8
3.
Find a suitable δ for the following limit.
limx→−2
(2x2 − x+ 5) = 15
4.
Find a suitable δ for the following limit.
limx→3
(x2 − 6x+ 2) = −7
5.
Find a suitable δ for the following limit.
limx→1
(7x2 + 9x− 11) = 5
6.
Find a suitable δ for the following limit.
limx→−2
(2x2 + 9x− 1) = −11
7.
Find a suitable δ for the following limit.
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limx→−2
5 = 5
8.
Find a suitable δ for the following limit.
limx→4
√x = 2
9.
Find a suitable δ for the following limit.
limx→1
1
x= 1
10.
Show that δ =√
εa is a suitable δ of a quadratic with leading coefficient of
a and x approaching the vertex.
11.
Evaluate the limit or show it does not exist.
limx→5
x2 − 25
x− 5
12.
Evaluate the limit or show it does not exist.
limx→1
x5 − 1
x− 1
13.
Evaluate the limit or show it does not exist.
limx→3
4x4 − 12x3 + 2x2 − x− 15
x− 3
14.
Evaluate the limit or show it does not exist.
limx→2
x4 − 3x3 + x2 + 2x
x− 2
15.
Evaluate the limit or show it does not exist.
limx→0
e2x − 5ex + 4
ex − 1
16.
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Evaluate the limit or show it does not exist.
limx→π
2
sin(2x)
cos(x)
17.
Evaluate the limit or show it does not exist.
limx→5
√4x+ 16− 6
x− 5
18.
Evaluate the limit or show it does not exist.
limx→8
x2 − 64√x+ 1− 3
19.
Evaluate the limit or show it does not exist.
limx→0
√1 + x−
√1− x
x
20.
Evaluate the limit or show it does not exist.
limx→0
√1 + sin(x)−
√1 + x
x
21.
Evaluate the limit or show it does not exist.
limx→0
√ex + 3− 2
ex − 1
22.
Evaluate the limit or show it does not exist.
limx→5
1x+1 −
16
x− 5
23.
Evaluate the limit or show it does not exist.
limx→1
(1
x− 1+
1
x2 − 3x+ 2
)24.
Evaluate the limit or show it does not exist.
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limx→3
√x+ 6− xx2 − 9
25.
Evaluate the limit or show it does not exist.
limx→2
sin(x− 2)
x2 − 4
26.
Evaluate the limit or show it does not exist.
limx→3
x− 3
sin(x2 − 9)
27.
Evaluate the limit or show it does not exist.
limx→0
x sin(x)
(x+ sin(x))2
28.
Evaluate the limit or show it does not exist.
limx→π
4
cos(2x)
cos(x)− sin(x)
29.
Evaluate the limit or show it does not exist.
limx→4
sin(√x− 2)
x2 − 7x+ 12
30.
Evaluate the limit or show it does not exist.
limx→1
sin(1− 1x )
x− 1
31.
Evaluate the limit or show it does not exist.
limx→0
sin(e2x − 1)
ex − 1
32.
Evaluate the limit or show it does not exist.
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limx→0
sin(1− cos(x))
x
33.
Evaluate the limit or show it does not exist.
limx→9
√x− 5− 2√x− 3
34.
Evaluate the limit or show it does not exist.
limx→4
1√x− 1
2√x+ 5− 3
35.
Evaluate the limit or show it does not exist.
limx→0
sin(sin(x))
x
36.
Evaluate the limit or show it does not exist.
limx→0
1− cos(x)
x2
37.
Evaluate the limit or show it does not exist.
limx→π
4
sin(x)− cos(x)
1− tan(x)
38.
Evaluate the limit or show it does not exist.
limx→ 3π
4
sin(x) + cos(x)
1 + tan(x)
39.
Find values of a and b so that
limx→0
√ax+ b− 2
x= 1
40.
Evaluate the limit or show it does not exist.
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limx→0
sin(x) arctan
(1
x
)41.
If f(x) < g(x) < h(x) for x 6= 2 and
f(x) = ex2−4x−2 h(x) = ex
2
Find
limx→2
g(x)
42.
Find the value of a that makes the following limit exist
limx→−2
ax2 + 15x+ 15 + a
x2 + x− 2
43.
Show the following limit Does not Exist
limx→0
sin |x|x
44.
Find the value of a that makes the following limit exist
limx→2
f(x) f(x) =
x+ 5 x ≤ 2
ax− 2 2 < x
45.
Find the value of a that makes the following limit exist
limx→a
f(x) f(x) =
x2−a2x−a x < a
ax+ 1 a ≥ x
46.
Find values of a, b and c so that the function f(x) has the following proper-
ties.
limx→1
f(x) exists limx→4
f(x) exists f(2) = 3
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f(x) =
x2−1x−1 x < 1
ax2 + bx+ c 1 ≤ x ≤ 4x2+3x−28
x−4 x > 4
47.
A
Evaluate the limit or show it does not exist.
limx→0
√1 + x+
√1− x− 2
x2
Hint:
√1 + x+
√1− x− 2
x2=
1
x2
((√
1 + x− 1) + (√
1− x− 1)
)48.
Find all Vertical Asymptotes of the function.
f(x) =x
x− 3
49.
Find all Vertical Asymptotes of the function.
f(x) =x− 1
x3 − 2x2 − x+ 2
50.
Find all Vertical Asymptotes of the function.
f(x) =
√x+ 1
x2 − 4
51.
Find all Vertical Asymptotes of the function.
f(x) =ln(x)
9− x252.
Find all Vertical Asymptotes of the function.
f(x) =sin(x)
x2 − x53. Show that x = 0 is a vertical asymptote.
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f(x) =sin(x)
1− cos(x)
54.
Find all Vertical Asymptotes of the function on the interval [0, 2π].
f(x) =sin(x)
sin(2x) + sin(x)
55.
For what values of A does the following equation have 0, 1 or 2 vertical
asymptotes?
f(x) =1
x2 +Ax+ 9
56.
The following function has a vertical asymptote at x = 0. Use the definition
of a vertical asymptote to find a suitable δ to show
limx→0
1
x2=∞
57.
Find all values of x that make the function discontinuous. Label each dis-
continuity as removable or nonremovable.
f(x) =x− 1
x3 − x2 − 4x+ 4
58.
Find all values of x that make the function discontinuous. Label each dis-
continuity as removable or nonremovable.
f(x) =x
ln(x)− 1
59.
f(x) =
2x+ 1 x < −1
x2 − 1 < x ≤ 4
3x x > 4
Sketch this curve for the students and discuss the following:
limx→−1−
f(x) limx→−1+
f(x) limx→4−
f(x) limx→4+
f(x)
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Discuss the continuity of the function
60.
Show the following function is continuous at x = 0
f(x) =
x arctan
(1x
)x 6= 0
0 x = 0
61.
Find all values of x that make the function discontinuous. Label each dis-
continuity as removable or nonremovable.
f(x) =
2x+ 3 x ≤ 2
5x− 3 2 < x < 4
3x x ≥ 4
62.
Find all values of x that make the function discontinuous. Label each dis-
continuity as removable or nonremovable.
f(x) =
x2−4x−2 x < 2
2x 2 ≤ x < 6
x2 x ≥ 6
63.
Find all values of k that make the function continuous.
f(x) =
x x ≤ 1
2 sin(kx) 1 < x < 9
7− x x ≥ 9
64.
For the given function show that x = a is a nonremovable discontinuity.
f(x) =|x− a|x− a
65.
Find all values of a and b that make the function continuous.
f(x) =
x2−4x+2 x < −2
ax+ b − 2 ≤ x ≤ 5x2−25x−5 x > 5
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66.
Find all values of a and b that make the function continuous.
f(x) =
sin(x2−9)x+3 x < −3
ax+ b − 2 ≤ x ≤ 6x2−36x−6 x > 6
67.
Find all values of a and b that make the function continuous.
f(x) =
x2+6x+8x+2 x < −2
ax+ b |x| ≤ 2sin(x2−4)x−2 x > 2
68.
Find the equation of a circle with center (3, 0) that will make the function
continuous
f(x) =
4− x x ≤ 0
circle 0 < x < 6
x− 2 x ≥ 6
69.
Show that it is impossible to find a quadratic function g(x) with a vertex at
x = 3 that will make f continuous.
f(x) =
x+ 1 x < 2
g(x) 2 ≤ x ≤ 4
3−√x x > 4
70.
Show the following polynomial has roots on each of the following intervals
f(x) = x3 − 3x2 − 10x+ 24
I1 = [−4,−2] I2 = [−2, 3] I3 = [3, 5]
71.
Use consecutive approximations to estimate the root of
f(x) = x3 + 2.777x2 − 1.892x− 3.669
between x = 1 and x = 2
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Chapter The Second: Derivatives
72.
Use the definition to find f ′(x).
f(x) = 3x2 − 5x+ 4
73.
Use the definition to find f ′(x).
f(x) =√
6x+ 1
74.
Use the definition to find f ′(x).
f(x) =6x
x+ 1
75.
Use the definition to find f ′(x).
f(x) = sin(2x)
76.
Use the definition to find f ′(x).
f(x) = cos(4x)
77.
Use the definition to find f ′(x).
f(x) = tan(4x)
78.
Find the equation of the tangent line to f when x = 4. Find the equation
of the normal line to f when x = 4.
f(x) =√
2x+ 1
79.
Show that f is not differentiable at x = 2.
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f(x) = |x− 2|
80.
Show that f is not differentiable at x = 2.
f(x) = (x− 2)23
81.
Use the definition of the derivative to find f ′(0)
f(x) =
x2 x ≤ 0
x3 x > 0
82.
Use the definition of the derivative to find f ′(0)
f(x) =
x2 x ≤ 0
x x > 0
83.
Find values of α and β to make the function differentiable for all real num-
bers.
f(x) =
α− βx2 x < 1
x2 x ≥ 1
84.
Use the definition of the derivative to find f ′(0)
f(x) =
x2 sin
(1x
)x 6= 0
0 x = 0
85.
Use the alternate definition of the derivative to find f ′(8)
Definition: f ′(c) = limx→c
f(x)− f(c)
x− c
f(x) = 3√x
86.
If a function is differentiable at x = c then
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f ′(c) = limh→0
f(c+ h)− f(c− h)
2h
Show that if the function is not differentiable at x = c by then the above
equality does not hold by calculating the above limit with
f(x) = |x| c = 0
87. Use the differentiation rules to find f ′.
f(x) = 4x3 − 5x2 − 5√x+ sin(x)− 8 cos(x)
88. Use the differentiation rules to find f ′.
f(x) =(x− 3)(x− 2)√
x
89.
For the two functions below find points on f where the tangent line to f is
parallel to g.
f(x) = x3 − 15
2x2 − 32x+ 1 g(x) = 10x+ 2
90.
The position of a particle is given by s(t) = t3 − 3t2 − 9t− 1. Find intervals
where the particle is moving forward and backward. Find the distance the
particle traveled backwards.
91.
Find the values of x where the tangent line to f(x) = 2x2 + 12x is parallel
to the tangent line to g(x) = x2 + 203 x
32
92.
Find the two points on the curve f(x) = x2 where the tangent line passes
through the point (1,−3). Hint: make a sketch of what this may look like.
93.
Let f(x) be a quadratic with roots at (a, 0) and (b, 0). Show that the slope
tangent line to f at these two points are the negatives of each other.
94.
Show that the tangent line to y = cx2 at any point P (a, b) Crosses the x-axis
at (a2 , 0).
95.
Let
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f(x) = −1 +1
(x− a)2
Find the two roots of f and the tangent line at each root. Show that the
two roots along with the intersection of the tangent lines form a triangle of area
2.
96.
Consider the function f(x) = 1x . Show that the tangent line to the curve
together with the x and y axis form a triangle with area 2
97.
Consider the function f(x) = x2. Let P be a point on the parabola. Let H
be a horizontal line passing through P and Let N be the normal line to f at P .
Show that the distance between the y-intercepts of H and N is always 12 .
98.
Find values of A and B so that the two curves fit together smoothly at x = 1
y1 = A+Bx2 + x4 y2 = x2
99.
Use calculus to show that y defines a parabola with a vertex at x = c
y = (x− c− h)2 + (x− c)2 + (x− c+ h)2
100.
The lines y1 = x2 + ax+ b and y2 = cx−x2 share a common tangent line at
(1, 0). Find a, b and c.
101.
Let f(x) = x2. Find equations of the tangent line when x = t and x = t+ 1.
Show these tangent lines intersect when x = t+ 12
102.
Show that if f(x) is defined at x = 0 and bounded for x near zero then
g(x) = x2f(x) is differentiable at zero and g′(0) = 0
103.
Use the differentiation rules to find f ′.
f(x) = x4 sin(x)− x2 cos(x)
104.
Use the differentiation rules to find f ′.
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f(x) =tan(x)
x
105.
Use the differentiation rules to find f ′.
f(x) =x cot(x)
x2 + 1
106.
Use the differentiation rules to find f ′.
f(x) = x sec(x)− x csc(x)
107.
Use the differentiation rules to find y′(0).
y(x) = exf(x) f(0) = 4 f ′(0) = 2
108.
Use the differentiation rules to find y′(0).
y(x) =f(x)g(x)
h(x)f(0) = 1 f ′(0) = 2 g(0) = 4 g′(0) = 3 h(0) = 1 h(0) = 6
109.
Use the differentiation rules to find f ′′.
f(x) =x2
x4 + 1
110.
Use the differentiation rules to find f ′′.
f(x) =(x3 − 2x) csc(x)
x+ cot(x)
111.
Use the differentiation rules to find f ′.
f(x) = (x3 + x2 − sin(x))4
112.
Use the differentiation rules to find f ′.
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f(x) =√
1 + x tan(x)
113.
Use the differentiation rules to find f ′.
f(x) =cos(√x)
x
114.
Use the differentiation rules to find f ′.
f(x) = csc
(x
x4 + π
)115.
Use the chain rule to find g′(0).
y(x) = f(cos(g(x))) g(0) =π
2y′(0) = 16 f ′(0) = 4
116.
If f ′ = g and g′ = f . Show f2 − g2 is a constant.
117.
Find f ′(0)
f(x) =3√
1 + 3x · 5√
1 + 5x · 7√
1 + 7x . . . 101√
1 + 101x2√
1 + 2x · 4√
1 + 4x · 6√
1 + 6x . . . 100√
1 + 100x
118.
Find dydx
xy = x2 + y2
119.
Find dydx √
x2 + y2 = x4 + 2x
120.
Find dydx
sin(x+ y) = x2y4 + x
121.
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Find the equations of the tangent and normal lines to the Folium of Descartes
at(2, 4).
x3 + y3 = 9xy
122.
Find two equations of the tangent lines when x = 1.
xy2 − y + 2xy = 2x
123.
Find two equations of the tangent lines when x = 0.
(x2 + 1)y2 + 3(x+ 1)y + 2 = 0
124.
Show that the tangent lines are parallel at P (1, 1) and Q(−1,−1)
(x2 + y2)2 = 4xy
125.
Use implicit differentiation to find the asymptotes to
x2 − y2 = 1
126.
Consider the curve x = y2. Show that for there to be three normal lines to
the curve that intersect the point (a, 0) then a > 12 . One of the normal lines is
the x-axis.
127.
Find d2ydx2 implicitly
x2 + y + y2 = 1
128.
Find d2ydx2 implicitly
y4 + 2y2 = x4
129.
Find the derivative of y = arctan(x) by differentiating
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x = tan(y)
130.
Prove the circle tangent theorem from geometry: The line connecting the
center of the circle to a point P on the is perpendicular to the circle’s tangent
line at P
131. Find two points on the the ellipse:
x2
4+y2
16= 1
where the tangent line passes through the point (0, 8√3)
132.
