Math 1220-004
Homework #13
Solutions
1. For the following power series, nd the interval of convergence.
(a)
n=0
nxn
3n+1
Absolute Ratio Test: We compute
= lim
n
(n + 1)xn+1 3n+1
n + 1 xn+1 3n+1
x
1
= lim
n n+2 = 1 x
=
n+2
n
n
3
nx
n
x
3
3
3
x
If < 1, t
Math 1220-004
Homework #2
Solutions
1. Let f (x) = xex+1 .
(a) Compute the rst derivative f (x) to determine where f (x) is increasing and where f (x) is
decreasing. If there are maximums or minimums, nd the points (x, y ) where they occur.
Using the prod
Math 1220-004
Homework #3
Solutions
1. The following problem deals with logistic growth. Let y be the population of a bacterial colony
counted in thousands. Suppose that the initial population is 5,000; that is, y = 5 when t = 0.
Assume that y can be mode
Math 1220-004
Homework #4
Solutions
1. A tank initially contains 50 gallons of water and 50 pounds of salt. Salt water at a concentration
of 2 pounds per gallon enters the tank at a rate of 1 gallon per minute. The mixture is stirred
continuously and the
Math 1220-004
Homework #5
Solutions
Compute all of the following integrals, each of which can be done with a simple substitution.
1. (a)
(b)
(x + 5)6 dx =
1
(x + 5)7 + C
7
(e)
1
dx = tan1 (x 3) + C
(x 3)2 + 1
2
(x + 5)3/2 + C
3
(f)
3x+1 dx =
x + 5 dx =
1
Math 1220-004
Homework #6
Solutions
1. (a) For any constants and , use integration by parts to derive the formula
x ex dx =
x ex
x1 ex .
Well start by performing integration by parts on the left-hand side of the equation
x ex dx. Well let u = x and dv =
Math 1220-004
Homework #7
Solutions
1. Use a trigonometric substitution to compute the following integral.
x2 1
dx
x4
Well use the substitution x = sec . Then
x2 1 = tan
and
The integral transforms as follows.
x2 1
dx =
x4
=
=
=
dx = sec tan d.
tan
sec
Math 1220-004
Homework #8
Solutions
1. Compute the following limits.
ln(x2 )
x 1 x 2 + 3 x 4
(a) lim
This limit is of type
0
, so we can use lHpitals rule.
o
0
2/x
2
ln(x2 )
L
= lim
=
2 + 3x 4
x1 2x + 3
x1 x
5
lim
ex ex
x 0
sin x
(b) lim
This limit is of
Math 1220-004
Homework #9
Solutions
1. For the following sequences, write out the rst 5 terms of the sequence, determine whether the
sequence converges or diverges, and, in case of convergence, compute the limit.
3n
2
The rst 5 terms are
(a) an =
3
a1 = ,
Math 1220-004
Homework #10
Solutions
1. Determine whether the following innite series converge or diverge. If they converge, compute
the sum.
(a)
5
n=1
1
3
n
3
1
5
n+1
The individual pieces are each a geometric series. They both converge since the
rs are
Math 1220-004
Homework #11
due Tuesday, November 13th
1. Determine whether the following innite series converge or diverge. You may use any test/method
you choose, but you must justify your conclusion.
2
nen
(a)
n=1
2
The two best tests for this series ar
Math 1220-004
Homework #12
due Tuesday, November 20th
1. Determine whether the following alternating series converge absolutely, converge conditionally,
or diverge. Justify your conclusion.
(a)
2
(1)n
nn
n=1
The series of positive terms is
2/n3/2 which i
Math 1220-004
Homework #1
Solutions
1. Let b(x) = ln (x2 5)2 . Find the equation of the line tangent to b(x) at the point (2, 0).
The point-slope form of the equation of a line through the point (2, 0) with slope m
is y = m(x 2). The slope m is the deriva