calculus problems for cutting and pasting chapter the ...€¦ · calculus problems for cutting and...

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.


limx→−2

(2− 3x) = 8

3.


limx→−2

(2x2 − x+ 5) = 15

4.


limx→3

(x2 − 6x+ 2) = −7

5.


limx→1

(7x2 + 9x− 11) = 5

6.


limx→−2

(2x2 + 9x− 1) = −11

7.


limx→−2

5 = 5

8.


limx→4

√x = 2

9.


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.


limx→1

x5 − 1

x− 1

13.


limx→3

4x4 − 12x3 + 2x2 − x− 15

x− 3

14.


limx→2

x4 − 3x3 + x2 + 2x

x− 2

15.


limx→0

e2x − 5ex + 4

ex − 1

16.


limx→π

2

sin(2x)

cos(x)

17.


limx→5

√4x+ 16− 6

x− 5

18.


limx→8

x2 − 64√x+ 1− 3

19.


limx→0

√1 + x−

√1− x

x

20.


limx→0

√1 + sin(x)−

√1 + x

x

21.


limx→0

√ex + 3− 2

ex − 1

22.


limx→5

1x+1 −

16

x− 5

23.


limx→1

(1

x− 1+

1

x2 − 3x+ 2

)24.


limx→3

√x+ 6− xx2 − 9

25.


limx→2

sin(x− 2)

x2 − 4

26.


limx→3

x− 3

sin(x2 − 9)

27.


limx→0

x sin(x)

(x+ sin(x))2

28.


limx→π

4

cos(2x)

cos(x)− sin(x)

29.


limx→4

sin(√x− 2)

x2 − 7x+ 12

30.


limx→1

sin(1− 1x )

x− 1

31.


limx→0

sin(e2x − 1)

ex − 1

32.


limx→0

sin(1− cos(x))

x

33.


limx→9

√x− 5− 2√x− 3

34.


limx→4

1√x− 1

2√x+ 5− 3

35.


limx→0

sin(sin(x))

x

36.


limx→0

1− cos(x)

x2

37.


limx→π

4

sin(x)− cos(x)

1− tan(x)

38.


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.


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.


limx→2

f(x) f(x) =

x+ 5 x ≤ 2

ax− 2 2 < x

45.


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

f(x) =

x2−1x−1 x < 1

ax2 + bx+ c 1 ≤ x ≤ 4x2+3x−28

x−4 x > 4

47.

A


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.


f(x) =x− 1

x3 − 2x2 − x+ 2

50.


f(x) =

√x+ 1

x2 − 4

51.


f(x) =ln(x)

9− x252.


f(x) =sin(x)

x2 − x53. Show that x = 0 is a vertical asymptote.

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.



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)

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.



f(x) =

2x+ 3 x ≤ 2

5x− 3 2 < x < 4

3x x ≥ 4

62.



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

66.


f(x) =

sin(x2−9)x+3 x < −3

ax+ b − 2 ≤ x ≤ 6x2−36x−6 x > 6

67.


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

Chapter The Second: Derivatives

72.

Use the definition to find f ′(x).

f(x) = 3x2 − 5x+ 4

73.


f(x) =√

6x+ 1

74.


f(x) =6x

x+ 1

75.


f(x) = sin(2x)

76.


f(x) = cos(4x)

77.


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.

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.


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.


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

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

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.


f(x) =tan(x)

x

105.


f(x) =x cot(x)

x2 + 1

106.


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.


f(x) = (x3 + x2 − sin(x))4

112.


f(x) =√

1 + x tan(x)

113.


f(x) =cos(√x)

x

114.


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.

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

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

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.

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?

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.


f(x) = x5 − 5x3 − 20x− 4

154.


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.



f(x) = x3 − 7x2 + 15x

160.



f(x) =x2 − 3x+ 5

x+ 2

161.

Show the hypothesis of Rolle’s theorem apply to the function on [0, 2π]. Find


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.


[0, 2]. Find all values guaranteed by the theorem.

f(x) = x2 + x+ 1

165.



f(x) =√

2x+ 1

166.



f(x) = x3 − 6x2 + 9x+ 2

167.



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.



f(x) = x√x+ 1

173.



f(x) =x2 − 2x+ 1

(x− 3)4

174.



f(x) = 3x2/3(x2 − 36)

175.



f(x) = 12x5/2 − 45x2 − 20x3/2 + 90x

176.



f(x) = 2 cos(x)− sin(2x)

177.


is increasing and decreasing and all relative extrema on [0, 2π].

f(x) = sin(x)− cos(x)

178.



f(x) = 2e3x − 15e2x + 24ex

179.



f(x) = x√a− x

180.

