Feb 711 
 §7.4: 2, 4, 8, 46
 §11.3: 3, 4, 6, 13, 14, 20, 24, 26
 §11.4: 2(b), 4 (determine convergence), 6, 10, 17, 23

Solutions

Feb 1418

 §11.6: 12, 14, 16, 18, 26, 30, 44, 46
 §11.7: 5, 8, 10, 22, 28, 48
 A: Find all x such that the following series converges:

Solutions

Feb 2125 
 §11.8: 5, 6, 8, 10, 14, 18, 30
 §11.5: 1, 7, 10, 12, 16, 22, 24
 A: If
has radius of convergence 5, what
is the radius of convergence of
?

Solutions

Feb 28  Mar 3 
 §11.10: 6, 16, 26, 33(a)(b), 34(a)(b)
 A: Find the unique power series
so that
if and f(0) = 1.
What is its radius of convergence?
 §11.9: 3, 4, 6 (ignore ``calculating utility'' business
in these problems)

B: Find the Maclaurin series for
.

Solutions

March 6  10 
 §11.9: 16 (ignore ``calculating utility'' business in this
problem), 18(a), 20(a), 22
 A: Find the Maclaurin series for
.

B: Use the identity
to find the first four nonzero terms in the Maclaurin series for
.

Solutions

March 1317 
 §7.6: 6, 10, 12, 36
 §7.3: 2, 8, 10, 14, 16, 20, 28, 50
 §8.2: 2, 4, 12, 15, 20, 35(a)(b), 45, 46

Solutions

March 2024 

§8.4: 4, 8, 10, 19^{*}
 §8.5: 2, 4, 8, 18
 §8.6: 2, 4, 11, 12, 14, 17
 §8.7: 4, 8, 12
*Note that problem 19 in §8.4 has been modified. The problem is
now to do the following:

a) Graph the curve y=x^{2/3} between x=0 and x=2.

b) Find the arclength of this curve between x=0 and x=2.

Solutions

March 2731 
SPRING BREAK! 
no solutions 
April 37 
 §9.2: 2, 8, 10, 12, 28, 30, 32, 41
 §9.5: 10, 12, 14, 16, 18, 20
Also, do the following problems not originally listed on the syllabus:
 §9.5: 28, 31
 A: Find the antiderivative of the function
1/(x^{2} + a)
when
1. a is less than 0
2. a is greater than 0
3. a is equal to 0
NB  Problem A appeared on this webpage in two different manifestations.
Apologies for the inconsistency. The solutions to both problems are
available to the right.

Solutions
Solutions to A (version 1)
Solutions to A (version 2)
Solutions to 28,31


April 1014

 §9.8: 1, 4, 6, 10, 14, 16, 18, 26, 49, 52(a), 55
 A: Sketch the graph of
for ,
making sure to label when the graph crosses the xaxis.
In terms of this picture, explain how the alternating series
test can be used to show that the improper integral
is convergent.
Of course, you do not need to compute the value of this integral
(the exact value is
, but the usual proofs of this
requires methods based on complex numbers)

Solutions

April 1721

 §10.1: 2, 8, 10, 12, 16, 20, 23, 26, 28, 44, 45
 Pages 910 in Supplement: 2, 4, 8, 9, 10, 14
 A: Solve the differential equation
.

Solutions

April 2428

 Pages 1718 in Supplement: 2, 6, 7, 10, 12, 22, 24
 Pages 2829 in Supplement: 1, 2, 3, 4
 Page 37 in Supplement: 1
 A: Do the following multipart problem:
1. Graph
f(x) = (x+3)^{2}(x1) (justify!).
2. Assume that P = P(t) is a function of time t
which satisfies
3. Graph the particular solutions P(t) with
P(0) = 4, 2, 2,
and label all equilibrium solutions, indicating for each whether or
not it is stable.
4. For which real numbers a is the solution P with
initial value P(0) = a an increasing function of t?
5. By using the identity
find an explicit formula for
the time t in terms of the value of P(t) (i.e., given
the value of P at some time, give a formula which computes what
time it is).
Your formula should depend upon the initial condition value P(0),
which you should assume is not equal to 3 or 1.
How does this formula relate to the sketches in the third
part of the problem?
You may not use this formula in any other part of the problem.

Solutions

May 15

 Page 44 in Supplement: 1, 2, 3, 4

A: Let the function y satisfy the differential equation
y'''=y.
Suppose that y is represented by its MacLaurin series,
with infinite radius of convergence,
.
1) Given that y(0)=1, y'(0)=0, and y''(0)=0,
what can you say
about the values of a_{0}, a_{1},
and a_{2}.
2) Assuming the same initial conditions as in part 1, find the
general form of the coefficient a_{k}. (That is, write a formula for
a_{k} in terms of k.)
3) Verify that the series with the coefficients you derived in part 2
has infinite radius of convergence.
4) Check that the function
is a solution to the
differential equation y'''=y with y(0)=1, y'(0)=0, and
y''(0)=0.
5) Use the uniqueness of the solution of the differntial equation
y'''=y with the given initial values in order to deduce the MacLaurin
series for the function
(Note that it would have been considerably more
difficult to compute the entire MacLaurin series for
this function directly.)

Solutions
