some basic substitution problems
1.
1
e4x+3 dx = e4x+3 + C
4
2. Dierentiate your answer to #1.
1
d
d 1 4x+3
e
+ C = e4x+3
[4x + 3] + 0 =
dx 4
4
dx
1 4x+3
4 = e4x+3
e
4
3.
3x2 cos x3 dx = sin x3 + C
4. Dierentiate your answer to #3.
d
d
sin x3 + C = (cos
. . . last tests . . .
1. Use the ratio test to determine whether or not the series
converges. If the test is inconclusive, then say so.
(a)
3k
k2
k=1
3k+1
(k+1)2
lim
3k
k+
k2
3k 3
k2
3k 2
k = lim 2
=
k+ k 2 + 2k + 1 3
k+ k + 2k + 1
= lim
6k
6
= lim
=3>1
2/22/13
Exam 2 Review, Chapter 11, Section 1 - 7
1. True or False?
(a) A sequence can be both increasing and decreasing. TRUE
(b) All monotonic sequences converge. FALSE - example?
(c) A monotonic sequence can oscillate. FALSE
(d) Given any real number, t
2/21/13
. . . more tests . . .
1. Use the comparison test to determine whether or not the series
converges.
(a)
1
3k + 2
k=1
Note the following:
1
1
< k for all k 0 =
3k + 2
3
k=1
Therefore
k=1
1
3k + 2
k=1
1
3k
1
1
is a convergent geometric series, with
04/23/14
Final Exam Preview - Chapters 8, 9 & 10
1. Solve the initial value problem below. Note that the dierential equation is a separable equation.
2x + sec2 x
dy
=
, y(0) = 5
dx
2y
2. Solve the initial value problem below. Note that the dierential equa
some basic substitution problems
1.
e4x+3 dx =
2. Dierentiate your answer to #1.
3.
3x2 cos x3 dx =
4. Dierentiate your answer to #3.
5.
esin x cos x dx =
6. Dierentiate your answer to #5.
7.
2
ln x dx
x
8. Dierentiate your answer to #7.
9.
2x
dx
+1
x2
10
Exam Review, Chapter 11 Sections 8 - 11
1.
1
1
ex = 1 + x + x2 + x3 + =
2
6
1
1 5
sin x = x x3 +
x =
6
120
+
n=0
1
1
cos x = 1 x2 + x4 =
2
24
+
n=0
(1)n 2n+1
x
for < x < +
(2n + 1)!
+
n=0
1
1
tan1 x = x x3 + x5 =
3
5
1 n
x for < x < +
n!
(1)n 2n
x for < x
some standard trig problems
1.
cos2 x dx
2.
sin3 x dx
3.
cos3 x sin2 x dx
4.
cot x dx
5.
csc x dx
6.
sin2 x cos2 x dx
7.
sec x tan x dx
8.
tan2 x dx
9.
sec2 x tan2 x dx
10.
sec3 x tan5 x dx
some standard trig problems
1.
cos2 x dx =
1
(1 + cos 2x) dx
2
1
1
= x + sin 2x + C
2
4
2. Use u = cos x
sin3 x dx =
=
sin2 x sin x dx
1 cos2 x sin x dx = cos x +
1
cos3 x + C
3
3. Use u = sin x
cos3 x sin2 x dx =
=
cos2 x sin2 x cos x dx
1 sin2 x sin2 x