If p and q are both integers greater than or equal to 2 and

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.



f(x) = e1−x2

72

183.



f(x) = x2 − 1

6x3

184.



f(x) = cos2(x)− 2 cos(x)− x2

185.



f(x) =x3

x2 − 4

186.



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.

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.


f(x) =x arctan(x)√

4x2 + 1

199.


f(x) =x+ 1√

4x2 + 2x

200.


f(x) =7e3x + 4

2e3x + 2

201.

Find all vertical and horizontal asymptotes.

f(x) =2ex − e−x

ex − e−x202.

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.




f(x) = |x2 − 6x+ 5|

209.




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?

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.

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.

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.


0

3x2 + 3dx

243.


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.

∫ 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

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.

∫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.


1

(√x+ 1)(

√x+ 2)√

xdx

270.


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

∫ 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.


f(x) =3x3 − x2 − 2x

x− 1[2, 4]

279.


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]

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]



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

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

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.

∫ √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

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.


y = x1x (1, 1)

323.

Find dydx

y = xxx

324.


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.

∫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 .

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.

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.

∫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)

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.

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


f(x) = C x > 0

380.

Let

f(x) = arcsin

(x√

1 + x2

)+ arctan

(1

x

)Show

f ′(x) = 0


f(x) = C x > 0

381.

Let

f(x) = arcsec(√x) + arcsin(x)

Show

f ′(x) = 0


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.

∫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

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.

∫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

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.


f(x) = x3 − 6x+ 1

g(x) = 3x+ 1

424.


f(x) = 4− x2

g(x) = |x|

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.



y =√x(x2 + 1)10 x = 0 x = 1




y =1 + tan(x)

cos(x)x = 0 x =

π

4




y = tan(x) + sec(x) x = 0 x =π

4




y = tan(x) + cot(x) x =π

6x =

π

3




y =sec(x) + tan(x)

cos(x)x = 0 x =

π

4

Revolve about the x-axis. TI



y =x

1 + x2x = 0 x = 1

Revolve about the x-axis. TS



y = ln(x) x = 1 x = e




y = x2 + x x = 0 x = 3

Revolve about the line y = −2.



y = e3x x = 0 x = 1

Revolve about the line y = −3.



y = 3√x x = 1 x = 27

Revolve about the y-axis.



y = ln(x) x = e x = e2




y =√x x = 1 x = 25

Revolve about the line x = −2.



y =√x y = x2




y = x y = tan(πx

4)


444.

When the ellipse: x2

a2 + y2

b2 = 1 is rotated about the x-axis it forms a ellipsoid.

Find its volume.


the graphs of the equations about the given lines using the method of cylindrical

shells.

y = x2 − 4x y = 0 x = 0 x = 2




shells.

y =√

9− x2 y = 0 x = 0




shells.

y = ex2

y = 1 y = e9 x = 0




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)

451.


f(x) = 2√x

452.


f(x) =x4

8+

1

4x2

453.


f(x) =x5

10+

1

6x3

454.


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


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).

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.


interval 3 < x < 4.

f(x) =√

25− x2

466.


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.


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

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.

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.


2(1 + y2)xdx = (1 + x2)dy y(0) = 1

481.


1

3x2 + 1

dy

dx= cos2(y) y(1) = 0

482.


xdy =√−5 + 6y − y2dx y(1) = 5

483.


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

xe3x

y= 2(3x+ 1)2

dy

dx

487.


1 +√y = (1 + sin(x))

dy

dxy(0) = 0

488.

Solve

(y2 + 1)dx = (x34 + x

54 )dy

489.


√1− x2 = 2xy

dy

dxy(1) = 2

490.


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?

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

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.

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

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.

∫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

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

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

∫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.

∫ √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.

∫ √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.

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)

∫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.

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

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

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

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.


√1

x+

1

x3dx

633.


4√x

x+ 1dx

634.


e−x−4√xdx

635.


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

Chapter The Eighth: Sequences and Series

638.

