Assignment 2
Complete this assignment after you have finished Unit 5, and submit your work to
your tutor for grading. Remember to attach a tutor-marked exercise form (paper or
electronic). Be certain to keep a copy of your work, in case the original is lost in the
mail. This exercise is worth 15 per cent of your final grade.
Total points: 180 (plus 12 bonus points)
1. Determine the values of r, if any, such that y = e
rx 9 pts is a solution of the
differential equations given below.
a. y
00 + 6y
0 + 9y = 0
b. y
00 - 5y
0 + 6y = 0
c. y
000 - 3y
0 + 2y = 0
12 pts 2. Use Euler’s method with the given step size (?x or ?t) to approximate the
solution of the initial-value problem over the stated interval. Present your answer
as a table and as a graph.
a.
dy
dx =
v3 y, y(0) = 1, 0 = x = 4, ?x = 0.5
b. cos y, y(0) = 1, 0 = t = 2, ?t = 0.5
6 pts 3. A tank with a 1000-gal capacity initially contains 500 gal of water that is
polluted with 50 lb of particulate matter. At time t = 0, pure water is added at a
rate of 20 gal/min and the mixed solution is drained off at a rate of 10 gal/min.
How much particulate matter is in the tank when it reaches the point of
overflowing?
9 pts 4. Polonium-210 is a radioactive element with a half-life of 140 days. Assume that
20 milligrams of the element are placed in a lead container and that y(t) is the
number of milligrams present t days later.
a. Find a formula for y(t).
b. How many milligrams will be present after 12 weeks?
c. How long will it take for 75% of the original sample to decay?
12 pts 5. Determine whether the following sequences are (eventually) decreasing,
(eventually) increasing, or neither. Explain.
a.
4 -
(-1)n
n
b.
(n!)2
(2n)!
c.
n
22
n
n!
d.
1 · 3 · XXXXXXXXXX2n - 1)
(2n)
n
Introduction to Calculus II
24 pts 6. Evaluate the limit (if it exists) of each of the following sequences. Indicate the
results (definition, theorems, etc.) you use to support your conclusion.
a. an =
n - 3
n
n
b. an =
(n!)2
(2n)!
c. an =
n
22
n
n!
d.
1
3
5
, -
1
3
6
,
1
3
7
, -
1
3
8
, . . . .
e. an =
v
n2 + 3n - n
f. an =
(-1)n
2n
3
n3 + 1
6 pts 7. A bored student enters the number 0.5 in her calculator, and then repeatedly
computes the square of the number in the display. Taking a0 = 0.5, find a
formula for the general term of the sequence {an} of the numbers that appear in
the display, and find the limit of the sequence {an}
9 pts 8. Give an example of two sequences {an} and {bn} such that
a. {an} and {bn} are divergent, but {an + bn} is convergent.
b. {an} is convergent, {bn} is divergent, and {anbn} is divergent.
c. {an} is divergent, and {|an|} is convergent.
12 pts 9. Find the sum of each of the convergent series given below.
a. X8
k=1
1
2
k
-
1
2
k+1
b. X8
k=2
1
k
2 - 1
c. X8
n=5
e
p
n-1
d. X8
i=1
5
3i
7
1-i
24 pts 10. Determine whether each of the series below is divergent, absolutely convergent
(hence convergent) or conditionally convergent. Indicate the test, result or results
you use to support your conclusion.
a. X8
k=1
v
k
v
k + 3
b. X8
k=1
(-1)k
p
k(k + 1)
Mathematics 266 /
c. X8
k=1
3k
2 - 1
k
4
d. X8
k=1
tan-1 k
k
2
e. X8
n=1
(-1)n+1 3
2n-1
k
2 + 1
f. X8
k=1
5
k + k
k! + 3
11. Use an appropriate Taylor polynomial of degree 2 to approximate tan 61o 4 pts .
8 pts 12. Give the Taylor polynomial of order n about x = x0
a. sin(px); x0 =
1
2
b. ln x; x0 = e
15 pts 13. Find the radius of convergence and the interval of convergence for each of the
series listed below.
a. X8
k=1
5
k
k
2
x
k
b. X8
n=1
(-1)nx
2n
(2n)!
c. X8
k=1
(ln k)(x - 1)k
k
8 pts 14. Find the first four terms of the Malaurin series for each of the functions given
below.
a. e
x
sin x
b. ln(1 + x)
1 - x
8 pts 15. Use Maclaurin series to find
a. limx?0
tan-1 x - x
x
3
b. Z
e
x - 1
x
dx
14 pts 16. a. Use the relationship
Z
1
v
1 + x
dx = sinh-1
x + C
to find the first four nonzero terms in the Maclaurin series for sinh-1
x.
b. Express the series in sigma notation.
c. What is the radius of convergence?