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Quiz 12 for Math 142: Calculus Problems, Quizzes of Mathematics

A math quiz consisting of 20 problems for a calculus 142 class. The problems cover various topics such as the fundamental theorem of calculus, volume of solids of revolution, arc length, and integration. Students are required to find exact answers and show their work.

Typology: Quizzes

Pre 2010

Uploaded on 02/12/2009

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Math 142, Quiz 12
Name: Student #:
Be neat and show your work in all problems. Circle your answers.
(1) [6.6]State the Fundamental Theorem of Calculus (Part 1).
(2) [6.8]Everyone completed this one!!
(3) [6.9]Assuming ln(a) = 3 and ln(b) = 10 evaluate Za
b
1
1
tdt
(4) [7.2]Find the volume of the solid that results when the region enclosed by the given
curves is revolved around the x-axis. y=x1/2, x = 0, x =1
7, y = 0
(5) [7.3]Use the cylindrical shell method to find the volume of the solid that results when
the region enclosed by the given curves is revolved around the y-axis. y= 2x3, y =
3x1/2, x = 0, x = 1
(6) [7.4]Find the exact arc length of the curve y= 2x3
2โˆ’1 from x= 0 to x= 1.
(7) [7.6]Find the average value of g(x) = cos(x) on [0, ฯ€].
(8) [8.2]Evaluate: Zx2cos(x)dx
(9) [8.3]Evaluate: Zcos5(x) sin3(x)dx
(10) [8.4]Evaluate: Z1
โˆš64 โˆ’x2dx
(11) [8.5]Evaluate: Zxโˆ’1
(x+ 1)(x2โˆ’xโˆ’12) dx
(12) [8.8]Evaluate: Z+โˆž
1
4x2
x3+ 1 dx
(13) [10.1]Determine whether the sequence converges, and if so find its limit. ๎˜š(โˆ’1)n(n2)
n3+ 1 ๎˜›โˆž
n=1
(14) [10.2]Show that the given sequence is eventually strictly increasing or eventually
strictly decreasing. ๎˜š(2n)!
(n+ 1)!๎˜›โˆž
n=1
(15) [10.3]Determine whether the series converges and if so find its sum.
โˆž
X
n=1 ๎˜’4
ฯ€๎˜“n
.
(16) [10.4]Determine whether the series converges or diverges.
โˆž
X
n=1
enn2.
(17) [10.5]Determine whether the series converges or diverges.
โˆž
X
n=1
(n!)2
(2n)!.
(18) [10.6]Determine whether the series converges absolutely, converges conditionally or
diverges.
โˆž
X
n=1
(โˆ’1)n(n+ 2)
n3+ 2nโˆ’2.
(19) [10.7]Find the first 4 terms of the Taylor series of sin(x) at x=ฯ€.
(20) [10.8]Find the interval of convergence of the following power series.
โˆž
X
n=1
(โˆ’1)n(xโˆ’1)n
en(n+ 1)
Donโ€™t forget to check the endpoints.
1

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Math 142, Quiz 12 Name: Student #: Be neat and show your work in all problems. Circle your answers.

(1) [6.6]State the Fundamental Theorem of Calculus (Part 1). (2) [6.8]Everyone completed this one!!

(3) [6.9]Assuming ln(a) = 3 and ln(b) = 10 evaluate

โˆซ ab

1

t

dt (4) [7.2]Find the volume of the solid that results when the region enclosed by the given curves is revolved around the x-axis. y = x^1 /^2 , x = 0, x = 17 , y = 0 (5) [7.3]Use the cylindrical shell method to find the volume of the solid that results when the region enclosed by the given curves is revolved around the y-axis. y = 2x^3 , y = 3 x^1 /^2 , x = 0, x = 1 (6) [7.4]Find the exact arc length of the curve y = 2x

(^32) โˆ’ 1 from x = 0 to x = 1. (7) [7.6]Find the average value of g(x) = cos(x) on [0, ฯ€]. (8) [8.2]Evaluate:

x^2 cos(x) dx

(9) [8.3]Evaluate:

cos^5 (x) sin^3 (x) dx

(10) [8.4]Evaluate:

64 โˆ’ x^2

dx

(11) [8.5]Evaluate:

x โˆ’ 1 (x + 1)(x^2 โˆ’ x โˆ’ 12)

dx

(12) [8.8]Evaluate:

1

4 x^2 x^3 + 1

dx

(13) [10.1]Determine whether the sequence converges, and if so find its limit.

(โˆ’1)n(n^2 ) n^3 + 1

n= (14) [10.2]Show that the given sequence is eventually strictly increasing or eventually strictly decreasing.

(2n)! (n + 1)!

n= (15) [10.3]Determine whether the series converges and if so find its sum.

โˆ‘^ โˆž

n=

ฯ€

)n .

(16) [10.4]Determine whether the series converges or diverges.

โˆ‘^ โˆž

n=

enn^2.

(17) [10.5]Determine whether the series converges or diverges.

โˆ‘^ โˆž

n=

(n!)^2 (2n)!

(18) [10.6]Determine whether the series converges absolutely, converges conditionally or

diverges.

โˆ‘^ โˆž

n=

(โˆ’1)n(n + 2) n^3 + 2n โˆ’ 2

(19) [10.7]Find the first 4 terms of the Taylor series of sin(x) at x = ฯ€.

(20) [10.8]Find the interval of convergence of the following power series.

โˆ‘^ โˆž

n=

(โˆ’1)n(x โˆ’ 1)n en(n + 1) Donโ€™t forget to check the endpoints. 1