Determine if the sequence converges or diverges.

sn = n sin

(1

n

)639.


sn = sin(n) arctan

(1

n

)640.


sn =√

4n2 + 6n− 2n

641.


sn =

(1 +

3

n

)2n

642.


sn =2n

2n + n

643.


sn =e

1n + e

2n + e

3n + ...+ e

2nn

n

644.


sn =n3√

n6 + n3 + 1

645.

Determine if the series converges or diverges. Clearly state the series test

used. Find the sum if possible.

∞∑n=1

(en + n

) 1n

646.



∞∑n=1

(1 +

1

n

)2n

647.



∞∑n=1

24

n2 + n

648.



∞∑n=3

6

n2 − 4

649.



∞∑n=1

ln

(n

n+ 1

)650.



∞∑n=1

n√n2 + 3

651.



∞∑n=1

n2 sin

(4

n2

)652.



∞∑n=1

(en + n

) 1n

653.



∞∑n=1

23n+1

32n

654.



∞∑n=1

3n + 22n+1

5n

655.

Solve for K

∞∑n=2

36

(K − 2

3

)n= 6

656.



∞∑n=1

4

n2 + 4n+ 3

657.



∞∑n=1

9

n2 + 3n

658.



∞∑n=1

ln

(n

n+ 1

)659.



∞∑n=2

ln

(1− n2

n2

)660.



∞∑n=0

(arctan(n+ 2)− arctan(n)

)661.



∞∑n=0

(arctan(n)− arctan(n+ 2)

)662.


used.

∞∑n=1

(arcsin

(1

n+ 1

)− arcsin

(1

n

))663.


used. Find the sum if convergent.

∞∑n=2

23n

32n

664.

Use a geometric series to express the repeating decimal as a fraction.

.11

665.


.123

666.


.99

667.



∞∑n=2

1

n ln(n)

668.



∞∑n=2

1

n(ln(n))2

669.



∞∑n=3

1

n ln(n) ln(ln(n))

670.



∞∑n=1

arctan

(1

n

)671.

Find the values of p > 0 that make the series converge.

∞∑n=3

1

n(ln(n))p

672.


∞∑n=3

ln(n)

np

673.


∞∑n=3

n2

(n3 + 1)p

674.



∞∑n=2

n2

n4 + n− 1

675.



∞∑n=2

n√n5 + 1

676.



∞∑n=1

1

n1+1n

677.

Show the series diverges for any value of p > 0

∞∑n=2

1

n(1+1np )

678.



∞∑n=2

2n

3n − n

679.



∞∑n=2

2n

4n − ln(n)

680.



∞∑n=2

2n

ln(n)4n − 1

681.



∞∑n=2

sin

(1

n

)682.



∞∑n=2

n sin

(1

n3

)683.



∞∑n=2

sin

(2n

3n

)684.

Show that if

∞∑n=2

an

converges then so does

∞∑n=2

sin(an)

685.



∞∑n=2

√n sin

(1

n3

)686.



∞∑n=2

sin

(n

n3 − ln(n)

)687.



∞∑n=2

n3 ln(n)

2n

688.



∞∑n=2

arctan(n) ln(n)

n2

689.



∞∑n=2

1

nln(n)

690.



∞∑n=2

n3n

n!

691.



∞∑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.



∞∑n=2

(−1)n ln(n)

n

694.



∞∑n=2

sin(

(2n+1)π2

)n

695.



∞∑n=2

(−1)n2n

3n + n

696.

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.



∞∑n=2

(−1)n1 · 3 · 5 · ... · (2n+ 1)

3nn!

700.



∞∑n=2

n23n

4n

701.



∞∑n=2

2nn!

(2n)!

702.



∞∑n=2

3nn!

1 · 3 · 5 · ... · (2n+ 1)

703.



∞∑n=2

(−1)nn!

nn

704.



∞∑n=2

(2n)!

(n!)2

705.



∞∑n=2

(n2

2n2 − 1

)n706.



∞∑n=2

(1− 1

n

)n2

707.



∞∑n=2

(e2n

e3n + n

)n708.



∞∑n=2

(arctan(n)

2

)n709.



∞∑n=2

(3n − n

2n

)n710.



∞∑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

717.


vergence.

∞∑n=1

xn

3n

718.


vergence.

∞∑n=1

xn

3nn

719.


vergence.

∞∑n=2

(x− 3)n

5nn ln(n)

720.


vergence.

∞∑n=2

xn sin

(1

n

)721.


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.

∞∑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


726.

Find a power series for f(x) = 3x2−3x centered at x = 3. Give the interval of


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


729.