Show that the length of the tangent line to the curve x23 + y
23 = 9 bounded
by the coordinate axis is a constant.
133.
Show that the tangent lines to the curve x2 − xy + y2 = 3 where the curve
touches the x-axis are parallel
134.
For the curve√x+√y =√k. Show that the sum of the x and y intercepts
of the tangent line is k
135.
Find the rate of change of the distance from the origin to a point on the
graph of y = x2 + 1 if the x coordinate is increasing at a rate of 3 meters per
second.
136.
A ladder is 25 ft. long and is leaning against a house. The base of the ladder
is pulled away from the wall at 3 ft. per second. How fast is the top of the
ladder moving down the wall when the base of the ladder is 7 ft. from the wall?
137.
A ladder is leaning against a house. At the moment the base of the ladder
is 4 ft from the wall the base of the ladder is being pulled away from the wall
at 1 ft. per second and the top of the ladder is sliding down the wall at 2 ft per
second. How long is the ladder?
138.
Water is being drained from a downward pointing cone at a rate of 5 cubic
meters per minute. If the cone’s height is 20 meters and its diameter is 10
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meters find the rate of change in the height of the water level when the height
is 10 meters.
139.
A particle moves along the path y = x2. x is increasing at at rate of 2 m/s.
How fast is the angle from line connecting the particle to the origin and the
x-axis changing when x = 3
140.
Two cars start at a common point. One car travels West at 40 mph while
the other car travels South at 30 mph. After 2 hrs of driving how fast is the
distance between the cars changing?
141.
A plane is flying at a constant altitude of 5 miles towards an air traffic control
tower. IF the velocity of the plane is 400 mph how fast is the angle of elevation
from the tower to the plane changing when the angle is 30?
142.
A person stands 100m from a rocket launch. The rocket’s velocity is 100
m/s. How fast is the angle of elevation from the observer to the rocket changing
when the rocket is 100m high?
143.
A balloon is rising at a constant rate of 4 m/s as a person rides a bike below
the balloon at 8 m/s. When the person passes directly under the balloon the
balloon is 36 m high. What is the rate of change of the distance between the
person and the balloon 3 seconds later?
144.
The height of a box is increasing at 2 in/s while the volume of the box is
decreasing at 3 cubic inches per second. If the base of the box is square how
fast must the length and width of the box decrease?
145.
For a Right Triangle with one leg twice as long as the other and the shorter
leg increasing at 3 inches per second. How fast is the area of the triangle
changing when the shorter leg is 10 inches.
146.
A person is standing on a pier and is pulling in a boat at 1 meter per second.
The person’s hand is 4 meters above the water where the rope is connected to
the boat. How fast is the boat approaching the pier when the ropes length is 6
meters.
147.
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A 6 ft. tall man is walking away from a 15 ft. light pole at 5 ft. per second.
When he is 10 feet from the pole how fast is the distance from the base of the
pole to the tip of his shadow changing and how fast is the length of his shadow
changing?
148.
The area between two concentric circles is a constant 9π. The rate of change
of the area of the large circle is 10π. How fast is the circumference of the smaller
circle changing when it has area of 16π?
149.
Consider an isosceles triangle whose two equal sides have length of 5 inches.
Let h be the length of the line from the vertex between these sides and the
midpoint on the third side. If h = 3 and increasing at 1 inch/min how fast is
the area of the triangle changing?
150.
A point moves along the curve y = 1x with a horizontal velocity of 5. How
fast is the distance between the point and the origin changing when x = 1 What
is the vertical velocity of this point when x = 1
151.
A clock has a 12 inch minute hand and a 8 inch hour hand. At 3:25 what is
the rate of change of the distance between the tips of the two hands?
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Chapter The Third: Applications of Differentiation
152.
Find the absolute max and min of the given function on the interval [−2, 3].
f(x) = x4 − 2x2
153.
Find the absolute max and min of the given function on the interval [−3, 4].
f(x) = x5 − 5x3 − 20x− 4
154.
Find the absolute max and min of the given function on the interval [−11, 24].
f(x) = 6x2(x− 16)23
155.
Find the absolute max and min of the given function on the interval [0, 2π].
f(x) = cos(2x) + 2 cos(x)
156.
Find the absolute max and min of the given function on the interval [−π3 ,π3 ].
f(x) = tan(x)− 2x
157.
Find the absolute max and min of the given function on the interval [−1, ln(10)].
f(x) = e2x − 12ex + 10x
158.
Show the hypothesis of Rolle’s theorem apply to the function on [0, 2]. Find
all values guaranteed by the theorem.
f(x) = x4 − 4x3 + 5x2 − 2x
159.
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Show the hypothesis of Rolle’s theorem apply to the function on [1, 3]. Find
all values guaranteed by the theorem.
f(x) = x3 − 7x2 + 15x
160.
Show the hypothesis of Rolle’s theorem apply to the function on [1, 3]. Find
all values guaranteed by the theorem.
f(x) =x2 − 3x+ 5
x+ 2
161.
Show the hypothesis of Rolle’s theorem apply to the function on [0, 2π]. Find
all values guaranteed by the theorem.
f(x) = sin(2x) + 2 cos(x)
162.
Show the hypothesis of Rolle’s theorem apply to the function on [a, b]. Show
the value of c guaranteed by the theorem is the geometric mean of a and b
f(x) = ln
(x2 + ab
x(a+ b)
)163.
Show the hypothesis of the Mean Value Theorem apply to the function on
[ 12 , 2]. Find all values guaranteed by the theorem.
f(x) =x+ 1
x
164.
Show the hypothesis of the Mean Value Theorem apply to the function on
[0, 2]. Find all values guaranteed by the theorem.
f(x) = x2 + x+ 1
165.
Show the hypothesis of the Mean Value Theorem apply to the function on
[0, 4]. Find all values guaranteed by the theorem.
f(x) =√
2x+ 1
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166.
Show the hypothesis of the Mean Value Theorem apply to the function on
[0, 4]. Find all values guaranteed by the theorem.
f(x) = x3 − 6x2 + 9x+ 2
167.
Show the hypothesis of the Mean Value Theorem apply to the function on
[2, 4]. Find all values guaranteed by the theorem.
f(x) = x+1
x− 1
168.
Show that the value of c guaranteed by the Mean Value Theorem on the
interval [a, b] of the quadratic f(x) = Ax2 + Bx + C is the midpoint of the
interval [a, b].
169.
Prove the following inequality using the Mean Value Theorem.
√x+ 1 <
1
2x+ 1 x > 0
170.
Prove the following inequality using the Mean Value Theorem.
x
1 + x2< arctan(x) < x x > 0
171.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
f(x) = 3x4 − 16x3 + 18x2
172.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
f(x) = x√x+ 1
173.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
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f(x) =x2 − 2x+ 1
(x− 3)4
174.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
f(x) = 3x2/3(x2 − 36)
175.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
f(x) = 12x5/2 − 45x2 − 20x3/2 + 90x
176.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
f(x) = 2 cos(x)− sin(2x)
177.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema on [0, 2π].
f(x) = sin(x)− cos(x)
178.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
f(x) = 2e3x − 15e2x + 24ex
179.
For the given function find all critical numbers, intervals where the function
is increasing and decreasing and all relative extrema.
f(x) = x√a− x
180.
If p and q are both integers greater than or equal to 2 and
Page 25
f(x) = (x− 1)p(x+ 1)q
Find the relative extrema if:
1. p and q are both even
2. p and q are both odd
3. p is even and q is odd
4. q is even and p is odd
181.
For the given function find all second order critical numbers, intervals where
the function is concave up and down and all points of inflection.
f(x) = 3x5 + 5x4 − 20x3
182.
For the given function find all second order critical numbers, intervals where
the function is concave up and down and all points of inflection.
f(x) = e1−x2
72
183.
For the given function find all second order critical numbers, intervals where
the function is concave up and down and all points of inflection.
f(x) = x2 − 1
6x3
184.
For the given function find all second order critical numbers, intervals where
the function is concave up and down and all points of inflection.
f(x) = cos2(x)− 2 cos(x)− x2
185.
For the given function find all second order critical numbers, intervals where
the function is concave up and down and all points of inflection.
f(x) =x3
x2 − 4
Page 26
186.
For the given function find all second order critical numbers, intervals where
the function is concave up and down and all points of inflection.
f(x) = 4 sin(x)− sin(2x)
187.
Use the second derivative test to find all relative max and mins.
f(x) = 3x4 − 16x3 + 6x2 − 48x− 1
188.
Use the second derivative test to find all relative max and mins.
f(x) = 3x4 − 8x3 + 24x2 − 96x
189.
Find a third degree polynomial with the following properties: Relative ex-
trema when x = −1 and x = 4, Point of inflection when x = 32 and a y-intercept
of 2
190.
Find a fourth degree polynomial with the following properties: Increasing
on (0, 3) ∪ (3,∞), Decreasing on (−∞, 0). Concave up on (−∞, 1) ∪ (3,∞).
Concave down on (1, 3). f(1) = 12
191.
Evaluate the limit.
limx→∞
2x3
x3 − 2x− 1
192.
Evaluate the limit.
limx→∞
3x√4x2 + 1
193.
Evaluate the limit.
limx→∞
(√4x2 − 6x− 1− 2x
)194.
Page 27
Evaluate the limit.
limx→−∞
(√9x2 + 4x+ 1 + 3x
)195.
Evaluate the limit.
limx→∞
(√x2 + 4x+ 1−
√x2 + 2x
)196.
Evaluate the limit.
limx→π
2+
4 tan2(x) + 2 tan(x)
tan2(x) + 1
197.
Find all horizontal asymptotes.
f(x) =4x3 + x+ 1
x3 + 2x+ 2
198.
Find all horizontal asymptotes.
f(x) =x arctan(x)√
4x2 + 1
199.
Find all horizontal asymptotes.
f(x) =x+ 1√
4x2 + 2x
200.
Find all horizontal asymptotes.
f(x) =7e3x + 4
2e3x + 2
201.
Find all vertical and horizontal asymptotes.
f(x) =2ex − e−x
ex − e−x202.
Page 28
Find all vertical and horizontal asymptotes for the given function. Then
find the function’s inverse and find all of its vertical and horizontal asymptotes.
What do you notice.
f(x) =ax+ b
cx+ d
203.
Find the value of k so that
limx→∞
(√x2 + kx+ 1− x
)= 10
204.
Find the values of A and B so that the f(x) has a slant asymptote of g(x).
f(x) =Ax3 +Bx2
x2 + 2x+ 1g(x) = x+ 1
205.
Show that if the rational function f(x) has a slant asymptote of y = mx+ b
with m 6= 0 then f ′(x) will be a rational function without a asymptote.
206.
Show that if the rational function f(x) has a horizontal asymptote of y = K
with K 6= 0 then f ′(x) will be a rational function with a horizontal asymptote
of y = 0
207.
Sketch the curve paying attention to where the function is increasing, de-
creasing, concave up and concave down. Label all relative extrema, points of
inflection and all asymptotes.
f(x) =x3
x2 − 16
208.
Sketch the curve paying attention to where the function is increasing, de-
creasing, concave up and concave down. Label all relative extrema, points of
inflection and all asymptotes.
f(x) = |x2 − 6x+ 5|
209.
Page 29
Sketch the curve paying attention to where the function is increasing, de-
creasing, concave up and concave down. Label all relative extrema, points of
inflection and all asymptotes.
f(x) = arctan
(1
x
)
210.
Show that the sum of a number and its reciprocal is at least 2.
211. Find the point on f(x) =√x closest to the point (4,0)
212. Find the maximum area of a square inscribed in a circle of radius 10.
213. Find the maximum area of an isosceles triangle inscribed in a circle of
radius r.
214.
A solid is formed by adjoining two hemispheres to a cylinder which is to
have a volume of 20π. Find the dimensions that minimize the surface area.
215. Bob is building a box with a square base using two types of wood. The
top and bottom of the box will be made from wood costing 5 dollar per square
foot, while the sides will be made from wood costing 3 dollars per square foot.
What is the largest volume this box can have if Bob only has 250 dollars to
spend on wood.
216. A square piece of paper is originally 15 inches by 15 inches. Square
corners of this paper are cut out and the sides folded up to make an open box.
What is the maximum volume this box can have?
217. Find the maximum volume of a cylinder inscribed in a cone with radius
R and height H.
218.
A car rental agency has 30 identical cars to rent. The owner believes that at
25 dollars a day all the cars will be rented but for each 2 dollar increase in the
price one car is not rented. What price should be charged to maximize revenue.
219.
An apple orchard currently has 35 trees per acre with an average yield of
500 apples per tree. For each additional tree planted per acre the average yield
decreases by 10 apples. How many trees per acre should be planted to maximize
yield?
Page 30
220. An offshore oil well is 3 miles off the coast. The oil refinery is 5 miles
down the coast. Laying pipe under water costs 3 times as much as it does on
land. What path should the pipe follow to minimize the cost.
221. There are two flagpoles 30 meters apart. The first pole is 12 meters high
while the second pole is 28 meters high. A wire will be connected to the top of
each pole and to the ground between the two poles. What is the shortest length
of the wire.
222. Show for a triangle whose base has length of 1 and whose perimeter is 4
has max area of 1√2
223. Find the volume of the largest cone that can be inscribed in a sphere of
radius R.
224. Kelli is building an enclosure for her cute and cuddly cockroach com-
munity. Her beloved cockroaches voted and wish to live in a right triangular
enclosure with one of the sides of the enclosure being a wall in Kelli’s room. She
has only 6 feet of fence to build this enclosure. What is the maximum area of
this enclosure?
225. A person must commute from one side of a 10 mile wide river to a
location 15 miles down the coast. The boat used to cross the river can travel 20
mph and the car use to traverse the land can travel 50 mph. What path should
the person take to minimize the total travel time?
226. Consider the angle θ made by connecting (0, 2) to (x, 0) to (4, 3). Find
x ∈ (0, 4) that maximizes θ.
227.
Find the linearization of f(x) = x√x near x = 100 and use it to approximate
the quantity√
99
228.
Use differentials to approximate the quantity√
101
229.
Use differentials to approximate the quantity3√
1242
230.
Use differentials to approximate the quantity 1.13 ln(1.1)
231.
The radius of a sphere is measured to be 10 in. with maximum possible
error of 1/2 in. Use differentials to approximate the maximum possible error in
measuring the volume and surface area of the sphere.
Page 31
232.
The included angle of an isosceles triangle is measured to be 30 with possible
error of 1. If the sides are correctly measured to be 10 in. what is the maximum
possible error in measuring the area of the triangle.
233.
A person is standing 100 feet from a tall building. If the person measures the
angle of elevation to the top of the building to be 60 with maximum possible
error of 1 calculate the maximum possible error in measuring the height of the
building.
234.
Bob orders a pizza 24 inches in diameter and tries to cut it into 6 equal
slices. Starting at the center and cutting towards the crust he cuts 6 slices with
each slice having a 60 angle with maximum error of 1. Use differentials to
approximate the error in measuring the area of each slice of pizza.
235.
The surface area of a sphere is measured to be 36π with maximum error of
π. Approximate the error in measuring the volume.
236.
If you are 1000m from a rocket launch and you want to measure the height
of a rocket accurate within 10 m when the rocket is 1000 m high how accurate
must you measure the angle of elevation.
237.
The radius of a sphere is increased by 2 percent. Find the percent increase
in the volume and surface area of the sphere.
238.
The kinetic energy of a object is given by K = 12mv
2 where m is the objects
mass and v is its velocity. Approximate the percent increase in kinetic energy
due to a 2 percent increase in velocity.
239.
The period of a pendulum with length L is given by T = 2π√
Lg where g
is the acceleration due to gravity. Due to thermal expansion the length of the
pendulum is increased by 1 percent. Approximate the percent change in the
pendulum’s period.
