Review Problems for Part I of the Final Exam in MA112
1. Evaluate
2. Evaluate
3. Evaluate
Z
Z
Z
x2 ln x dx
(x + 6) cosh(x2 + 12x) dx
x2
Z √π
2
4. Evaluate
10
dx
−x−6
x cos(x2 ) dx
0
5. Evaluate
6. Evaluate
7. Evaluate
8. Evaluate
9. Evaluate
10. Evaluate
11. Evaluate
Z
Z
Z
Z
Z
Z
Z
x sin(2x) dx
√
e x
√ dx
x
x2
x−2
dx
+ 5x + 4
x2
dx
1 − x3
x cosh(x) dx
4
3
1
dx
(x − 1) (x + 2)
∞
1
1
dx. Show all the steps involved.
x2 + 1
12. Solve the initial value problem x0 = 15 x2 t, x(1) = 10.
13. What is the solution to the differential equation
14. Find the general solution to
dy
= xy 2 . Put your answer in the form y = ...
dx
15. Solve the following initial value problem:
16. Evaluate
17. Evaluate
Z
Z
dx
= −3x + 6, x(0) = 4?
dt
dy
= ex (y 2 + 1), y(0) = 1
dx
xex dx.
2x3 + x − 2
dx
x+1
1
18. Determine the convergence or divergence of the improper integral
its value.
19. Find the general solution to the differential equation
dy
dx
Z
∞
2
3
dx. If convergent, find
x2
= y 2 cos(3x).
20. A flower vase is in the shape of a solid of revolution obtained by revolving a curve x = g(y),
0 ≤ y ≤ 24, about the y-axis. The value of g is measured at several different y values and tabled
below (all units in cm):
y
0 8.0 16.0 24.0
g(y) 7.0 4.0 5.0 6.0
(a) Set up an integral (involving g) that describes the volume of the vase
(b) Use the trapezoidal rule to estimate the volume of the vase. You may leave your answer in
an unsimplified form (e.g., a multiple of π.)
21. Find the general solution to the differential equation
dy
= 9t2 + sin t,
dt
22. Find the general solution to the differential equation
dy
y−3
= 2
.
dx x + 1
Express your solution explicitly, giving y as a function of x.
23. SET UP the appropriate initial value problem for which the solution is the function y(x) described
in the sentence below. DO NOT SOLVE the initial value problem!
“y(x) is the function whose graph passes through the point (0, 2) and whose derivative equals
twice the product of x and the square of y(x).”
24. A certain population grows exponentially so that the population size P (t) (in thousands) at time
t months satisfies the differential equation
dP
= kP , where k is a positive constant.
dt
Assume that the initial population size is P (0) = 2.5 (thousand), and the population size after 5
months is P (5) = 20 (thousand).
Show your steps as you use separation of variables, followed by whatever computation needed,
to show that
1
P (t) = 2.5e( 5 ln 8)t .
25. Determine the convergence or divergence of the infinite series
∞ µ ¶n
X
2
n=0
find the sum.
2
3
. If the series converges,
26. Determine the convergence or divergence of the infinite series
find the sum.
∞
X
1
. If the series converges,
n(n
+
1)
n=1
27. Determine the convergence or divergence of the infinite series
the sum.
∞
X
1
n
n=1
. If the series converges, find
∞
X
n
. If the series converges,
28. Determine the convergence or divergence of the infinite series
n+1
n=1
find the sum.
∞
X
2
.
29. Determine the convergence or divergence of the infinite series
n3
n=1
∞
X
2
√ .
30. Determine the convergence or divergence of the infinite series
n
n=1
31. Determine the convergence or divergence of the improper integral
its value.
Z
32. Determine the convergence or divergence of the improper integral
its value.
33. Determine the convergence or divergence of the improper integral
its value.
∞
0
Z
Z
e−x dx. If convergent, find
∞
1
1
0
2
dx. If convergent, find
x
2
√ dx. If convergent, find
x
34. Find the first 3 non-zero terms of the Taylor series for f (x) = cos(2x) centered about c = 0.
35. Find the first 3 non-zero terms of the Taylor series for f (x) = ln(x) centered about c = 1.
36. Use L’Hôpital’s Rule to find (a) lim x ln(x),
x→0
2
and (d) lim (1 + )x
x→∞
x
37. Evaluate the given indefinite integrals.
Z
(a) x8 ln(x)dx
Z
2−x
(b)
dx
5 − 6x + x2
3
(b) lim
x→0
µ
¶
1
1
−
,
x sin(x)
(c) lim xe−2x ,
x→∞
38. (a) Evaluate
Z2
0
x3
√
dx
1 + x4
(b) An object is moved along the x-axis from x = 0 feet to x = 1 feet by the application of a
force of f (x) = 2 + 3e−2x pounds at each point x. Calculate the work done during this motion.
39. Find the particular solution for
dy
3x2 + 1
=
, y(1) = 2.
dx
3y 2
Write the solution in explicit form as y = ...
40. Find the 3rd Taylor polynomial centered about c = 1 for the function f (x) =
Z∞
41. Evaluate the improper integral
1
.
x5
x2
dx
(1 + x3 )2
0
sin x + x3 − x
x→0
x sin x
42. Use l’Hôpital’s rule to find lim
43. (a) Find the sum of the convergent series
∞
X
5(−1)n
n=0
2n
(b) Determine, with appropriate explanation, convergence or divergence of the given infinite series
∞
X
n+2
.
3n
+
5
n=1
(c) Determine, with appropriate explanation, convergence or divergence of the given infinite series
∞
X
1
.
2
n +n+1
n=1
44.
4