Final exam
Spring 2005
MATH 1220-01
Instructor: Oana Veliche
Time: 2 hours
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) There are 20 problems, each worth 10 points.
(3) Justify all you answers. Correct answers with no justification will not be given any credit.
(4) Write your answers in the boxes provided at the end of each problem.
(5) No books, notes or calculators may be used.
1
2
3
4
5
6
7
8
9
10
11
12
Pts.
Maximum points:
Total:
1
200
13
14
15
16
17
18
19
20
2
Problem 1. Compute:
Problem 2. Compute
Z
Z
2
xex dx.
sinh(ln x)
dx. Simplify your answer!
x
3
Problem 3. Using a trigonometric substitution compute
Z
x
√
dx.
8 + 2x − x2
4
Problem 4. Compute
Problem 5. Compute
Z
Z
π
2
sin2 x cos3 x dx.
0
2
1
x2
1
dx.
+x
5
Problem 6. Compute lim x2x .
x→∞
Problem 7. Compute
Z
2
x ln x dx.
1
6
Problem 8. Compute the improper integral by breaking it up into two integrals
Z 2
1
dx.
2/3
0 (x − 1)
7
Problem 9. Determine whether the sequences converge or diverge. When they converge, find the limit.
(a) an = e−n cos n.
(b) an = √
2n
.
n2 + 1
8
Problem 10. Applying the integral test decide if the series
∞
X
tan−1 n
k=1
is divergent or convergent.
n2 + 1
9
Problem 11. Compute
∞
X
2k + (−3)k−1
k=1
5k
, in case it is convergent.
Problem 12. Determine whether the series
∞
X
(−1)n−1
n=1
n!
diverges, converges conditionally, or converges absolutely.
10
Problem 13. Find the power series representation for f (x) =
Z
x
ln(1 + t) dt.
0
Problem 14. Find the convergence set of the following power series:
∞
X
(−1)n (x − 1)n
√
.
n
n=1
11
π 3
π
Problem 15. Find the coefficient of x −
in the Taylor series of f (x) = cos x based at a = .
3
3
Problem 16. Find the Maclaurin polynomial of order 5 of the function
2
f (x) = xe−x .
12
Problem 17. Write the integral that gives the area inside the cardioid r = 1 + cos θ. Do not compute the
integral!
13
Problem 18. Solve the differential equation: (D 2 + 2D + 1)(D 2 − D)y = 0.
Problem 19. Find a particular solution yp , of the following differential equation:
y ′′ + y ′ − 2y = 4x2 .
14
Problem 20. A 12 pound weight stretches a spring 3 inches. The weight is raised 2 inches and given an
initial velocity of 2 feet per second upward. Find the equation of the motion (neglect friction).
15
Useful formulas
(1) sin2 x =
(2)
1 − cos 2x
,
2
cos2 x =
1 + cos 2x
.
2
1
[sin(m + n)x + sin(m − n)x]
2
1
sin mx sin nx = − [cos(m + n)x − cos(m − n)x]
2
1
cos mx cos nx = [cos(m + n)x + cos(m − n)x]
2
sin mx cos nx =
(3) 1 + tan2 x = sec2 x.
Z
u
du
√
+ C.
(4)
= sin−1
a
a2 − u2
Z
du
a
1
1
−1 |u|
−1
√
(5)
+ C = cos
+ C.
= sec
2
2
a
a
a
|u|
Z u u −a
(6)
csc2 du = − cot u + C.
Z
(7)
csc u cot u du = − csc u + C.
Z
(8)
cot u du = ln | sin u| + C.
Z
(9)
sec x dx = ln | sec x + tan x| + C.
x2 x3 x4 x5
+
−
+
− · · · , for −1 < x ≤ 1.
2
3
4
5
x3 x5 x7 x9
+
−
+
+ · · · , for −1 ≤ x ≤ 1.
tan−1 x = x −
3
5
7
9
x2 x3 x4 x5
ex = 1 + x +
+
+
+
+ ···.
2!
3!
4!
5!
x3 x5 x7 x9
sin x = x −
+
−
+
− ···.
3!
5!
7!
9!
x2 x4 x6 x8
+
−
+
− ···.
cos x = 1 −
2! 4! 6! 8! p
p
p
p
p
p
2
3
4
(1 + x) = 1 +
x+
x +
x +
x +
x5 + · · · , for −1 < x < 1.
1
2
3
4
5
(10) ln(1 + x) = x −
(11)
(12)
(13)
(14)
(15)