Page 32
Chapter The Fourth: Integrals
240.
Approximate the area under the curve f(x) = 4x2 + 2x from x = 0 to x = 2
using 4 rectangles in 3 different ways.
a) By using the left end points.
b) By using the right end points.
c) By using the midpoint
241.
Evaluate the following integral using a Riemann integral.∫ 4
0
(4x+ 8)dx
242.
Evaluate the following integral using a Riemann integral.∫ 1
0
3x2 + 3dx
243.
Evaluate the following integral using a Riemann integral.∫ 2
0
(6x2 + 4x)dx
244.
Evaluate the following integral by interpreting it in terms of Area.∫ 5
−5
√25− x2dx
245.
Evaluate the following integral by interpreting it in terms of Area.∫ 6
0
√6x− x2dx
246.
Evaluate the following integral by interpreting it in terms of Area.
Page 33
∫ 8
0
(√8x− x2 − (4− |x− 4|)
)dx
247.
Let
f(x) =
|x| x ≤ 2√
8− x2 2 < x < 2√
2
Compute:
∫ 2√2
−4f(x)dx
248.
Find the value of c so that∫ 5
−3
(|x|x
+|x− c|x− c
)dx = 0
249. ∫4x(x3 − 2x)dx
250. ∫cos(2x)
cos(x)− sin(x)dx
251.
∫ (sin
(x
2
)+ cos
(x
2
))2
dx
252. ∫x4 + x3 + 7x2 + 4x+ 12
x2 + 4dx
253. ∫6x4 − 2x3 + 6x2 − 2x+ 1
x2 + 1dx
254. ∫3x4 + 2x3 − 4x+ 4
x2 + 2x+ 2dx
Page 34
255. ∫ √1− x2
(√1− x2 +
1
1− x2
)dx
256. ∫dx√
(1 + x)(1− x)
257. ∫sec(x) + tan(x)
cos(x)dx
258. ∫(sec(x) + csc(x))(sec(x)− csc(x))dx
259. ∫cos(x)
(1− sin(x))(1 + sin(x))dx
260. ∫1 + cos3(x)
cos2(x)dx
261. ∫(tan(x) + cot(x))2dx
262. ∫(tan(x) + sec(x))2dx
263. ∫ (1
cos(x)− 1
sin(x)
)(1
cos(x)+
1
sin(x)
)dx
264. ∫sec(x)
(1 + sin(x)
cos(x)
)dx
265.
Page 35
∫dx
1 + sin(x)
266.
The acceleration of a particle is given by
a(t) = 6et − 3 cos(t) v(0) = 7 s(0) = 9
Find the position function s(t)
267.
Find a fourth degree polynomial with Critical Numbers at x = 0, 1, 2 a
y-intercept of 2 and an x-intercept when x = 4.
268.
Evaluate the following integral using the fundamental theorem of calculus.∫ 2
0
(3x2 + 8x)dx
269.
Evaluate the following integral using the fundamental theorem of calculus.∫ 4
1
(√x+ 1)(
√x+ 2)√
xdx
270.
Evaluate the following integral using the fundamental theorem of calculus.∫ 3
1
6x4 − 8x3 + 13x2 − 12x+ 6
2x2 + 3dx
271.
Evaluate the following integral using the fundamental theorem of calculus.∫ π4
0
(1
1 + sin(x)+
1
1− sin(x)
)dx
272.
Evaluate the following integral using the fundamental theorem of calculus.∫ π4
π6
(tan2(x) + cot2(x))dx
273.
Find the values of a and b that maximize the integral
Page 36
∫ b
a
(14 + 5x− x2)dx
274.
Show ∫ b
a
(x− a)(x− b)dx = (a− b)3
275.
Show ∫ b
a
xf ′′(x)dx = bf ′(b)− af ′(a) + f(a)− f(b)
276.
Let f(x) be an even function with the following properties∫ 10
0
f(x)dx = 20
∫ −15−10
2f(x)dx = 6
∫ 20
15
f(x)dx = 10
find ∫ 0
−20f(x)dx
277.
Find the average value of the function on the interval
f(x) =sin(2x)
sin(x)
[π
4,π
2
]278.
Find the average value of the function on the interval
f(x) =3x3 − x2 − 2x
x− 1[2, 4]
279.
Find the average value of the function on the interval
y =√a2 − x2 [−a, a]
280.
Find k so that the average value of f(x) = 3x2 + x is 10 on the interval
[k, 2k]
Page 37
281.
Find t so that the average value of f(x) = 3x2 is 37 on the interval [t, t+ 1]
Then use the mean value theorem for integrals to find the values of c in the
interval (t, t+ 1) guaranteed by the theorem.
282.
Let f be a quadratic with roots at x = n and x = 2n and f(0) = 2n2. Find
n so that the average value of f is -6 on [n, 2n].
283.
Find the Average Value of f(x) = x2 − 4x + 6 on [0, 3] and then find all
values guaranteed by the mean value theorem for integrals.
;
284.
Find a quadratic with the following properties:
Average Value of 4 on [0, 1]
Average Value of 9 on [0, 2]
Average Value of 16 on [0, 3]
285.
Find the derivative of the following function
f(x) =
∫ sin(x)
x2
et2
dt
286.
Find the derivative of the following function
f(x) = x2∫ x4
x2
1
1 + t4dt
287.
Let f(x) and g(x) be functions given by:
f(x) =
∫ x3
x2
et2
dt
h(x) = x3
Find the derivative of h(f(x)).
288.
Let
Page 38
g(y) =
∫ y
0
f(x)dx f(x) =
∫ x3
0
√1 + t4dt
Find g′′(y)
289.
Sketch the graph of f(x) = ex
x and find an interval [t, t + 1] with t > 0 so
that the area under the graph of f(x) is a minimum.
290.
Find a function f(x) such that the average value of f on [0, t] is t2
291. ∫(x2 − 3)(x3 − 9x− 2)5dx
292. ∫(x)(x2 + 1)4dx
293. ∫(x3)(x2 + 1)4dx
294. ∫sec(√x) tan(
√x)√
xdx
295.
∫ sin
(1x
)− cos
(1x
)x2
dx
296. ∫x2
cos2(x3)dx
297. ∫e√x
√xdx
298. ∫(1 +
√x))5√xdx
Page 39
299. ∫x2 − 3
x3 − 9x− 1dx
300. ∫ e
1
1
x(1 + ln(x))dx
301. ∫sin(x) + cos(x)
sin(x)− cos(x)dx
302. ∫1
cos2(x)(1 + tan(x))dx
303.
∫1
x2
√x− 1
xdx
304. ∫1√√
x− 1 + (x− 1)54
dx
305. ∫sin(2x)√
1 + cos2(x)dx
306. ∫ √1 + sin(x)dx
307. ∫1
sec(x)− 1dx
308. ∫sec2(x)
sec2(x) + 2 tan(x)dx
309.
Page 40
∫ √tan(x)− cot(x)
sin2(x) cos2(x)dx
310. ∫sin(x) + cos(x)√
sin(x) + sec(x) cos2(x)dx
311.
∫4x
52 + x2 + 8x
32 + 2x+ 6
√x+ 1√
x(x+ 1)2dx
312. ∫dx√
1 +√xdx
313. ∫dx√
1 +√
1 +√xdx
314.
Show ∫ b
a
f(x)dx =
∫ b
a
f(a+ b− x)dx
Making ∫ b
a
f(x)dx =1
2
∫ b
a
(f(x) + f(a+ b− x))dx
315.
Use the results of the previous problem to calculate∫ 7
3
ln(x+ 2)
ln(24 + 10x− x2)dx
316.
Use the results of the previous problem to calculate∫ 7
1
sin(x+ 3)
sin(x+ 3) + sin(11− x)dx
Page 41
Chapter The Fifth: Exponential, Logarithmic and Inverse Trig
Functions
317.
Find dydx
y = ln
√(x2 + 1)3
x cos(x)
318.
Find dydx
y =ex
3√x2 + 1
x sin(x)
319.
Find dydx
y = (x4 + x)sin(x)
320.
Find dydx
y = (1 + x ln(x))tan(2x)
321.
Find the equation of the tangent line to the curve at the indicated point
yx = xy (1, 1)
322.
Find the equation of the tangent line to the curve at the indicated point
y = x1x (1, 1)
323.
Find dydx
y = xxx
324.
Page 42
Find the equation of the tangent line to the curve at the indicated point
x ln(y) = x+ y (−1, 1)
325.
Let f be a polynomial of degree 20 with roots at x = 1, 2, 3, ..., 20 and leading
coefficient of π find:
d
dxln(f(x))
326.
Use logarithmic differentiation to derive a formula for the derivative of the
product of n different functions
327.
Find points on y = ln(x) where the tangent line passes through (0, 0)
328.
Find where the function is increasing and decreasing and all relative ex-
tremum
f(x) = x2 − 14x+ 6 ln(x− 3)
329.
Find the relative extremum of the function and use the results to determine
the larger eπ and πe
f(x) =ln(x)
x
330. ∫x2 − 3
x3 − 9x− 1dx
331. ∫ e
1
1
x(1 + ln(x))dx
332. ∫sin(x) + cos(x)
sin(x)− cos(x)dx
333.
Page 43
∫tan(x) + 1
tan(x)− 1dx
334. ∫1
cos2(x)(1 + tan(x))dx
335. ∫1
x+√xdx
336. ∫1
1 +√xdx
337. ∫x3 − x√
1− x2 + 1− x2dx
338.
Find dydx .
y = ex sin(x)
339.
Find dydx .
y = eex
2
ex2+1
340.
Find dydx .
y = cot(ex3−x)
341.
Find dydx .
exy = ln(x+ y)
342.
Find dydx .
Page 44
ex2+y2 = ln(x2 + y2) + x
343.
Find f ′.
f =
∫ ex4
ex2
dt
1 + t4
344.
Find the critical number of f .
f =
∫ ex3
ex2
ln(t)
tdt
345.
Hermite polynomial of degree n is defined as
Hn = (−1)nex2 dn
dxne−x
2
Find H1, H3, and H3
Then Calculate the following:∫H1
H2dx and
∫H2
H3dx
346.
Two hyperbolic trig functions are defined as follows
cosh(x) =ex + e−x
2sinh(x) =
ex − e−x
2
Show
d
dxcosh(x) = sinh(x) and
d
dxsinh(x) = cosh(x)
347.
Find a point on the curve f(x) = e4x + x where the tangent line passes
through the origin
348.
A triangle is formed with the positive x-axis, the positive y-axis and the
tangent line to y = e−x for x > 0. Find the maximum area of this triangle.
349.
Page 45
Show that the largest rectangle you can inscribe under f(x) = e−x2
has two
vertices at the points of inflection of f .
350. ∫(e2x − e−x)(e2x + 2e−x + 4)10dx
351.
∫ e3
1
√1 + ln(x)
xdx
352.
∫ e2
1
(1 + ln(x))(2 + ln(x))
xdx
‘
353. ∫x+ 1
x2 + 2x+ 2dx
354. ∫e2x − e−2x
e2x + e−2xdx
355. ∫1
1 + exdx
356. ∫1
ex + 2 + e−xdx
357. ∫e√x
√xdx
358. ∫ 1
0
e2x
(ex + 1)2dx
359.
Page 46
∫etan(x)
cos2(x)dx
360.
∫ e3
e
1
ln(xx)dx
361. ∫(1 + ln(x))(3− ln(x))
xdx
362. ∫tan(x)
sec(x)− 1dx
363. ∫6x4 − 4x3 + 6x2 + 4x
1 + x2dx
364. ∫6e6x + 2e5x + 6e2x + 2ex
e4x + 1dx
365. ∫4√
x+ 8 +√xdx
366.
Use implicit differentiation to find dydx .
y = arcsin(x)
367.
Find dydx .
y = arcsin(e4x)
368.
Find dydx and simplify.
y = arctan(√x2 − 1)− arcsec(x)
Page 47
369. Find f ′ and simplify.
f = arcsec(√
1 + x2)
370.
Find dydx .
y = arctan(x2) ln(1 + x4)
371.
Find dydx .
y = (x3 + x)arcsec(x)
372.
Find dydx .
arcsin(x+ y) = xy
373.
Find dydx .
arcsin(xy) = ex+y
374.
Find dydx .
xy2 = arctan(y)
375.
Find dydx .
xy = xarcsec(y)
376.
Find f ′ and simplify.
f = arctan
(x− 1
x+ 1
)377.
Find f ′ and simplify.
Page 48
f = arcsin(√
1− x)
378.
Let
f(x) = arctan(x) + arctan
(1
x
)Show
f ′(x) = 0
And find C such that
f(x) = C x > 0
379.
Let
f(x) = 2 arctan
(√1− xx+ 1
)+ arcsin(x)
Show
f ′(x) = 0
And find C such that
f(x) = C x > 0
380.
Let
f(x) = arcsin
(x√
1 + x2
)+ arctan
(1
x
)Show
f ′(x) = 0
And find C such that
f(x) = C x > 0
Page 49
381.
Let
f(x) = arcsec(√x) + arcsin(x)
Show
f ′(x) = 0
And find C such that
f(x) = C x > 0
382.
Sketch the graph of
f(x) = arctan
(1
x
)383.
Let f(x) = x3 − 6x2 + x. Without computing f−1(x) calculate (f−1)′(−4)
384.
Let f(x) = x5 + x3. Without computing f−1(x) calculate (f−1)′(2)
385.
Let f(x) = x3 − 2x2 + 3x. Without computing f−1(x) calculate (f−1)′(6)
386.
Let f(x) = x4 − 2x3 + 1. Without computing f−1(x) calculate (f−1)′(1) at
the value where it exists.
387.
Let f(x) = ax+bcx+d . Without computing f−1(x) calculate (f−1)′(a+bc+d )
388.
Find the values of C so that the function in invertible for all real numbers.
f(x) =x3
3+ 3x2 + Cx
389.
∫ 1
0
√arctan(x)
1 + x2dx
390.
Page 50
∫1
arctan(x) + x2 arctan(x)dx
391. ∫arcsin(x)√
1− x2dx
392. ∫ ∞√2
x
4 + x4dx
393. ∫2ex√
9− e2xdx
394. ∫eeex
eex
exdx
395. ∫sec(x) tan(x)
2 + tan2(x)dx
396. ∫sec2(x)√
2− sec2(x)dx
397. ∫2x3 + 5x2 + 12x+ 10
x2 + 2x+ 5dx
398. ∫dx
x23 + x
43
399. ∫1√
e2x − 4dx
400. ∫cos(x)√
1 + 2 sin(x)− cos2(x)dx
Page 51
401. ∫2
x2 + 8x+ 65dx
402. ∫x+ 3
x2 + 6x+ 34dx
403. ∫2x+ 10
x2 + 10x+ 106dx
404. ∫2
x2 + 10x+ 106dx
405. ∫4x− 2
x2 + 10x+ 106dx
406. ∫3√
−x2 + 6x− 5dx
407. ∫2x+ 1√
−x2 + 6x− 5dx
408. ∫e2x
e4x + 4e2x + 5dx
409. ∫1
√x√
1− xdx
410. ∫1
x12 + x
32
dx
411.
Page 52
∫4x3 + 10x
x4 + 4x2 + 8dx
412. ∫1
x√x− 1
dx
413. ∫1
(2− x)√x− 1
dx
414. ∫1
x√x6 − 1
dx
415. ∫ √x
1 + x3dx
416. ∫1√
x(x+ 1)dx
417. ∫1√
x+ 1√xdx
418. ∫1
2√x+ 2x+ x
32
dx
419. ∫1
(x− 1)√x2 − 2x
dx
Page 53
Chapter The Sixth: Applications of Integration
420.