Find a power series for f(x) = ln(1−x2) centered at x = 0. Give the interval


730.

Find a power series for f(x) = ln(x+1x−1

)centered at x = 0. Give the interval


731.

Find a power series for f(x) = x(1−x)2 centered at x = 0. Give the interval


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.

∫ 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

Chapter The Ninth: Parametric and Polar

742.

Eliminate the parameter and sketch the curve.

x = t+ 1 y = t2 + 2t+ 5

743.


x = et y = e2t + 2et + 6

744.


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)

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


their tangent lines

x = t3 − 3t y = 2t3 − 9t2 + 12t


their tangent lines

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.


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.


x = arctan (t) y = ln√

1 + t2 0 6 t 6 1

763.


x = arcsin (t) y = ln√

1− t2 0 6 t 61

2

764.


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.


surface created by the parametric equations about the y-axis.

x = t y =t2

4+

1

2ln(t) 1 6 t 6 2

771.


surface created by the parametric equations about the y-axis.

x =√

1− t2 y = arcsin(t) 0 6 t 6 1

772.


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.


x = 10

777.


x2 + 2y = 1

778.


4x+ 7y = 10

779.


(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.


r = θ

783.


r = 4 csc (θ)

784.


r =√

tan(θ) sec(θ)

785.


r =sin(θ)− cos(θ)

sin2(θ)

786.


r =cot(θ)− 1

cos2(θ)

787.


tangent lines

r = 1 + sin (θ)

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.


r = sin(θ) and r = cos(θ)

792.


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

r = 1 + 2 cos θ

797.


r = 3− 2 sin (θ) and r = 3 + 2 sin (θ)

798.


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 .

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.

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.

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

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.

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.

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.

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.

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.

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.

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

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.


⟨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.


⟨12 t

2, 3t, 4t

⟩, t=0 , t=5

892.


⟨23 t

3, t2, t

⟩, t=0 , t=1

893.


⟨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

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.

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]

916.


r(t) =< et,√

2t, e−t >

on [0, 1]

917.


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]

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.


lim(x,y)→(0,0)

x2 − y2

x2 + y2

922.


lim(x,y)→(0,0)

exy − 1

xy

923.


lim(x,y)→(0,0)

x3 + y3

x2 + y2

924.


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.

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

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.


z = sin(x2 + y2)− cos(x2 + y2)

940.


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:

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?

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?

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

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.

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.

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:

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)

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.


0

∫ 2

3√y

ex4

dxdy

1010.


0

∫ 2

√x

1

1 + y3dydx

1011.


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.

∫ 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

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

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.



y =4

xy = 4 y = 1 ρ(x, y) = ky

1037.



y = ex y = 0 x = 0 x = 1 x = 0 ρ(x, y) = ky

1038.



y = sinx y = 0 x = π x = 1 x = 0 ρ(x, y) = k

1039.



y = ln(x) x = 1 x = e y = 0 ρ(x, y) = k

1040.



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 )

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.

∫ ∫ ∫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

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.


(x2 + y2)dA

R is region in first quadrant bounded by the curves

y =1

xy =

3

xx2 − y2 = 1 x2 − y2 = 4

1060.


xy

1 + x2y2dA

R is region in first quadrant bounded by the curves

y =1

xy =

4

xx = 1 x = 4

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.


(x2 + y2)ds

C(t) =< sin(t), cos(t), t > t ∈ [0, 2π]

1063.


(x2 + y2)dx− 2xydy

C(t) is the parabola y = 2t2 connecting the points(0, 0) and (2, 8)

1064.


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

∮(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.

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.


Theorem ∮(2x2y + y3)dx+ (xy3 − x)dy

the path is the the triangle with vertices at (0, 0) (2, 0) and (2, 2)

1074.


Theorem ∮(ye−x)dx+

(1

2x2 − e−x

)dy

the path is the circle of radius 1 centered at (2, 0)

1075.


Theorem ∮(2xy sin(x2) + ey)dx+ (cos(x2))dy


1076.

Evaluate∮F · dr along the closed path by using Green’s Theorem

F =< xy − 2x2y3, 3xy >


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

1084.


∫F ·NdS

F (x, y, z) = x2i− 2xyj + xyz2k

S: z =√

4− x2 − y2, z = 0

1085.


∫F ·NdS

F (x, y, z) = xi + y2j + zk

S: x2 + y2 = 25, z = 0, z = 5

1086.


∫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

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