See the attached file for questioins
Thehonest
2.doc1:
d dx
3t2 + 2( )dt x
9
∫ ⎛ ⎝⎜
⎞ ⎠⎟
d
dx
3t
2
+2
()
dt
x
9
ò
æ
è
ç
ö
ø
÷
Pblem 2:
Find the antiderivative
x + 3 x
dx∫
x+3
x
dx
ò
Pblem 3:
Find the surface area when the line segment A in the figure below is rotated about the lines:
(a) y = 1
(b) x = -2
(a) The line segment follows the function f (x) = x + 1. The integral for the surface area of revolution is:
(a) The line segment follows the function f (y) = y – 1. The integral for the surface area of revolution is:
Pblem 4:
A sphere of radius 2 foot is filled with 2000 pounds of liquid. How much work is done pumping the liquid to a point 5 feet above the top of the sphere?
Pblem 5:
Find the integral
2 x2 −1
dx∫
2
x
2
-1
dx
ò
Pblem 6:
Find the integral
2x + 5 x2 − 4x +13
dx∫
2x+5
x
2
-4x+13
dx
ò
Pblem 7:
Use the definition of an improper integral to evaluate the given integral:
3x2
8− x3 dx
0
2
∫
3x
2
8-x
3
dx
0
2
ò
Pblem 8:
Find the indefinite integrals and evaluate the definite integrals. A particular change of variable is suggested.
Pblem 9:
Evaluate the integral
xln x +1( )dx∫
xlnx+1
()
dx
ò
Pblem 10:
Evaluate the integral
7x2 + 8x − 2 x2 + 2x∫ dx
7x
2
+8x-2
x
2
+2x
ò
dx
Pblem 11:
Evaluate the integral
1 x 3− x2
dx∫
1
x3-x
2
dx
ò
Pblem 12:
Show that if m and n are integers, then
sin mx( )cos nx( )dx 0
2π
∫ = 0
sinmx
()
cosnx
()
dx
0
2p
ò
=0
. (Consider m = n andm ≠ n.)
Pblem 13:
Use derivatives to determine whether the sequence below is monotonic increasing, monotonic decreasing, or neither:
1+ 1 n
⎛ ⎝⎜
⎞ ⎠⎟ 3⎧
⎨ ⎪
⎩⎪
⎫ ⎬ ⎪
⎭⎪
1+
1
n
æ
è
ç
ö
ø
÷
3
ì
í
ï
î
ï
ü
ý
ï
þ
ï
Pblem 14:
Each special washing of a pair of overalls removes 80% of the radioactive particles attached to the overalls. Represent, as a sequence of numbers, the percent of the original radioactive particles that remain after each washing.
Pblem 15:
Calculate the value of the partial sum for n = 4 and n = 5, and find a formula for sn. (The patterns may be more obvious if you do not simplify each term.)
Pblem 16:
In the proof of the Integral Test, we derived an inequality bounding the values of the partial sums
sn = ak k=1
n
∑
s
n
=a
k
k=1
n
å
between the values of two integrals:
f x( )dx 1
n+1
∫ ≤ ak k=1
n
∑ ≤ a1 + f x( )dx1 n
∫
fx
()
dx
1
n+1
ò
£a
k
k=1
n
å
£a
1
+fx
()
dx
1
n
ò
Pblem 17:
Use any of the methods learned from this MATH141 class to determine whether the given series converge or diverge. Give reasons for your answers.
sin2 1 n
⎛ ⎝⎜
⎞ ⎠⎟n=1
∞
∑
sin
2
1
n
æ
è
ç
ö
ø
÷
n=1
¥
å
Pblem 18:
Determine whether the given series Converge Absolutely, Converge Conditionally, or Diverge, and give reasons for your conclusions.
−1( )n+1 n 2 + 7
n4 +10n=1
∞
∑
-1
()
n+1
n
2
+7
n
4
+10
n=1
¥
å
Pblem 19:
Find the interval of convergence for the series below. For x in the interval of convergence, find the sum of the series as a function of x. (Hint: You know how to find the sum of a geometric series.)
xn n=0
∞
∑
x
n
n=0
¥
å
Pblem 20:
Represent the integral as a numerical series:
e −x2( ) dx
0
1
∫
e
-x
2
()
dx
0
1
ò
Use the series representation of these functions to calculate the limits.
lim x→0
x −sin x( ) x3
lim
x®0
x-sinx
()
x
3
Determine how many terms of the Taylor series for f(x) are needed to approximate f to within the specified error on the given interval. (For each function use the center c = 0.)
f x( ) = ex
fx
()
=e
x
within 0.001 on [-1, 4].