Find the area between the curves.
f(x) = 3x2 + 4x− 10
g(x) = 2x2 + 5x− 4
421.
Find the area between the curves from x = 0 to x = π2 .
f(x) = e2x
g(x) = cos(x)
422.
Find the area between the curves from x = 0 to x = 4.
f(x) = x2 + 2x+ 2
g(x) = 4x+ 5
423.
Find the area between the curves.
f(x) = x3 − 6x+ 1
g(x) = 3x+ 1
424.
Find the area between the curves.
f(x) = 4− x2
g(x) = |x|
Page 54
425.
Sketch the region bounded by and find the area under
f(x) =e
1x
x2
on the interval (1
ln(K), 1
)K > e
Show this Area becomes unbounded as K →∞426.
Find the line y = b that divides the region bounded by
f(x) = 9− x2
y = 0
into two regions of equal area
427.
Find k so that the line y = 3 divides the region bounded by
f(x) = k − x2
y = 0
into two regions of equal area
428.
Let f(x) = −3x2 +27 find the equation of another parabola g that intersects
f at x = −3 and x = 3 and the area between f and g is 144.
429. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y =√
3x2 + 1 x = 0 x = 2
Revolve about the x-axis.
430. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
Page 55
y =√x(x2 + 1)10 x = 0 x = 1
Revolve about the x-axis.
431. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y =1 + tan(x)
cos(x)x = 0 x =
π
4
Revolve about the x-axis.
432. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = tan(x) + sec(x) x = 0 x =π
4
Revolve about the x-axis.
433. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = tan(x) + cot(x) x =π
6x =
π
3
Revolve about the x-axis.
434. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y =sec(x) + tan(x)
cos(x)x = 0 x =
π
4
Revolve about the x-axis. TI
435. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y =x
1 + x2x = 0 x = 1
Revolve about the x-axis. TS
436. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = ln(x) x = 1 x = e
Revolve about the x-axis.
Page 56
437. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = x2 + x x = 0 x = 3
Revolve about the line y = −2.
438. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = e3x x = 0 x = 1
Revolve about the line y = −3.
439. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = 3√x x = 1 x = 27
Revolve about the y-axis.
440. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = ln(x) x = e x = e2
Revolve about the y-axis.
441. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y =√x x = 1 x = 25
Revolve about the line x = −2.
442. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y =√x y = x2
Revolve about the x-axis.
443. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines.
y = x y = tan(πx
4)
Page 57
Revolve about the x-axis.
444.
When the ellipse: x2
a2 + y2
b2 = 1 is rotated about the x-axis it forms a ellipsoid.
Find its volume.
445. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines using the method of cylindrical
shells.
y = x2 − 4x y = 0 x = 0 x = 2
Revolve about the y-axis.
446. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines using the method of cylindrical
shells.
y =√
9− x2 y = 0 x = 0
Revolve about the y-axis.
447. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines using the method of cylindrical
shells.
y = ex2
y = 1 y = e9 x = 0
Revolve about the y-axis.
448. Find the volume of the solid formed by revolving the region bounded by
the graphs of the equations about the given lines using the method of cylindrical
shells.
y = x2 − 4x y = 0
Revolve about the line x = 5.
449. Find the Arc Length of the function on the the interval 0 < x < π3 .
f(x) = ln(cos(x))
450.
Find the Arc Length of the function on the the interval 1 < x < 3.
f(x) = ln(x)
Page 58
451.
Find the Arc Length of the function on the the interval 1 < x < 3.
f(x) = 2√x
452.
Find the Arc Length of the function on the the interval 1 < x < 3.
f(x) =x4
8+
1
4x2
453.
Find the Arc Length of the function on the the interval 1 < x < 2.
f(x) =x5
10+
1
6x3
454.
Find the Arc Length of the function on the the interval 0 < x < 1.
f(x) = 1 + 6x32
455. Find the Arc Length of the function on the the interval π4 < x < π
2 .
f(x) = −1
2ln | csc(x) + cot(x)|+ 1
2cos(x)
456. Find the Arc Length of the function on the the interval π6 < x < π
4 .
f(x) = −1
2cot(x)− 1
2tan(x)
457. Find the Arc Length of the function on the the interval 0 < x < 1.
f(x) =1
2ln |x+ 1| − 1
4(x+ 1)2
458.
A
Find the Arc Length of the function on the the interval 0 < x < 1.
f(x) =1
2arctan(x)− 1
2(x+
1
3x3)
459.
A
Find the Arc Length of the function on the the interval ln(2) < x < ln(3).
Page 59
f(x) =1
2ln |ex − e−x| − 1
2ln |ex + e−x|
460.
A A
Find the Arc Length of the function on the the interval 0 < x < π6 .
f(x) = −1
2ln | cos(x)− sin(x)| − 1
2ln | cos(x) + sin(x)|
461.
A AA
Find the Arc Length of the function on the the interval 0 < x < π6 .
f(x) = ln
((1 + sin(x))4
(sec2(x) + sec(x) tan(x))116
)462.
Let f(x) be differentiable on (0, 1) with a vertical asymptote of x = 1.
Calculate the following integral:∫ 1
0
√1 + (f ′(x))2dx
463.
Let Ln be the Arc Length of the function
fn =xn+1
2(n+ 1)− x1−n
2(1− n)
on [1, 2]
Find
limn→∞
Ln
464.
Find the surface area obtained by rotation the curve about the x-axis on the
interval 0 < x < 2.
f(x) = x3
465.
Find the surface area obtained by rotation the curve about the x-axis on the
interval 3 < x < 4.
Page 60
f(x) =√
25− x2
466.
Find the surface area obtained by rotation the curve about the x-axis on the
interval 1 < x < 2.
f(x) =x3
6+
1
2x
467.
Find the surface area obtained by rotation the curve about the y-axis on the
interval 0 < y < 1.
f(x) = 1− x2
468.
Find the surface area obtained by rotation the curve about the x-axis on the
interval 0 < x < 1.
f(x) = ex
469.
Let f(x) = 1x . Revolve f about the x-axis and consider the interval I =
(1,∞). Show the volume of the solid produced has finite volume and infinite
surface area.
470.
Find the centroid of the region bounded by:
y = x+ 2 y = x2
471.
Find the centroid of the region bounded by:
y =√
4 + 3x2 y = x2
472.
Show
y = cosh(x) =ex + e−x
2
is a solution to
Page 61
K =1
y2
where K is the curvature
K =|y′′|(
1 + (y′)2) 3
2
473.
Find the values of r so that y = erx is a solution to
y′′′ − 6y′′ + 11y′ − 6y = 0
474.
Find the values of k so that y = sin(kx) is a solution to
y′′ + 100y = 0
475.
Find the values of n so that y = xn is a solution to
x2y′′ + 7xy′ + 8y = 0
476.
Show
y =
ex − 1 x ≥ 0
1− e−x x < 0
is a solution to
y′ = |y|+ 1
Remember, you must use the definition of the derivative to calculate y′(0).
477.
If:
dx
dt− .1x+ .0001xy
dy
dt= .05y − .001xy
represents a predator-prey model which variable x or y represents the preda-
tor.
478.
Page 62
Use Euler’s method with step size j = .2 to approximate y(2)
y′ = x√y y(1) = 4
479.
Solve the initial value problem
(y + 2)dy
dx= −x ln(x) y(1) = 1
480.
Solve the initial value problem
2(1 + y2)xdx = (1 + x2)dy y(0) = 1
481.
Solve the initial value problem
1
3x2 + 1
dy
dx= cos2(y) y(1) = 0
482.
Solve the initial value problem
xdy =√−5 + 6y − y2dx y(1) = 5
483.
Solve the initial value problem
15e−y sin3(x) = cos6(x)dy
dxy(0) = ln(2)
484.
Solve
2(x3y + x2y + xy + y)dy = (y2 + 1)(3x2 + 2x+ 5)dx
485.
Solve
(ex − e−x)(y2 + 1) = (2yex + 2ye−x)dy
dx
486.
Solve
Page 63
xe3x
y= 2(3x+ 1)2
dy
dx
487.
Solve the initial value problem
1 +√y = (1 + sin(x))
dy
dxy(0) = 0
488.
Solve
(y2 + 1)dx = (x34 + x
54 )dy
489.
Solve the initial value problem
√1− x2 = 2xy
dy
dxy(1) = 2
490.
Solve the initial value problem
4x ln(x) + 4xy ln(x) = ydy
dxy(e) = 0
491.
Use the substitution u = yex to transform the equation into a separable
equation and then solve it
ydx+ (1 + y2e2x)dy = 0
492.
Use the substitution y = zex to transform the equation into a separable
equation and then solve it
dy
dx= y +
√e2x − y2
493.
A population is modeled by the differential equation:
dP
dt= 1.25P
(1− P
5000
)P (0) = 1000
For what values of P is the population increasing?
Page 64
For what values of P is the population decreasing?
Find P (t)
Find limt→∞
P (t)
494.
Another type of population model is the Gopertz growth model. It is similar
to the logistic equation in that the model assumes the population will increase
at a rate proportional to the size of the population.That means the population
will increase a a rate of kP (t). The like the logistic model the Gopertz growth
model also takes into account the maximum population a species can have in
an environment of fixed size and resources. Instead of using (M − P (t)) as a
factor like the logistic model does the Gopertz growth model uses ln
(MP (t)
)as a factor, with M being the maximum population (carrying capacity). The
Gopertz growth model is
dP
dt= kP ln
(M
P
)k > 0
We also see from the differential equation if a population P is less than M
then dPdt > 0 and the population will increase and approach the carrying capacity
and if P is greater than M then dPdt < 0 and the population will decrease and
approach the carrying capacity.
Show the population is increasing fastest when the population is Me and then
solve this differential equation for the population P (t) as a function of time and
show:
limt→∞
P (t) = M
495.
Solve:
xy′ − 3y = x4 y(1) = 1
496.
Solve:
y′ + exy = ex y(0) = 2e
Page 65
497.
Solve:
y′ + tan(x)y = tan(x) y
(π
4
)= 1
498.
Solve:
y′ + 4 sec(x)y = sec(x)(sec(x) + tan(x))
499.
Solve:
y′ + cos(x)y = sin(2x)
500.
Solve:
y′ + exy = ex
501.
Solve:
√1− x2 dy
dx+ y = 1 y(0) = 4
502.
Solve:
xdy
dx+ 3y = xex
4
503.
Solve:
dy
dx− cos(x)
1 + sin(x)y = 1 y(0) = 1
504.
Solve:
dy
dx+
6x2 − 4x+ 8
x3 − x2 + 4x− 4y =
ex3+12x
(x− 1)2(x2 + 4)
505.
Page 66
Solve:
dy
dx+
cos(x)− sin(x)
cos(x) + sin(x)y = sec3(x) y(0) = 4
506.
Solve:
(1− x2)dy
dx− xy = 1 y
(1
2
)=
√3
2
507.
Solve:
dy
dx+
4x+ 1
xy = ex y(1) = 0
508.
Solve:
(1 + x2)dy
dx+ (4x2 − 4x+ 2)y = 9 ln(x) y(1) = 0
509.
Solve:
dy
dx+ sin(x)y = sin(2x)
510.
Solve:
dy
dx+
y
1 + e−x=
1
e2x + 2xex + x2y(0) = 0
511.
Solve:
dy
dx− 2xy = (2 + x−2) y(1) = 0
Page 67
Chapter The Seventh: Techniques of Integration
512. ∫(9x2 + 4x) ln(x)dx
513. ∫(ln(x))2dx
514. ∫(4x− 8) sin(2x)dx
515. ∫(27x− 9)e3xdx
516. ∫ln(1 + x2)dx
517. ∫9x2 arctan(x)dx
518. ∫2x ln(1 + x4)dx
519. ∫e2x cos(x)dx
520. ∫e√xdx
521.
Page 68
∫e
4√xdx
522. ∫arctan(
√x)dx
523. ∫arcsin(x)dx
524. ∫arcsin(
√x)√
1− xdx
525. ∫xe3x
(3x+ 1)2dx
526. ∫x√
1− x2 arcsin(x)dx
527. ∫sec2(x) csc(x)dx
528. ∫(2 + x−2)e−x
2
dx
529. ∫cos(ln(x))dx
530. ∫sec3(x)dx
531. ∫ex sin(x)dx
Page 69
532. ∫arctan
(1 +
1
x
)dx
533.
Find the volume of the solid formed after rotating the function about the
x-axis.
f(x) = ln(x) 1 ≤ x ≤ e
534.
If f(π) = 1 find f(0).∫ π
0
(f(x) + f ′′(x)
)sin(x)dx = 2
535.
Let u(x) be an even function and v(x) be odd function with the following
properties:
u(1) = 3 v(1) = 5
∫ 1
−1udv = 12
Find ∫ 1
−1vdu
536.
A ∫x2 cos(x) + x cos(x) + sin(x)
(x+ 1)2dx
537. ∫cos3(x) sin2(x)dx
538. ∫sec4(x) tan3(x)dx
539. ∫sec4(x) tan6(x)dx
Page 70
540.
Evaluate the integral in two different ways∫sin2(x)dx
1. Using a power reducing formula
2. Using cyclic Integration by Parts
541. ∫sin2(x) cos2(x)dx
542. ∫tan4(x)dx
543. ∫ π4
0
tan6(x)dx
544. ∫tan3(x)
cos3(x)dx
545. ∫x sin2(x)dx
546. ∫tan3(x)√
cos(x)dx
547. ∫x cos3(x)dx
548. ∫sin(x) ln(sin(x))dx
549.
A
Page 71
∫dx
sec2(x) + sec(x) tan(x)
550. ∫1
(9− x2)32
dx
551. ∫ √4− x2x
dx
552. ∫1
x4√
9 + x2dx
553. ∫dx√
9 + x2
554. ∫dx
x√
1 + x2
555. ∫dx
x3√x2 − 1
556. ∫ √x2 − 4
xdx
557. ∫dx
(1 + x2)52
558. ∫ √x2 − 4
x2dx
559.
Page 72
∫ √1− x2x2
dx
560. ∫ √x2 + 1
xdx
561. ∫x2
(x2 + 9)2dx
562. ∫x2
(x2 + 1)32
dx
563. ∫ √9x2 + 16
x4dx
564.
∫(x2 − 4)
32
xdx
565. ∫1
(x2 − 1)2dx
566. ∫2 arctan(x)
x3dx
567. ∫arcsec(x)dx
568. ∫ln
(x+
√x2 − 1
)dx
569.
Page 73
∫ √x3 − 1
xdx
570. ∫x√
1 + x4dx
571. ∫ √1 + x2dx
572.
∫(1 + x2)
52
x2dx
573. ∫ √x2n − 1
xdx
574. ∫dx
2x√x− 1
dx
575.
Show: if f is continuous on [0, 1] then∫ π2
0
f(sin(x))dx =
∫ π2
0
f(cos(x))dx
by applying two different trig substitutions to the following∫ 1
0
f(u)√1− u2
du
Try u = sin θ and u = cos θ
576. ∫−3x2 + 8x+ 9
x3 − 6x2 + 9xdx.
577. ∫−3x− 1
x3 − 3x2 + x− 3dx.
Page 74
578. ∫x2 − 2x− 5
x3 − x2 + x− 1dx.
579. ∫10x4 + 2x2 + 2
x5 + xdx.
580. ∫5x2 − 11x− 2
x3 − 3x2 + x− 3dx.
581. ∫2x5 + 18x3 + 10x2 + 36
x4 + 9x2dx.
582. ∫2x4 − 3x3 + x2 − 7
x3 − 2x2 + x− 2dx.
583. ∫2x4 − 2x3 + 17x2 − 19x− 33
x3 − 2x2 + 9x− 18dx.
