Prob 1:
A number of people had problems with algebra or evaluating cos 0 or
cos (pi). Everyone already saw this problem on midterm I.
Prob 2:
Several people did not see that they could have used the substitution
u = x2 to get an integral which leads to the arc tan u
rule. Everyone already saw this problem on midterm II. A few people tried
(contrary to basic arithmetic rules) to split this into two fractions.
Prob 3:
This needed a repeated use of integration by parts. A couple of people
did not realize the difference between the derivative of ln x and the
integral of ln x. Everyone already saw this problem on midterm III.
Prob 4:
Some had difficulty correctly multiplying the two
e expressions defining cosh 2x and sinh x. Some people
had difficulty correctly integrating e3x or
e-3x.
Prob 5:
A few people found the volume of rotation rather than merely the
area between the two curves.
A couple of people forgot
that e0 is 1 rather than 0.
Prob 6:
This topic (solving a differential equation)
was covered at the end of last quarter, and at the beginning
of this quarter. There was a very similar problem on the sample final
from last year.
You must separate variables and then integrate. Then use the
extra conditions to evaluate the "+C" and then put the equations
together. Several people tried to use the extra conditions before
integrating.
Note that the integral of 1/y is ln |y| and not merely
ln y. Without the absolute value signs, one gets only half of
the complete curve that is the solution to the differential equation.
Prob 7:
Some people did not realize that the point of the problem was to perform
the integration (i.e., some people broke up the expression
into separate fractions but then failed to integrate)!
Prob 8:
There were two major ways to do the problem -- one via a
u = x2-4 substitution
and the other via a secant substitution. The u substitution was
shorter. A third way was to use integration by parts.
Prob 9:
No major problems. Some people forgot that the derivative of
sine is NEGATIVE cosine. Some people, surprisingly< forgot that the derivative
of ln u is 1/u times the derivative of u.
Prob 10:
Some did not recognize this as an improper integral.
Prob 11:
In each case the method chosen should be the "EASIEST" (or "BEST" or
"SIMPLEST") of the various methods. Sometimes people indicated a
complicated "parts" method, when a simpler substitution method
would have also worked.
The denominator of a) can be factored as (x)(x+1)(x-1).
NOTE: In a), 1/(x3-x2) CANNOT
be separated into two fractions 1/x3-1/x2.
If you do not believe this, note that 1/(2-1) equals 1/1 = 1, but
1/2-1/1 = -1/2!
Part b) needs algebraic substitution first (the contents of the
radical is a perfect square -- so "completing the square" is not
a real option).
Also, in b), some people algebraically simplified
the expression without indicating that this was "method 8".
Part e) should have the substitution
u = 1-x2 and NOT u = x2.
Part f) cannot be done by partial fractions since the denominator
is in a square root!
NOTE 1: Some people confused when to use which substitution, i.e.,
some indicated secant substitution instead of sine for d) and f).
NOTE 2: Some people suggested that f) could be done by partial
fractions. But partial fractions can NEVER be used if the
denominator has a square root in it!
final nfinal
240 78
231 73
195 55
192 53
186 50
186 50
177 45
167 40
151 32
135 24
MAXIMUM 250 100
x
x x x x
x x x x x
130- 150- 170- 190- 210- 230-
149 169 189 209 229 250
(1) (2) (3) (2) (0) (2)
This page is maintained by Dennis C. Smolarski, S.J.
dsmolarski "at" scu.edu<
© Copyright 2011 Dennis C. Smolarski, SJ, All rights reserved.
Last changed: 17 March 2011.