Review Problems for Final Exam
April 08, 2015
Review Problems for Chapter 12
1. Let the point P (1, 2) and the points Q(4, 1), compute the distance between P and Q.
1.
2. Let u = h3, −4i and v = h1, 2i, then
(a) Compute u + 2v.
(a)
(b) Find the angle between u and v?
(b)
3. Let P be the plane which passes through the point (3, 0, 1) and is parallel to the plane 3x − 2y − z = 0, then
(a) Write down the equation of the plane P .
(a)
(b) Find the value of a such that the point (a, 2, 0) is on the plane P .
(b)
(c) Find the value of b such that the plane P is orthogonal to the plane 2x − by = 1.
(c)
4. Let f (x, y) = ex and g(x, y) = y sin(xy), then
(a) Compute fx (x, y).
(a)
(b) Compute fy (x, y).
(b)
(c) Compute gy (x, y).
(c)
5. (a) Let f (x, y) = y sin(xy), compute fyx (x, y).
(a)
(b) Can you find a function h(x, y) such that hxx = y 2 cos(xy 2 ) and hxy = xy 2 cos(xy 2 )? (Do not need to
find h(x, y), just put ‘Yes’ or ‘No’)
(b)
6. Let g(x, y) = 2x2 + y 4 + 1, find a local minimum point of g(x, y).
6.
7. Let R = {(x, y) : x2 + y 2 ≤ 4} and f (x, y) = 2x2 + 2y 2 + 1, then
(a) Find the absolute maximum value of f (x, y) on R.
(a)
(b) Find the absolute minimum value of f (x, y) on R.
(b)
8. (a) Find the value of a such that the gradient of f (x, y, z) = aex + xy + sin(z) at point (0, 0, 0) is orthogonal
to a normal vector of the plane 2x − z = 1.
(a)
(b) Can you find a function h(x, y) such that ∇h(x, y) = hcos(y), sin(x)i? (Do not need to find h(x, y), just
put ‘Yes’ or ‘No’)
(b)
(c) Solve the system
x3
= xy
.
y−x=2
(c)
9. Let f (x, y) = x2 + xy 2 − 2x + 1, then
(a) Find all critical points of f (x, y).
(a)
(b) Compute the Hessian matrix of f (x, y).
(b)
(c) Compute the discriminant D(x, y) of f (x, y).
(c)
(d) Classify all critical points of f (x, y). (Make a table)
10. Sketch two level curves of the surface z =
x2
y2
+ .
9
16
11. Find and sketch the level curves of the function f (x, y) = y − x2 − 1.
12. Use the Lagrange multipliers to find the maximum and minimum values of f (x, y) = xy subject to x2 +y 2 −xy =
9.
13. Find the maximum and minimum values of the objective function f (x, y) = 2x2 + y 2 + 2, where x and y lie
on the ellipse C given by g(x, y) = x2 + 4y 2 − 4 = 0.
14. Let f (x, y) = 2x2 + 2y 2 − 6x and R := {(x, y) : x2 + y 2 ≤ 9}, then
(a) Use Lagrange multipliers to find the maximum and minimum values of f (x, y) on the boundary of R.
(b) Find the absolute maximum and minimum values of f (x, y) on R.
Review Problems for Chapter 5 and Chapter 7
Z
15. (a) Write down the midpoint Riemann sum with n = 50 to approximate
1
2
1
dx using sigma notations.
x
(a)
Z
(b) Let f and g be two integrable functions on [1, 2], if
Z 2
[2f (x) − 3g(x)] dx.
2
Z
1
2
g(x) dx = 1, compute
f (x) dx = 2 and
1
1
(b)
√
1 − x2 , if 0 ≤ x ≤ 1,
, and R be the region bounded by the graph of f (x) and x-axis between
16. Let f (x) =
2 − 2x,
if x > 1,
x = 0 and x = 3, then
(a) Find the area of R.
(a)
(b) Find the net area of R.
(b)
Z
3
(c) Compute
f (x) dx.
0
(c)
17. Let f (x) =
1, if x is a rational number
, then
0, if x is not a rational number
(a) What’s the value for the left Riemann sum for any regular partition of [0, 1]?.
(a)
(b) For any partition of [0, 1], is it true that we can always find a Riemann sum whose value is 0? (Just put
‘Yes’ or ‘No’).
(b)
(c) Is f integrable on [0, 1]? (Just put ‘Yes’ or ‘No’).
(c)
18. (a) Find an antiderivative of |x|.
(a)
(b) Compute
d
dx
Z
d
dx
Z
x
sin2 (t) dt.
1
(b)
(c) Compute
e2x
sin(t)
dt.
t
x−1
(c)
(d) Compute lim
x→0
1
x
Z
x2
cos(t2 ) dt.
x
(d)
Z r q
√
(e) Evaluate
x x x dx.
(e)
Z
(f) Evaluate
10x · 32x dx.
(f)
Z
(g) Evaluate
cos(2x)
dx.
cos(x) − sin(x)
(g)
Z
(h) Evaluate
cos(2x)
dx.
· sin2 (x)
cos2 (x)
r
Z
(i) Evaluate
(h)
!
r
1+x
1−x
dx.
+
1−x
1+x
(i)
Z
(j) Evaluate
√
dx
√ .
x+ 3x
(j)
Z
(k) Evaluate
1
dx.
ex + e−x
(k)
Z
(l) (0.5 points) Evaluate
√
dx
.
−x2 − 8x − 7
(l)
Z
(m) Evaluate
tan−1 (x) dx.