584. ∫6x3 + x2 + 18x+ 7
x4 + 5x2 + 4dx.
585. ∫2x2 − 11x+ 197
(x2 − 10x+ 106)(3x− 5)dx.
586. ∫3x5 − 12x4 + 42x3 − 62x2 + 8x− 28
(x2 − 2x+ 10)(x− 2)dx.
587.
Evaluate the following integral in 2 different ways:
1) Substituting u = cos(x) and then using partial fractions
2) Using trig to convert all cos(x) terms into sin(x)
Page 75
∫cos4(x)
sin(x)dx.
588.
Evaluate the following integral in 3 different ways:
1) Using Partial Fractions
2) Substituting x = tan2 θ
3) Substituting u = 1x ∫
dx
x(1 + x)dx
t = tan
(x
2
)dx =
2dt
1 + t2cos(x) =
1− t2
1 + t2sin(x) =
2t
1 + t2
589. ∫dx
2 + cos(x)
590. ∫dx
1 + sin(x)− cos(x)
591.
A ∫ √tan(x)dx
592.
limx→0
e2x − 2x+ 1
xex − x
593.
limx→1
x ln(x)
x3 − e1−x
594.
limx→∞
(√e2x + x− ex
)
595.
Page 76
Evaluate:
limx→0
arcsin(x)
ln(x+ 1)
596.
Evaluate:
limx→0
arctan(x)
ln(x2 + 1)
597.
Evaluate:
limx→0
x− arctan(x)
arcsin(x)− x598.
limx→0
(1
x− 1
sin(x)
)
599.
limx→0
(1
x− 1
ex − 1
)
600.
limx→∞
x3 sin
(4
x3
)
601.
limx→∞
x arctan
(1
x
)
602.
limx→∞
(e3x + x)2x
603.
limx→∞
(1
1 + ex
) 1x
604.
limx→∞
(1 +
2
x
)4x
605.
limx→∞
(ex
1 + ex
)x
Page 77
606.
limx→∞
(1 +
1
x2
)x2
607.
limx→0+
e−1x
x
608.
limx→0+
(sin(x))x
609.
limx→0+
(arctan(x))x
610.
Show that if f is differentiable then
limh→0
f(x+ h)− f(x− h)
2h= f ′(x)
611. ∫ ∞√2
x
x4 + 4dx
612. ∫ ∞1
6x
x4 + 5x2 + 4dx
613. ∫ ∞1
x2
(1 + x2)2dx
614. ∫ ∞0
2x2 − 1
x4 + 5x2 + 1dx
615. ∫ ∞0
xe−x2
dx
Page 78
616. ∫ ∞0
√arctan(x)
1 + x2dx
617. ∫ ∞0
x
1 + x2dx
618. ∫ ∞1
arctan
(1
x
)dx
619. ∫ ∞1
√x
1 + x3dx
620.
The Gamma function is defined as
Γ(x) =
∫ ∞1
tx−1e−tdt
Show
Γ(x+ 1) = xΓ(x)
621.
The Laplace Transform of f(t) is defined as
F (s) =
∫ ∞0
f(t)e−stdt
Find the Laplace Transform of f(t) = t
622. ∫ π4
0
sin(x) + cos(x)
cos(x)− sin(x)dx
623. ∫ 1
0
x3√1− x2
dx
624. ∫ 1
0
1
1− exdx
Page 79
625. ∫ 2
1
1√x2 − 1
dx
626. ∫ 2
0
1√2− x
dx
627. ∫ e
1
1
x ln(x)dx
628. ∫ 2
0
1
(x− 2)2dx
629.
Find the values of p > 0 that make the improper integral converge and
diverge ∫ ∞e
dx
x(ln(x))p
630.
Find the value of C that makes the integral converge. Evaluate the integral
for this value. ∫ ∞0
(x
x2 + 1− C
x+ 1
)dx
631.
Use a comparison test to determine the convergence or divergence of∫ ∞1
1
x2 + arctan(x)dx
632.
Use a comparison test to determine the convergence or divergence of∫ ∞1
√1
x+
1
x3dx
Page 80
633.
Use a comparison test to determine the convergence or divergence of∫ ∞1
4√x
x+ 1dx
634.
Use a comparison test to determine the convergence or divergence of∫ ∞1
e−x−4√xdx
635.
Use a comparison test to determine the convergence or divergence of∫ ∞1
4x3 + 4x2 + 2x+ 1
x4 + x3 + x2 + xdx
636.
Show ∫ ∞0+
f(x)dx =
∫ ∞0+
f
(1
x
)(1
x2
)dx
Making ∫ ∞0+
f(x)dx =1
2
∫ ∞0+
(f(x) + f
(1
x
)(1
x2
))dx
637.
Use the results of the previous problem to calculate∫ ∞0+
ln(2x)
1 + x2dx
Page 81
Chapter The Eighth: Sequences and Series
638.
Determine if the sequence converges or diverges.
sn = n sin
(1
n
)639.
Determine if the sequence converges or diverges.
sn = sin(n) arctan
(1
n
)640.
Determine if the sequence converges or diverges.
sn =√
4n2 + 6n− 2n
641.
Determine if the sequence converges or diverges.
sn =
(1 +
3
n
)2n
642.
Determine if the sequence converges or diverges.
sn =2n
2n + n
643.
Determine if the sequence converges or diverges.
sn =e
1n + e
2n + e
3n + ...+ e
2nn
n
644.
Determine if the sequence converges or diverges.
sn =n3√
n6 + n3 + 1
645.
Page 82
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
(en + n
) 1n
646.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
(1 +
1
n
)2n
647.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
24
n2 + n
648.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=3
6
n2 − 4
649.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
ln
(n
n+ 1
)650.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
n√n2 + 3
651.
Page 83
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
n2 sin
(4
n2
)652.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
(en + n
) 1n
653.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
23n+1
32n
654.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
3n + 22n+1
5n
655.
Solve for K
∞∑n=2
36
(K − 2
3
)n= 6
656.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
4
n2 + 4n+ 3
657.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
Page 84
∞∑n=1
9
n2 + 3n
658.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
ln
(n
n+ 1
)659.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
ln
(1− n2
n2
)660.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=0
(arctan(n+ 2)− arctan(n)
)661.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=0
(arctan(n)− arctan(n+ 2)
)662.
Determine if the series converges or diverges. Clearly state the series test
used.
∞∑n=1
(arcsin
(1
n+ 1
)− arcsin
(1
n
))663.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if convergent.
Page 85
∞∑n=2
23n
32n
664.
Use a geometric series to express the repeating decimal as a fraction.
.11
665.
Use a geometric series to express the repeating decimal as a fraction.
.123
666.
Use a geometric series to express the repeating decimal as a fraction.
.99
667.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
1
n ln(n)
668.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
1
n(ln(n))2
669.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=3
1
n ln(n) ln(ln(n))
670.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
arctan
(1
n
)671.
Page 86
Find the values of p > 0 that make the series converge.
∞∑n=3
1
n(ln(n))p
672.
Find the values of p > 0 that make the series converge.
∞∑n=3
ln(n)
np
673.
Find the values of p > 0 that make the series converge.
∞∑n=3
n2
(n3 + 1)p
674.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
n2
n4 + n− 1
675.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
n√n5 + 1
676.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=1
1
n1+1n
677.
Show the series diverges for any value of p > 0
∞∑n=2
1
n(1+1np )
Page 87
678.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
2n
3n − n
679.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
2n
4n − ln(n)
680.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
2n
ln(n)4n − 1
681.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
sin
(1
n
)682.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
n sin
(1
n3
)683.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
sin
(2n
3n
)684.
Page 88
Show that if
∞∑n=2
an
converges then so does
∞∑n=2
sin(an)
685.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
√n sin
(1
n3
)686.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
sin
(n
n3 − ln(n)
)687.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
n3 ln(n)
2n
688.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
arctan(n) ln(n)
n2
689.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
1
nln(n)
Page 89
690.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
n3n
n!
691.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
ln(n)
n
692.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
(−1)nn√n3 + 1
693.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
(−1)n ln(n)
n
694.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
sin(
(2n+1)π2
)n
695.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
(−1)n2n
3n + n
696.
Page 90
Find the sum of the series with error less than .0001
∞∑n=2
(−1)n2n
n!
697.
Find the number of terms required to determine the sum of the series with
error less than .01
∞∑n=2
sin(
(2n+1)π2
)ln(n)
698.
Find the number of terms required to determine the sum of the series with
error less than .001
∞∑n=2
sin(
(2n+1)π2
)22
699.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
(−1)n1 · 3 · 5 · ... · (2n+ 1)
3nn!
700.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
n23n
4n
701.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
2nn!
(2n)!
702.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
Page 91
∞∑n=2
3nn!
1 · 3 · 5 · ... · (2n+ 1)
703.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
(−1)nn!
nn
704.
Determine if the series is conditionally convergent absolutely convergent or
divergent. Clearly state the series test used. Find the sum if possible.
∞∑n=2
(2n)!
(n!)2
705.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
(n2
2n2 − 1
)n706.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
(1− 1
n
)n2
707.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
(e2n
e3n + n
)n708.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
Page 92
∞∑n=2
(arctan(n)
2
)n709.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
(3n − n
2n
)n710.
Determine if the series converges or diverges. Clearly state the series test
used. Find the sum if possible.
∞∑n=2
(n sin
(2
n
))n711.
Find a third order Taylor polynomial for f(x) = e3x. Use this polynomial to
approximate the value of e6 and determine the accuracy of the approximation.
712.
Find a third order Taylor polynomial for f(x) = ln(x2 + 1). Use this poly-
nomial to approximate the value of ln(5) and determine the accuracy of the
approximation.
713.
Find a fourth order Taylor polynomial for f(x) = cos(3x). Use this poly-
nomial to approximate the value of cos(.3) and determine the accuracy of the
approximation.
714.
Determine the degree of the Maclaurin polynomial of the function f(x) = ex
needed to approximate the value of e−.1 with error less than .00001
715.
Determine the degree of the Maclaurin polynomial of the function f(x) =
cos(x) needed to approximate the value of cos(.2) with error less than .00001
716.
Find the interval of convergence of the power series and the radius of con-
vergence.
∞∑n=1
xn
n2
Page 93
717.
Find the interval of convergence of the power series and the radius of con-
vergence.
∞∑n=1
xn
3n
718.
Find the interval of convergence of the power series and the radius of con-
vergence.
∞∑n=1
xn
3nn
719.
Find the interval of convergence of the power series and the radius of con-
vergence.
∞∑n=2
(x− 3)n
5nn ln(n)
720.
Find the interval of convergence of the power series and the radius of con-
vergence.
∞∑n=2
xn sin
(1
n
)721.
Find the interval of convergence of the power series and the radius of con-
vergence.
∞∑n=2
n(x− 2)n
n3 − 1
722.
Find the radius of convergence of the power series.
∞∑n=1
1 · 3 · 5 · ... · (2n+ 1)xn
3nn!
723.
Find the radius of convergence.
Page 94
∞∑n=1
(n!)k+10xn
((k + 10)n)!
724.
Find a power series for f(x) = 32−x centered at x = 3. Give the interval of
convergence of this series.
725.
Find a power series for f(x) = 22−3x centered at x = 1. Give the interval of
convergence of this series.
726.
Find a power series for f(x) = 3x2−3x centered at x = 3. Give the interval of
convergence of this series.
727.
Find a power series for f(x) = 1−x2−2x centered at x = 1. Give the interval
of convergence of this series.
728.
Find a power series for f(x) = 3xx2−4 centered at x = 1. Give the interval of
convergence of this series.
729.
Find a power series for f(x) = ln(1−x2) centered at x = 0. Give the interval
of convergence of this series.
730.
Find a power series for f(x) = ln(x+1x−1
)centered at x = 0. Give the interval
of convergence of this series.
731.
Find a power series for f(x) = x(1−x)2 centered at x = 0. Give the interval
of convergence of this series.
732.
Use the power series for arctan(x) to approximate the value of the integral
with error less than .001. ∫ 1
0
arctan(x2)dx
733.
Use the power series arctan(x) to approximate the value of the integral with
error less than .001.
Page 95
∫ 1
0
arctan(x)dx
x
734.
Use the power series for e−x2
to approximate the value of the integral with
error less than .001. ∫ 1
0
e−x2
dx
735.
Find a power series for f(x) = ex2
centered at x = 0.
736.
Find a power series for f(x) = 3√x− 27 centered at x = 0.
737.
Find a power series for f(x) =√
4x− 64 centered at x = 0.
738.
A Find a power series for f(x) = sin2(x) centered at x = 0.
739.
A Find the power series for f(x) = arcsin(x).
740.
A Use the power series for f(x) = 3√
1 + x3 to approximate∫ 1
03√
1 + x3dx
with error less than .0001
741.
A A A Use the fact that ddx
(1
1−x
)= 1
(1−x)2 to find a power series for
f(x) = 1(1−x)2 . Use this information to calculate the sum of
∞∑n=1
n
3n
Continue to generalize and find the sum of
∞∑n=0
(n+ 2)(n+ 1)
3n
Page 96
Chapter The Ninth: Parametric and Polar
742.
Eliminate the parameter and sketch the curve.
x = t+ 1 y = t2 + 2t+ 5
743.
Eliminate the parameter and sketch the curve.
x = et y = e2t + 2et + 6
744.
Eliminate the parameter and sketch the curve.
x = 3− 6 sin θ y = 2 + 4 cos θ
745.
Eliminate the parameter
x = A sin θ +B cos θ y = B sin θ −A cos θ
746.
Eliminate the parameter
x = A tan θ +B sec θ y = B tan θ +A sec θ
747.
Eliminate the parameter.
x = 1 +√
sin t y = sec t
748. Eliminate the parameter.
x = ln (t2 + t) y = (ln (t+ 1) + ln (t))(t2 + t)
Page 97
749. Eliminate the parameter.
x =3t√t2 + 1
y =3√t2 + 1
750.
Sketch each parametric curve and eliminate the parameter. What is the
difference in these curves.
Curve 1: x = 3 cos θ, y = 3 sin θ
Curve 2: x =√t, y =
√9− t
Curve 3: x =√9t2−1|t| , y = 1
t
Curve 4: x = −√
9− e2t, y = et
751.
Parameterize the circle x2 + y2 = 1 in 4 different ways:
A) Clockwise Orientation and Period of π
B) Counterclockwise Orientation and Period of π
C) Clockwise Orientation and Period of π2
D) Counterclockwise Orientation and Period of π2
752. Find dydx and d2y
dx2 .
x = sin4 (t) y = cos4 (t)
753. Find dydx and d2y
dx2 .
x = t2 + 6t− 2 y = t+ 3
754. Find all points of vertical and horizontal tangency and the equations of
their tangent lines
x = 2t3 − 9t2 + 12t y = t3 − 6t2 + 9t
755. Find all points of vertical and horizontal tangency and the equations of
their tangent lines
x = t3 − 3t y = 2t3 − 9t2 + 12t
756. Find all points of vertical and horizontal tangency and the equations of
their tangent lines
Page 98
x = t3 + 6t2 − 36t+ 2 y = t3 − 18t2 + 96t− 4
757.
Find all points of vertical and horizontal tangency and the equations of their
tangent lines
x = sin4 (t) y = cos4 (t)
758.
Find all points of vertical and horizontal tangency and the equations of their
tangent lines
x = 3x53 − 15x
43 + 15x y = 6x
53 − 15x
43 + 10x
759.
Determine the t intervals on which the curve is concave up and down. Find
all points of inflection.
x = te4t y = t2e4t
760.
Determine the t intervals on which the curve is concave up and down. Find
all points of inflection.
x = t2 + 4t+ 6 y = t3 + 7t2 + 16t+ 12
761.
Find the arc length of curve.
x = t y =t5
10+
1
6t31 6 t 6 2
762.
Find the arc length of curve.
x = arctan (t) y = ln√
1 + t2 0 6 t 6 1
763.
Find the arc length of curve.
x = arcsin (t) y = ln√
1− t2 0 6 t 61
2
Page 99
764.
Find the arc length of curve.
x = e2t cos (t) y = e2t sin (t) 0 6 t 6π
2
765.
Use parametric equations to derive the formula for the circumference of a
circle.
766.
Find the Area bounded by and the x-axis
x = ln(1 + x6) y =1
6x30 6 t 6 1
767.
Find the Area bounded by and the y-axis
x = t3 y = arctan(t) 0 6 t 6 1
768.
Find the Area bounded by and the y-axis
x = et2
y = 3t2 0 6 t 6 1
769.
Find the Surface Area of the solid of revolution formed by rotating the
surface created by the parametric equations about the x-axis and then the y-
axis.
x =t4
8+
1
4t2y = t 1 6 t 6 2
770.
Find the Surface Area of the solid of revolution formed by rotating the
surface created by the parametric equations about the y-axis.
x = t y =t2
4+
1
2ln(t) 1 6 t 6 2
771.
Find the Surface Area of the solid of revolution formed by rotating the
surface created by the parametric equations about the y-axis.
x =√
1− t2 y = arcsin(t) 0 6 t 6 1
Page 100
772.
Find the Surface Area of the solid of revolution formed by rotating the
surface created by the parametric equations about the x-axis.
x = arcsin (t) y = ln√
1− t2 0 6 t 61
2
773.
Convert the rectangular Point to polar form in two ways. First with r > 0
and then with r < 0.
(x, y) = (−3√
3, 3)
774.
Convert the rectangular Point to polar form in two ways. First with r > 0
and then with r < 0.
(x, y) = (−4, 4)
775.
Convert the rectangular equation to polar form and sketch its graph
x2 + 6x+ y2 = 0
776.
Convert the rectangular equation to polar form and sketch its graph
x = 10
777.
Convert the rectangular equation to polar form and sketch its graph
x2 + 2y = 1
778.
Convert the rectangular equation to polar form and sketch its graph
4x+ 7y = 10
779.
Convert the rectangular equation to polar form and sketch its graph
Page 101
(x2 + y2)2 − 9(x2 − y2) = 0
780. Convert the polar equation to rectangular form and sketch its graph
r = 4
781.
Convert the polar equation to rectangular form and sketch its graph
r = 3 sec (θ)
782.
Convert the polar equation to rectangular form and sketch its graph
r = θ
783.
Convert the polar equation to rectangular form and sketch its graph
r = 4 csc (θ)
784.
Convert the polar equation to rectangular form and sketch its graph
r =√
tan(θ) sec(θ)
785.
Convert the polar equation to rectangular form and sketch its graph
r =sin(θ)− cos(θ)
sin2(θ)
786.
Convert the polar equation to rectangular form and sketch its graph
r =cot(θ)− 1
cos2(θ)
787.
Find all points of vertical and horizontal tangency and the equations of their
tangent lines
r = 1 + sin (θ)
Page 102
788.
Find the Area of
r = 1 + cos (θ)
789.
Sketch and find all tangents at the poles.
r = sin (3θ)
790.
Find the common area of
r = 4 sin (2θ) and r = 2
791.
Find the common area of
r = sin(θ) and r = cos(θ)
792.
Find the common area of
r2 = 4 sin(2θ) and r2 = 4 cos(2θ)
793.
Find the area inside r1 outside r2
r1 = 4 sin(θ) and r2 = 2
794.
Convert the rectangular equation to polar and find the area it incloses.
(x2 + y2)2 − 16(x2 − y2) = 0
795.
Convert the rectangular equation to polar and find the area it incloses.
(x2 + y2)2 − 8(xy) = 0
796.
Sketch and find the Area of the inner loop
Page 103
r = 1 + 2 cos θ
797.
Find the common area of
r = 3− 2 sin (θ) and r = 3 + 2 sin (θ)
798.
Find the common area of
r = 4 sin (θ) and r = 2
799.
Show the Area of r = a sin (nθ) does not depend on the value of n.
800.
Find the arc Length of r = 1 + sin (θ)
801.
Find the arc Length of r = sin (θ)
802.
Find the arc Length of r = sin2 θ2 from θ = 0 to θ = π
803.
Find the arc Length of r = e4θ from θ = 0 to θ = 1
804.
Find the Surface area of the solid formed when you rotate r = 4 cos(θ) about
the x-axis.
805.
Find the Surface area of the solid formed when you rotate r = 4 cos(θ) about
the y-axis.
806.
Find the Surface area of the solid formed when you rotate r = 1 + cos(θ)
about the x-axis.
807.
Find the Surface area of the solid formed when you rotate r = sin(θ)+cos(θ)
about the x-axis, 0 ≤ θ ≤ π2 .
Page 104
Chapter The Tenth: Vectors
808.
A plane is flying with direction N45W (45 West of North) at a speed of
450 mph. At some time the wind begins blowing 85 mph with direction S30E.
Find the resulting speed and direction of the plane.
809.
Consider the following points: (1,3), (2,7), (4,12). Do they form an acute,
obtuse or right triangle. Find the Area of said triangle. Remember Heron’s
formula for area of a Triangle with sides a, b and c says:
Area =√s(s− a)(s− b)(s− c) s =
a+ b+ c
2
810.
Prove the following theorem from geometry using vectors: The line segment
that connects the midpoints of two sides of a triangle is parallel to and half the
length of the third side.
811.
Let C be the point on the line segment AB that is twice as far from B as it
is from A. For any point O if u =−→OA v =
−−→OB and w =
−−→OC show:
w =2
3u +
1
3v
812.
Use Vectors to prove the Law of Sines
813.
Use vectors to prove the following theorem from geometry: For any paral-
lelogram the midpoints of the four sides are the vertices of a parallelogram.
814.
Find 2 unit vectors that make an angle of π3 with the vector v =< 4, 3 >
815.
Let u =< a, 3, 1 > and v =< a, a, 2 >. Find the values of a that make u
orthogonal to v
816.
Show (u|v|+ v|u|) is orthogonal to (u|v| − v|u|)817.
Page 105
Let u =< a, 1, 3a > and v =< 3, 0, 4 >. Find the values of a that cause the
angle between u and v to be 60.
818.
Find two orthogonal vectors u and v that are also orthogonal to w =<
1,−1, 1 >
819.
Prove that u and v are orthogonal iff |u + v| = |u− v|820.
If |u| = 2 and |v| = 3 and the angle between u and v is π3 find |u + v| and
|u− v|821.
Let u,v, and w be mutually orthogonal vectors in R3 prove u + v + w 6= 0
822.
Find the direction angles (direction cosin) for v =< 4, 2, 7 >.
823.
If two of the direction angles for a vector are cosα = 16 and cosβ = 1
4 find
cos γ
824.
Find a vector with a magnitude of 4 and the following direction angles:
α =π
3β =
π
40 ≤ γ ≤ π
2
825.
Prove that the sum of any two direction angles must be greater than π2
826.
Let u =< 2, 1, 2 > v =< 6, 0, 8 >. Find the component of u parallel to v.
Find the component of u orthogonal to v.
827.
Find the distance from the point (1, 3, 4) to the line Passing through (1, 2, 1)
and (0, 4,−3)
828.
If (u + 2v) is orthogonal to (u − v), |u| = 4 and u · v = 2 find |v| and the
cosine of the angle between u and v.
829.
Use vectors to prove the following theorem from geometry: the diagonals of
a rhombus are perpendicular.
830.
Page 106
If w = |u|v + |v|u show w bisects u and v.
831.
Show that the area of the parallelogram spanned by vectors u and v is given
by the equation
A = |v| · |u− Projv(u)|
832.
Find a unit vector orthogonal to both u =< 1, 2, 1 > and v =< 1, 4, 5 >
833.
Find the area of the parallelogram spanned by
u =< 2,−2, 1 > v =< 3, 4, 7 >
834.
Show that the parallelogram spanned by the two vectors has constant area
u =< 2, 0, x > v =< 2, 0, x− 1 >
835.
If u× v is a unit vector and |u| = 3 and |v| = 4 what is the angle between
u and v
836.
Find the magnitude of the torque produced by applying a 15 N force on a
.65 m wrench at an angle of 30.
837.
Find the volume of the parallelepiped spanned by:
u =< 1, 2, 1 > v =< 2, 4, 1 > w =< 1, 7, 2 >
838.
Show the volume of the parallelepiped spanned by:
u =< 1, x, 3x > v =< 1, 2 + x, 4x > w =< 2, 2x, 1 + 6x >
is constant.
839.
Show
|u× v|2 = |u|2|v|2 − (u · v)2
Page 107
840.
Prove the Jacobi identity
u× (v ×w) + v × (w × u) + w × (u× v) = 0
841.
If |u| = 4 and |v| = 5 Find the area of the parallelogram spanned by u+ 2v
and u− v.
842.
Find the parametric and symmetric equations of the line passing through
(0, 3, 1), and (4,−2, 6)
843.
Find the distance between the point (2, 2, 3) and line:
x− 4
2= y + 1 =
z − 1
3
844.
Find the point of intersection of the two lines:
Line 1: x−82 = y−1
1 = z+153
Line 2: x4 = y−6
−1 = z−3
845.
Determine weather the lines are parallel, identical or neither.
Line 1: x−12 = y+2
3 = z−4−1
Line 2: x+52 = y+11
3 = z−7−1
Line 3: x+34 = y+1
6 = z−3−2
Line 4: x−136 = y−16
9 = z+2−3
846.
Find the equation of a plane passing through equidistant from (1, 1, 1) and
(2, 4, 8)
847.
Find the equation of a plane passing through (0, 1, 0), (1, 9, 3) and (1,−2, 6)
848.
Find the equation of a plane passing through (1, 1, 0), and containing the
line: x+34 = y+1
6 = z−3−2 .
849.
Page 108
Find the point of intersection of the line and the plane:
Line: x−12 = y+2
3 = z−4−1
Plane: 2x− y − 3z = 16
850.
Show the line and the plane do not intersect
Line :x− 4
1=y − 2
3=z − 7
−1
Plane : −2x+ y + z = 0
851.
What value of D will make the line lie in the plane
Line :x− 4
1=y − 2
3=z − 7
−1
Plane : −2x+ y + z = D
852.
Find the line of intersection of the two planes:
Plane 1: 2x− 4y + 5z = 3
Plane 2: 4x− 2y − 4z = −2
853.
Find the distance between the planes 6x+3y−27z = 81 and 2x+y−9z = 210
854.
Find the distance from the point to the plane.
Point (4, 3, 5) Plane 2x+ 3y + z = 5
855.
The Plane 2x+ 3y− z = 12 is tangent to a Sphere with center (1, 1, 1). Find
the equation of this Sphere
856.
a) Find the equation of a sphere with center (0, 1, 0) and radius r = 7.
b) Show that points P1(2, 4, 6) and P2(−6, 4,−2) are on the sphere.
Page 109
c) Find a tangent plane to the sphere at P1 let us name this plane Π1.
d) Find a tangent plane to the sphere at P2 let us name this plane Π2.
e) Find the angle between Π1 and Π2
f) Find the line of intersection of Π1 and Π2 let us name this line L
g) Find the point(s) of intersection of L and sphere of radius 3 centered at
(−14,−3, 14)
857.
Find the distance between the following skew lines:
Line 1x− 1
1=y − 1
2=z
1
Line 2x− 1
2=y
1=z − 2
1
858.
Find the equation of the line passing through (6, 4, 5) intersecting and or-
thogonal to the line:
x = 2 + t y = 3− 3t z = 3 + 2t
859.
For the two parallel planes
Π1: ax+ by + cz = d1
Π2: ax+ by + cz = d2
Show that the distance between the two planes is given by
Dist
(Π2,Π2
)=
|d1 − d2|√a2 + b2 + c2
Use this to find a two planes parallel to and distance of 10 units from the
plane
2x+ y + 3z = 6
860.
Page 110
Consider the hyperbola y = 1x and two fixed points A(a, 1a ) and B(b, 1b ) on
the hyperbola, use vectors and differential calculus to find a point C(c, 1c ) with
a < c < b where the triangle between points A, B and C has a maximum area.
Page 111
Chapter The Eleventh: Vector Valued Functions
861.
Find the domain of
r(t) =
⟨ln(t− 1),
√25− t, arcsin
(6
t
)⟩862.
Find the value of t that maximizes the volume of the parallelepiped spanned
by the three vector valued functions:
r(t) =
⟨t, 1− t, t
⟩u(t) =
⟨1− t, t, 1− t
⟩v(t) =
⟨t3, t,
−1
2t2⟩
863.
Find a vector valued function representing the intersection of the paraboloid
z = x2 + y2 and the plane x+ y = 8
864.
Find a vector valued function representing the intersection of the cylinder
9 = x2 + y2 and the surface z = x2 in two ways: First by letting x = t and
solving for y and z. Second by trying to force a trig identity on the equations.
865.
Show that the vector valued function r(t) =< t, 2t cos(t), 2t sin(t) > lies on
the cone 4x2 = y2 + z2.
866.
Evaluate:
limt→0
(arcsin(πt)
t· i +
e2t − 2t− 1
t2· j +
sin(πt)
t· k)
867.
Find r′(t)
r(t) =
(t3et · i + arctan(t2) · j + (3t− 5) · k
)868.
Let r(t) =< t3 + t, t2 − t− 1, t4 > and u(t) =< t2 + t, 3t3 − 3t− 1, 2t3 − t >.
Show that these vector valued functions intersect when t = 1 and find the angle
between the vector valued functions where they intersect.
Page 112
869.
Prove that if r is continuous then ||r|| is also continuous
870.
Find a function r that is not continuous but ||r|| is continuous
871.
Show
d
dt||r(t)|| = r(t) · r′(t)
||r(t)||872.
Show
d
dt
(r(t)× r′(t)
)= r(t)× r′′(t)
873.
A projectile is fired at 200 m/s at an angle of 30 above the horizontal from
a height of 25 feet above ground. What is the maximum height of the projectile
and what is its range.
874.
A projectile is fired from the ground at an angle of 45 above the horizontal.
How fast must the projectile be fired to have a range of 500m.
875.
A projectile is fired from the ground at an angle of 45 above the horizontal.
How fast must the projectile be fired so that it can hit a target 200m above the
ground and 200m horizontal distance from the firing spot.
876.
A golfer hits a golf ball with an initial velocity of 160 feet per second at an
angle of 30 above the horizontal. Find the vertical and horizontal components
of of the velocity and position vectors. What is the maximum height the ball
reaches? What is the balls range?
877.
Find r(t)
r′′(t) =
(et · i + 6t2 · j + 24t3 · k
)
r′(0) =< 2, 2, 1 > r(0) =< 1, 5, 9 >
878.
Page 113
Find r(t)
r′′(t) =
(sin(2t) · i + cos(2t) · j + (6t− 8) · k
)
r′(0) =< 2, 4, 2 > r(0) =< 6, 5, 9 >
879.
The acceleration of a particle is given by:
a(t) =< 2, 6t, et >
If the initial velocity and position is given by:
v(0) =< 1, 2, 3 > r(0) =< 2, 4, 6 >
Find a vector valued function representing the position of the particle.
880.
Prove that if a particle moves with constant speed then its velocity and
acceleration vectors are orthogonal.
881.
The vector valued functionr(t) =< t3, t2, t > intersects the plane x + 2y +
4z = 24 at a point. Find the angle of intersection at this point.
882.
If a particle follows the path r(t) =< t3, 4t2, 6t > until it flies off on a tangent
at t = 1. Where does the particle hit the plane −x+ 2y − z = 15?
883.
If a particle follows the path r(t) =< t cos(t), t sin(t), t > until it reaches a
speed of√
2 + π2. At this time the particle flies off on a tangent. Where does
it hit the plane 4x+ 1πy + 4z = −2
884.
For the position vector r(t) =
⟨t3
3 , t2, 2t
⟩find: T(1), N(1), aT(1), aN(1),
K(1).
885.
For the position vector r(t) =< et, e2t, e3t > find: T(0), N(0), aT(0), aN(0).
886.
Find r(t) given:
T =< t,√
2t, 1 >
t+ 1N =
<√
2t, 1− t,−√
2t >
t+ 1
Page 114
aT = 1 aN =1√2t
r(0) =< 0, 0, 0 > r′(0) =< 0, 0, 1 >
887.
Find the Arc Length of r(t) =< t, t cos(t), t sin(t) > on the interval [0,√
2]
888.
Find the Arc Length of r(t) =
⟨12 t,√32 t,
16 t
3 + 12t
⟩on the interval [1, 2]
889.
Find the Arc Length of r(t) =
⟨t, 2√2
3 t32 , 12 t
2
⟩on the interval [0, 2]
890.
Find the Arc Length of r(t) =< t2, t3, 2t3 > from the point (0, 0, 0) to the
point (1, 1, 2)
891.
Find the Arc Length of r(t) =
⟨12 t
2, 3t, 4t
⟩, t=0 , t=5
892.
Find the Arc Length of r(t) =
⟨23 t
3, t2, t
⟩, t=0 , t=1
893.
Find the Arc Length of r(t) =
⟨14e
4t,√23 e
3t, 12e2t
⟩, t=0 , t=1
894.
Express the arc length:s(t) of the helix r(t) =< 4 cos(t), 4 sin(t), t > as a
function of t by integrating:
s(t) =
∫ t
0
||r′(u)||du
Then parameterize r(t) with the are length parameter s by solving the pre-
vious calculation for t and inserting it into r(t) to create r(s). Find the point
on the helix when s = 4. Show ||r′(s)|| = 1. Use the definition of Curvature:
K = ||T′(s)|| to find curvature K when s = 4.
895.
Express the arc length:s(t) of the path r(t) =< et, e−t,√
2t > as a function
of t by integrating:
s(t) =
∫ t
0
||r′(u)||du
Page 115
Then parameterize r(t) with the are length parameter s by solving the pre-
vious calculation for t and inserting it into r(t) to create r(s). Find the point
on the path when s = 4. Show ||r′(s)|| = 1.
896.
Find the curvature for r(t) =< et cos(t), et sin(t) > when t = π4 .
897.
Find the curvature for r(t) =< 2t3, t, t2 > when t = 1.
898.
Find the curvature for r(t) =< t3, t, t3 > when t = 1.
899.
Find the curvature of the ellipse x2
9 + y2
36 = 1 at the point (0, 6).
900.
Find the curvature of y = arctan(x).
901.
Find the point on the graph of y = ln(x) where curvature is a maximum.
What happens to the curvature as x→∞.
902.
Find the point on the graph of y = 1x where curvature is a maximum. What
happens to the curvature as x→∞.
903.
Show a parabola has maximum curvature at its vertex
904.
Show an ellipse has maximum curvature at points where the major axis
intersects the curve and a minimum curvature where the minor axis intersects
the curve.
905.
Let P be a point on the curve x23 + y
23 = a
23 . Let L be the tangent line to
the curve at P . Show that the curvature at P is 3 times the distance from the
origin to tangent line L
906.
Find a vector valued function representing the intersection of the paraboloid
z = x2 + y2 and the plane x + y = 8. At what point(s) does the vector valued
function intersect the plane 2x+ 2y + z = 54?
907.
Let r(t) be a vector valued function in R3 with the property r(t) · r′(t) = 0.
What does this tell us about the shape of the graph of r(t).
908.
Page 116
Let r(t) be a vector valued function with constant magnitude. Show r(t)
and r′(t) are orthogonal.
909.
Use the results of the previous problem to show that if the magnitude of
r(t) is constant then r′(t) can be written as the sum of two vectors: the first
parallel to r(t) and the second orthogonal to r(t).
910.
Show that if r′′(t) is parallel to r(t) then r(t)× r′(t) is constant.
911.
Show T’(t) ·T(t) = 0
912.
Show B’(t) ·B(t) = 0
913.
Show B’(t) ·T(t) = 0
914.
Show
y = cosh(x) =ex + e−x
2
is a solution to
K =1
y2
where K is the curvature
K =|y′′|(
1 + (y′)2) 3
2
915.
if k(t) is the curvature of a function then we define the Total Curvature on
the interval [t0, t1] to be
K =
∫ t1
t0
k(t)|v(t)|dt
Find the total curvature of
r(t) =< cos(t), sin(t), t >
on [0, 2π]
Page 117
916.
Find the total curvature of
r(t) =< et,√
2t, e−t >
on [0, 1]
917.
Find the total curvature of
r(t) =
⟨1
2t2,
4√
2
3t32 , 4t
⟩on [0, 1]
918.
It has been proven that the Total Curvature of of a vector valued function
on a closed path is always an integer multiple of 2π. Confirm this holds for the
circle:
r(t) =< R cos(t), R sin(t) > [0, 2π]
Page 118
Chapter The Twelfth:
919.
Find the domain and range of the function.
f(x, y) =√
16− x2 − y2
920.
Find the domain and range of the function.
f(x, y) = arcsin
(1
x2 + y2
)921.
Evaluate the limit or show it does not exist.
lim(x,y)→(0,0)
x2 − y2
x2 + y2
922.
Evaluate the limit or show it does not exist.
lim(x,y)→(0,0)
exy − 1
xy
923.
Evaluate the limit or show it does not exist.
lim(x,y)→(0,0)
x3 + y3
x2 + y2
924.
Evaluate the limit or show it does not exist.
lim(x,y)→(0,0)
sin(8x2 + 6y2)
20x2 + 15y2
925. Determine if the function is continuous. Find all points of discontinuity.
f(x) =
x2y2
x2+y2 (x, y) 6= (0, 0)
0 (x, y) = (0, 0)
926.
Find the values of C where level sets exist and describe them.
Page 119
z = sin(x2 + y2)
927.
Use the definition of partial derivatives to find ∂f∂x and ∂f
∂y
f(x, y) = 2x2 − 4xy + 3y2 − x+ y
928.
Use the definition of partial derivatives to find ∂f∂x and ∂f
∂y
f(x, y) =√xy − 4
929.
Use the differentiation rules to find ∂f∂x and ∂f
∂y
f(x, y) = x arctan(x3 − y3)
930.
Use the differentiation rules to find ∂f∂x and ∂f
∂y
f(x, y) =xey
2√x2 + y2
931.
Use the differentiation rules to find ∂f∂x , ∂f
∂y , ∂2f∂y2 , ∂2f
∂x2 , ∂2f∂x∂y and ∂2f
∂y∂x
f(x, y) = x4y3 − 6x3y + 4x5 − y2
932.
Use the differentiation rules to find ∂f∂x , ∂f
∂y , ∂2f∂y2 , ∂2f
∂x2 , ∂2f∂x∂y and ∂2f
∂y∂x
f(x, y) =
∫ y
x
et2
dt
933.
Show fxyz = fxzy = fzxy for
f(x, y, z) = x2y2z3 − 6x3y + xz4 − y
934.
Show the function satisfies the Laplace equation ∂2z∂x2 + ∂2z
∂y2 = 0
Page 120
z = e2x sin(2y)
935.
Show the function satisfies the Laplace equation ∂2z∂x2 + ∂2z
∂y2 = 0
z = arctan
(y
x
)936.
Show the function satisfies the wave equation ∂2z∂t2 = c2 ∂
2z∂x2
z =1
2(f(x− ct) + f(x+ ct))
937.
Show the function satisfies the wave equation ∂2z∂t2 = c2 ∂
2z∂x2
z = sin(ωct) sin(ωx)
938.
Find the total differential dz
z = 4x3y + 7xy3 − x− 3y
939.
Find the total differential dz
z = sin(x2 + y2)− cos(x2 + y2)
940.
Find the total differential dz
z = arctan
(x
y
)941.
Use the total differential to approximate the quantity:
√3.12 + 3.952
942.
Use the total differential to approximate the quantity:
Page 121
arctan
(.98
1.02
)943.
The height of a right circular cylinder is measured to be 10 with maximum
possible error of .1 units. The radius is measured to be 4 with maximum possible
error of .2 units. Use differentials to approximate the maximum possible error
of the volume and surface area of the cylinder.
944.
The major axis of an ellipse is measured to be 8 units long with maximum
possible error of .2 units. The minor axis is measured to be 5 units long with
maximum possible error of .3 units. Use differentials to approximate the maxi-
mum possible error of the Area of the ellipse.
945.
An engineer uses a digital multimeter to measure the resistance of a resister.
The engineer measures the resistance to be 20 Ω, but the engineer also knows
the digital multimeter is not accurate and can have an error of .05Ω. The
voltage source connected to the resister produces a 120 Volt with a possible
error in measuring the voltage of 1 Volt. Use differentials to estimate the error
in measuring the power dissipated by the resister.
P =V 2
R
946.
Two resistors R1 and R2 are in parallel. R1 is measured to be 10 Ohms
while R2 is measured to be 13 Ohms. The maximum error in each of these
measurements can be no more than .25 Ohms. Use differentials to estimate the
error in measuring the resistance of these resistors in parallel.
1
R=
1
R1+
1
R2
947.
The radius of a cylinder is increased by 2 percent while the height is decreased
by 3 percent. Use differentials to approximate the percent change in the volume
of the cylinder.
948.
The radius of a cone is increased by 3 percent. What percent decrease in
the height of the cone will produce no change in its volume?
Page 122
949.
If the velocity of a mass is increased by 5 percent what percent change in
mass will produce a 1 percent increase in kinetic energy.
950.
A gas is kept at a constant pressure and its temperature increases by 1 per-
cent. Use differentials to approximate the percent change in volume. Remember
PV = nrT .
951.
If x = t2, y = t3 and z = 2t and w = x2 + y2 + z2 find dwdt
952.
If x = t cos(t), y = t sin(t) and z = t3 and w = xy + x2z find dwdt
953.
If x = r cos(θ), y = r sin(θ) and z = (x2 + y2)2 − 16xy. Find ∂z∂r and ∂z
∂θ .
954.
If x = t cos(θ), y = t sin(θ) and z = t+ θ and w = x2 + y2 + z2 find ∂w∂t and
∂w∂θ
955.
If x = t3 arctan(θ), y = sin(θ+ t) and z = t2 + θ2 and w = x3 + xy+ z3 find∂w∂t and ∂w
∂θ
956.
Let w = x2 + 2y2 + 3z3 if y(t) = t2, z(t) = t3 and dwdt = 0 at the point
(3, 1, 1) find x(t)
957.
Let f(x, y) = xyx2+xy+y2 = 1. Use partial derivatives to calculate the dy
dx
implicitly.
958.
Let f(x, y, z) = xyz − xy − yz. Use partial derivatives to calculate the ∂z∂x
and ∂z∂y implicitly.
959.
Let r(t) =< x(t), y(t), z(t) > be a vector valued function representing the
velocity of a particle and let w = f(x, y, z). Show dwdt is zero at a point where
the velocity of the particle is zero. Can you also explain this in terms of the
rate of change of w?
960.
Let f(x, y, z) = 4 be a level set function with the property that ∇f(x, y, z) =
16(x, y, z). Let r(t) =< x(t), y(t), z(t) > be on the level set f(x, y, z) = 4. What
shape must the path of r(t) produce?
Page 123
961.
Find the gradient of f(x, y) = x2y + 3xy3 at the point (1, 1, 4) and the
equation of the tangent plane and normal line to the surface of f at this point.
962.
The Temperature of an object at any point in the x-y Plane is given by:
T (x, y) = x2y3 + xy
If you start at the point (1, 1) in what direction should you travel so that
the temperature of the object is increasing the most? In what direction should
you travel so that the temperature of the object is decreasing the most?
963.
Show that the following vector field cannot be the gradient of some functionf(x, y, z)
F =< xy, xz, yz >
964.
Let
f(x, y) = y2exy + 2y P (0, 2)
In what direction u is
(Duf
)∣∣∣∣P
= 1
965.
Show
P =
(1√2, 1, 2
)and Q =
(−1√
2, 1, 2
)are on the ellipse x2+
y2
4+z2
16= 1
Find the line of intersection of the tangent planes to the ellipse at P and Q.
966.
The two vector valued functions r1(t) =< t, t3, 2t2 > and r2(s) =< 2s, s3 +
7s, 3s2 + 5 > lie on a surface and share the point (2, 8, 8). Find the equation of
the tangent plane to this surface at this point
967.
Consider the surface x2+2y2+3z2 = 9. Find all points on the surface where
the tangent plane is parallel to x+ 2y + 3z = 10
968.
Let f(x, y) = x2 + y2. Show that all tangent planes to f that pass through
the point P (0, 0,−1) lie on a circle centered at (0, 0, 1) of radius 1
Page 124
969.
Let u be orthogonal to ∇f at P. Find Duf(P )
970.
Find the directional derivative of f(x, y) = 2x2y+ xy2 at (1, 2) in the direc-
tion of a vector that makes an angle of π6 with the positive x-axis.
971.
Find the directional derivative of f(x, y, z) = xz2 + 2xy3 + z3 at (1, 2, 1) in
the direction v =< 2, 3, 6 >
972.
Find the directional derivative of f(x, y, z) = x3 + xyz + y3 + z3 at (1, 2, 1)
in the direction of a vector that makes an angle of 4π3 with the positive x-axis
and an angle of 7π6 with the positive z-axis.
973.
The directional derivative of f(x, y) in the direction e1 =< 1, 0 > isDe1f(x, y) =
3x2y3 + 2x and the derivative of f(x, y) in the direction e2 =< 0, 1 > is
De2f(x, y) = 3x3y2 + 2y if f(1, 1) = 4 find f(x, y).
974.
At point P the directional derivative of f in the direction < 3, 4 > is 2 and
the directional derivative in the direction < −6, 8 > is 4. Find the directional
derivative of f in the direction < 9, 20 >
975.
Find all relative extrema of f(x, y) = 2x3 + xy2 + 5x2 + y2
976.
Find all relative extrema of f(x, y) = x3 − 12xy + 8y3
977.
Find all relative extrema of f(x, y) = x3 + 3x2y2 − 8y3
978.
Find all relative extrema of f(x, y) = x4y + x4 − 4x− 4xy
979.
Find all relative extrema of f(x, y) = xy + 27x + 27
y
980.
Find all relative extrema of f(x, y) = (2x− x2)(2y − y2)
981.
Find all relative extrema of f(x, y) = xe3xy−2y2
982.
Page 125
Find all relative extrema of f(x, y) = x3 + y2 − 6xy + 6x+ 3y
983.
Find all relative extrema of f(x, y) = x3 + y3 − 3x2 − 3y2 + 9x
984.
Show (0, 0) is a critical point of the function and find the values of k that
that make (0, 0) a max, a min and a saddle point.
f(x) = x2 + 4xy + ky2
985.
Show (0, 0) is a critical point of the function and find the values of k that
that make (0, 0) a max, a min and a saddle point.
f(x) = kx2 + 4xy + ky2
986.
Find all absolute extrema of f(x, y) = 3x2 + 2y2 − 4y on the region in the
x-y plane bounded by the curves y = x2 and y = 4.
987.
Find the Absolute Max and Minimum of the function on the region bounded
by y = 2x, x = 4 and y = 0
f(x, y) = 2x2 + y2 − 4xy + 4y
988.
Find all extrema of f(x, y) = x3y5 subject to x+ y = 8
989.
Find all extrema of f(x, y) = x14 y
34 subject to 2x+ 3y = 24
990.
Find all extrema of f(x, y, z) = x3 +y3 +z3 on the intersection of the planes
x+ y + z = 2 and x+ y − z = 3
991.
Find all extrema of f(x, y, z) = xyz on the plane x+ y + z = 15
992.
Find all extrema of f(x, y, z) = xy + xz + yz on the plane x+ y + z = 1
993.
Find all extrema of f(x, y, z) = xy2z subject to x2 + y2 + z2 = 4
994.
Page 126
Find all extrema of f(x, y, z) = x2y2z2 on 4x2 + y2 + z2 = 1
995.
Find all extrema of f(x, y, z) = xy−xz subject to x+2z = 6 and x−3z = 12
996.
Find all extrema of f(x, y, z) = xyz subject to x+y+z = 32 and x−y+x = 0
997.
Find all extrema of f(x, y, z) = 2xy + 2yz − 2x2 − 2y2 − 2z2 subject to
x2 + y2 + z2 = 4
998.
Find the minimum distance from the origin to the intersection of xy = 6
and 7x+ 24z = 0
999.
After graduating from medical school, Bubba decides to start his own trav-
eling circus with clowns, pirates and ninjas. Bubba only has enough money to
provide food for 12 employees and estimates the entertainment level of the show
to be governed by the equation E(C,P,N) = C3P 4N5 where C represents the
number of clowns, P represents the number of Pirates, N represents the number
of Ninjas. How many of each should Bubba employ?
1000.
In the chapter over vectors we learned how to find the distance from a point
(x0, y0, z0) to the plane ax + by + cz + d = 0 using projection. The general
formula for this distance is:
D =|ax0 + by0 + cz0 + d|√
a2 + b2 + c2
Use the method of Lagrange Multipliers to derive this formula
1001.
Find the distance from the point (1, 1, 1) to the plane x+ 2y + 3z = 20.
1002.
Let the sum of x, y and z be a constant C use the method of Lagrange
Multipliers to show:
3√xyz ≤ x+ y + z
3
1003.
Use the method of Lagrange Multipliers to show that the maximum area of
a triangle with fixed perimeter p and sides a, b, and c occurs when the triangle
is an equilateral triangle. Use Heron’s formula:
Page 127
A =√s(s− a)(s− b)(s− c) s =
p
2
1004.
Use Lagrange Multipliers to prove the following
f(x1, x2, x3, ..., xn) = xe11 · xe22 · x
e33 · ... · xenn
is maximized subject to the constraint
x1 + x2 + x3 + ...+ xn = e1 + e2 + e3 + ...+ en
when
xi = ei
1005.
Show that for three positive functions f , g and h whose product is a constant
then their sum is a minimum when the three functions are equal. Use the results
to minimize
f = x2 +100
x
1006.
Show that for three positive functions f , g and h whose sum is a constant
then their product is a maximum when the three functions are equal. Use the
results to maximize
f = xy(100− 2x− 5y)
Page 128
Chapter the Thirteenth: Double Integrals
1007.
Calculate the following integral:∫ ∫2xydA
Over the region bounded by y = x2, x+ y = 2 and x = 0
1008.
Calculate the following integral:∫ ∫sin(x2)√y
dA
Over the region bounded by y = x2, y = 4x2 and x =√π
1009.
Calculate the following integral:∫ 8
0
∫ 2
3√y
ex4
dxdy
1010.
Calculate the following integral:∫ 4
0
∫ 2
√x
1
1 + y3dydx
1011.
Calculate the following integral:∫ 1
0
∫ 1
x
exy dydx
1012.
Find the volume of the solid bounded by the planes z = x, y = x, x+ y = 2
and z = 0.
1013.
Set up double integral that would find the volume of the solid above the x-y
plane bounded by z = 16− x2 − y2, x2 + y2 = 4
1014.
∫ 1
0
∫ √xx2
xydydx
1015.
Page 129
∫ 1
0
∫ 1+x2
1
x+ 1
y2dydx
1016.
Find the volume under the graph of f(x, y) = ye√x
x over the region R.
R: region bounded by y = 0, x = 4 and y =√x.
1017.
Find the volume under the graph of f(x, y) = 11+x4 over the region R.
R: region bounded by x = 2, y = 0 and y = x3.
1018.
Find the volume under the graph of f(x, y) = 1ln(y) over the region R.
R: region bounded by y = exand y = e√x.
1019.
Find the volume under the graph of f(x, y) = 4y3 sin(x3) over the region R.
R: region bounded by y = 0, x = π and y =√x.
1020.
∫ 1
0
∫ 12
y2
e−x2
dxdy
1021.
∫ ln(5)
0
∫ 5
ex
1
ln(y)dydx
1022. ∫ π2
0
∫ 1
sin(y)
1
arcsin(x)dxdy
1023.
∫ π2
0
∫ π
√y
sin(x3)dxdy
1024.
Find the volume under the graph of f(x, y) = cos(x2 + y2) over the region
R.
R: region bounded by y = 0 and y =√
9− x2.
1025.
Find the volume under the graph of f(x, y) = yex over the region R.
R: region bounded by x2 + y2 = 1
Page 130
1026.
Find the region R that maximizes the integral∫R
∫(16− x2 − y2)dA
1027.
Find the region R in the x-y plane that maximizes the volume under the
surface and above the x-y plane
f(x, y) = 2x2 − 8x+ y2 + 6y
1028.
∫ 2
0
∫ √4−y2
0
(x2 + y2)32 dxdy
1029.
∫ 2
0
∫ √2x−x2
0
√x2 + y2dydx
1030.
∫ 2
0
∫ √4−y2
y
2
(arctan
(yx
))3
√x2 + y2
dxdy
1031.
∫ 2
0
∫ √2y−y2
0
3xey√
x2+y2 dxdy
1032.
∫ e
−e
∫ √e2−x2
√1−x2
1
x2 + y2dydx
1033.
∫ 2
0
∫ x
0
√x2 + y2dydx+
∫ 2√2
2
∫ √8−x2
0
√x2 + y2dydx
1034. ∫ ∞0
e−x2
dx
Page 131
1035.
Find the Center of mass of the lamina bounded by the graphs of the equations
with the given density function ρ(x, y):
y =√x y = 0 x = 1 ρ(x, y) = ky
1036.
Find the Center of mass of the lamina bounded by the graphs of the equations
with the given density function ρ(x, y):
y =4
xy = 4 y = 1 ρ(x, y) = ky
1037.
Find the Center of mass of the lamina bounded by the graphs of the equations
with the given density function ρ(x, y):
y = ex y = 0 x = 0 x = 1 x = 0 ρ(x, y) = ky
1038.
Find the Center of mass of the lamina bounded by the graphs of the equations
with the given density function ρ(x, y):
y = sinx y = 0 x = π x = 1 x = 0 ρ(x, y) = k
1039.
Find the Center of mass of the lamina bounded by the graphs of the equations
with the given density function ρ(x, y):
y = ln(x) x = 1 x = e y = 0 ρ(x, y) = k
1040.
Find the Center of mass of the lamina bounded by the graphs of the equations
with the given density function ρ(x, y):
y =√
4− x2 0 < y < x ρ(x, y) = k
1041.
Find k so that the Center of mass of the lamina bounded by the graphs of
the equations with the given density function ρ(x, y) = 1 is ( 32 ,
485 )
Page 132
y = x2 y = 0 x = k
1042.
Find the Surface Area of the plane in the first octant 2x+ 3y + 4z = 12.
1043.
Find the Surface Area of the paraboloid f(x, y) = 16 − x2 − y2 above the
x-y plane
1044.
Find the Surface Area of f(x, y) = ln(x2+y2) between the circles x2+y2 = 4
and x2 + y2 = 12.
1045.
Find the Surface Area of f(x, y) = x2 + 4xy − y2 over the region in the x-y
plane bounded by the circle x2 + y2 = 1
1046.
Find the Surface Area of the sphere x2 + y2 + z2 = 100 inside the cylinder
x2 + y2 = 4.
1047.
Let ax + by + cz = d be a plane that makes an acute angle of φ with the
positive z axis. Let R be a closed region in the x − y plane. Show that the
surface area of the plane over the region is:
A secφ where A is the area of the region
1048.
Find the volume of the solid inside both the sphere x2 + y2 + z3 = 16 and
the cylinder x2 + y2 = 4
1049.
Find the volume of the solid inside both the sphere x2 + y2 + z3 = 16 and
(x− 1)2 + y2 = 4.
1050.
Find the volume of the solid inside the sphere x2 + y2 + z3 = 16 and outside
the cone z =√x2 + y2.
1051.
Find the volume of the solid inside the sphere x2 + y2 + z3 = 16 and above
the upper nappe of the cone z =√x2 + y2.
1052.
Evaluate the following integral over the unit ball.
Page 133
∫ ∫ ∫ex
2+y2+z2dV
1053.
Evaluate the following integral over the given Solid.∫ ∫ ∫(9− x2 − y2)dV
Solid:
x2 + y2 + z2 ≤ 9 z ≥ 0
1054.
Evaluate the following integral over the region between the spheres ρ = 2
and ρ = 4 and above the cone φ = π4 .∫ ∫ ∫
xyzdV
1055.
Evaluate the following integral∫ ∫ ∫R
ln(1 + x2 + y2 + z2)dV
R is first octant of sphere x2 + y2 + z2 ≤ 9
1056.
m =
∫ ∫ ∫ρ(x, y, z)dV
Myz =
∫ ∫ ∫xρ(x, y, z)dV
Mxz =
∫ ∫ ∫yρ(x, y, z)dV
Mxy =
∫ ∫ ∫zρ(x, y, z)dV
x =Myz
my =
Mxz
mz =
Mxy
m
Find the center of mass for the unit cube with density function
ρ = 2x+ 2y + 2z
Page 134
1057.
Find the center of mass of the solid bounded by:
5x+ 3y + 3x = 15 x = 0 y = 0 z = 0
with density function
ρ = x
1058.
Evaluate the following integral∫ ∫R
(x+ y)dA
R is square with vertices (0, 0),(1, 2),(3, 1),(2,−1).
1059.
Evaluate the following integral∫ ∫R
(x2 + y2)dA
R is region in first quadrant bounded by the curves
y =1
xy =
3
xx2 − y2 = 1 x2 − y2 = 4
1060.
Evaluate the following integral∫ ∫R
xy
1 + x2y2dA
R is region in first quadrant bounded by the curves
y =1
xy =
4
xx = 1 x = 4
Page 135
Chapter the thirteenth: Integrals over paths and surfaces.
1061.
Evaluate the following line integral∫C
(x+ y + z)ds
C(t) =< sin(t), cos(t), t > t ∈ [0, 2π]
1062.
Evaluate the following line integral∫C
(x2 + y2)ds
C(t) =< sin(t), cos(t), t > t ∈ [0, 2π]
1063.
Evaluate the following line integral∫C
(x2 + y2)dx− 2xydy
C(t) is the parabola y = 2t2 connecting the points(0, 0) and (2, 8)
1064.
Evaluate the following line integral∫C
y2dx+ 2xdy + dz
C(t) connects the points (0, 0, 0) to(1, 1, 1) by following the circle from (0, 0, 0)
to (1, 1, 0) and then on a vertical line to (1, 1, 1)
1065.
Evaluate the following line integral along the closed path
Page 136
∮(x− y2)dx+ 2xydy
the path is the square with corners(0, 0) (1, 0) (1, 1)and (0, 1)
1066.
Evaluate the following line integral along the closed path∮(x2 + y2)dx+ 2xydy
the path is the circle centered at the origin of radius 1
1067.
Determine whether the vector field is conservative
F(x, y) = ex2y(y + 2x2y)i + ex
2y(x+ xy)j
1068.
Determine whether the vector field is conservative; if so find the potential
function.
F(x, y) = (6xy3 + 8y)i + (9x2y2 + 8x)j
1069.
Determine whether the vector field is conservative; if so find the potential
function.
F(x, y) =
(1
x+ y+ 1
)i +
(1
x+ y+ 3y2
)j
1070.
Evaluate∫CF · dr. Hint: see if F is conservative.
F(x, y) = (12x3y3 + 12xyex2
)i + (9x2y2 + 6ex2
)j
The path C is the semicircle centered at the origin of radius 1 connecting
the points (1, 0) to (−1, 0)
1071.
Evaluate∫CF · dr. Hint: see if F is conservative.
Page 137
F(x, y) = (ln(y) + y3)i +
(x
y+ 3xy2
)j
The path C consists of the parabola y = x2 connecting (0, 0) to (2, 4) and
then the line connecting (2, 4) to (5, 5).
1072.
Evaluate the following line integral along the closed path by using Green’s
Theorem ∮(2x2y + y3)dx+ (xy3 − x)dy
the path is the the triangle with vertices at (0, 0) (2, 0) and (2, 2)
1073.
Evaluate the following line integral along the closed path by using Green’s
Theorem ∮(2x2y + y3)dx+ (xy3 − x)dy
the path is the the triangle with vertices at (0, 0) (2, 0) and (2, 2)
1074.
Evaluate the following line integral along the closed path by using Green’s
Theorem ∮(ye−x)dx+
(1
2x2 − e−x
)dy
the path is the circle of radius 1 centered at (2, 0)
1075.
Evaluate the following line integral along the closed path by using Green’s
Theorem ∮(2xy sin(x2) + ey)dx+ (cos(x2))dy
the path is the square with corners(0, 0) (1, 0) (1, 1)and (0, 1)
Page 138
1076.
Evaluate∮F · dr along the closed path by using Green’s Theorem
F =< xy − 2x2y3, 3xy >
the path is the square with corners(0, 0) (1, 0) (1, 1)and (0, 1)
1077.
Use Green’s Theorem to find the area bounded by the hypocycloid:
x23 + y
23 = r
23
1078.
Find the Area of the helicoid defined by:
x = r cos(θ) y = r sin(θ) z = θ 0 ≤ θ ≤ 2π 0 ≤ r ≤ 1
1079.
Find the Area of the Torus defined by:
x = (R+cosφ) cos(θ) y = (R+cosφ) sin(θ) z = sinφ 0 ≤ θ ≤ 2π ≤ φ ≤ 2π R > 1 is constant
1080.
Find the Area of the portion of the unit sphere that is cut out by the cone
z =√x2 + y2
1081.
Let f(x, y, z) =√x2 + y2 + 1. Evaluate
∫Sf(x, y, z)dS where S is the heli-
coid:
x = r cos(θ) y = r sin(θ) z = θ 0 ≤ θ ≤ 2π 0 ≤ r ≤ 1
1082.
Let f(x, y, z) = z2. Evaluate∫Sf(x, y, z)dS where S is the unit sphere
1083.
Use the divergence theorem to calculate∫S
∫F ·NdS
F (x, y, z) = x2i + y2j + z2k
S: x = 0, y = 0, z = 0, x = 1, y = 1, z = 1
Page 139
1084.
Use the divergence theorem to calculate∫S
∫F ·NdS
F (x, y, z) = x2i− 2xyj + xyz2k
S: z =√
4− x2 − y2, z = 0
1085.
Use the divergence theorem to calculate∫S
∫F ·NdS
F (x, y, z) = xi + y2j + zk
S: x2 + y2 = 25, z = 0, z = 5
1086.
Use the divergence theorem to calculate∫S
∫F ·NdS
F (x, y, z) = xyi + zj + (x+ y)k
S: z = 5− x, z = 0,y = 0, y = 5,
1087.
Use Stokes’ theorem to calculate the line integral∫C
−y3dx+ x3dy − z3dz
C is the intersection of the cylinder x2 + y2 = 1 and the plane x+ y+ z = 1