Z (n) Evaluate
(m)
1
ln(ln x) +
dx.
ln x
(n)
(o) Find the area of the region bounded by the graph of ln x and x-axis between x = e−1 and x = e.
(o)
(p) Compute lim
n→∞
1
3
3
1
+
2
+
·
·
·
+
n
.
n4
(p)
(q) Compute lim n
n→∞
1
(r) Compute lim
n→∞ n
1
1
1
+
+ ··· + 2 .
n2 + 1 n2 + 2 2
2n
(q)
2π
n−1
π
sin + sin
+ · · · + sin
π .
n
n
n
(r)
Z
(s) Evaluate
sin3 (x) cos2 (x) dx.
(s)
Z
(t) Evaluate
tan(x) sec2 (x) dx.
(t)
Z
(u) Evaluate
sec3 (x) dx.
(u)
Z
(v) Evaluate
tan2 (x) sec3 (x) dx.
(v)
Z
19. (a) Evaluate
sec2 (x) ln(tan(x) + 2) dx.
(a)
Z
(b) Evaluate
x2 e3x dx.
(b)
Z
(c) Evaluate
x sin(x) cos(x) dx.
(c)
Z
(d) Evaluate
sin3 (x) dx.
(d)
Z
(e) Evaluate
sin−1 (x) dx.
(e)
π
(f) Let R be the region bounded by y = sin(x) cos(x), the x-axis, the line x = 0 and the line x = . Find
2
the volume of the solid that is generated when the region R is revolved about the x-axis. .
(f)
Z
20. (a) Evaluate
1
dx.
(x2 + 1)2
(a)
Z
(b) Evaluate
tan3 (x) dx.
(b)
Z
(c) Evaluate
dx
3
(4 − 4x2 ) 2
.
(c)
Z
(d) Evaluate
dx
3
(4 + 4x2 ) 2
.
(d)
Z
(e) Evaluate
dx
(4x2
3
− 4) 2
.
(e)
(f) Find the partial fraction decomposition of
x2
.
x3 − 16x
(f)
Z
(g) Evaluate
x3 + 3x2 − 3x + 2
dx.
x3 − 2x2
Z (g)
3
2
sec (2x) +
dx.
x
Z 2
(h) Evaluate
(h)
(i) Evaluate
3e2x −
x
2
dx.
Z (i)
1
1
sin(4x) + cos(3x) dx.
4
3
Z (j)
2
1
2
tan(x) − csc (5x) dx.
3
5
(j) Evaluate
(k) Evaluate
(k)
Z (l) Evaluate
−√
2
5
+
2
3
+
x2
2−x
dx.
(l)
Z (m) Evaluate
sec(x)
2 sec(x) tan(x) +
2
dx.
(m)
Z
21. Let f be a differentiable function on [2, 4] such that f (2) = 1 and f (4) = 5, compute
1
21.
Z
22. Use the right Riemann sum to compute
1
(2x + 1) dx. (Hint:
0
n
X
k=
k=1
n(n + 1)
.)
2
Z
23. Use the integration by parts to get a reduction formula of the integral
Z
24. Evaluate
2x3
dx.
x3 − 4x
ex sinn (x) dx.
2
f (2x)f 0 (2x) dx.
Z
25. Evaluate the improper integral
0
∞
√
3
1
√ dx.
x(x + 3 x)
y
,
+1 .
26. Solve the initial value problem

y(0) = 1.
Z 1
tf (t) dt.
27. Find the function f (x) such that f (x) = 1 +

 y0 =
x2
x
Review Problems for Probability Theory
28. The length of time X, needed by students in a course to complete a 1 hour exam is a random variable X with
k(x2 + x), if 0 ≤ x ≤ 1,
PDE given by f (x) =
, then
0,
otherwise.
(a) Find the value k.
(b) Find the CDF.
(c) Find the probability that a randomly selected student will finish the exam in less that half an hour.
(d) Find the mean time needed to complete in an 1 hour exam.
ax2 + b, if 0 ≤ x ≤ 1,
29. Let f (x) =
for some constants a and b such that f (x) is a probability density
0,
otherwise
function for some continuous random variable X.
(a) Find conditions for a and b.
(b) Compute a and b such that E(X) = 1/2.
Review Problems for Chapter 8 and Chapter 9
30. For a sequence
{an }∞
n=1
Z
1
such that an =
0
√
dx
.
n2 + x2
(a) Find the explicit formula of an .
(a)
(b) If {an }∞
n=1 has a limit, find this limit.
(b)
31. For a sequence {an }∞
k=1 such that a1 = 1 and an+1 = 4an for all n ≥ 1. Find the explicit formula of an .
31.
√
cos( n) 3 tan−1 (n)
.
32. (a) Compute lim 2 · √
+
n→∞
n3 + 3
n
(a)
n3 + 2n
23n−1
−1
(b) Compute lim tan
+ n−10 .
n→∞
n2 + 2n + 1
9
(c) Is the series
∞
X
√
k=100
(b)
k+1
convergent or divergent?
k
(c)
(d) Is the series
∞
X
4
convergent or divergent?
2
k
ln
k
k=9
(d)
33. Is the series
∞
X
k=1
√
3
k4 + 1
√
convergent or divergent?
k5 + 9
33.
34. Is the series
∞
X
k=1
(k!)2
convergent or divergent?
(2k)!
35. Evaluate the series
∞ h
X
cos
34.
π n
n=2
3
π n i
− 2 · − tan
.
6
35.
36. Is the series
∞
X
k 10 10k (k!)2
k=1
(2k)!
convergent or divergent?
36.
37. Consider the power series
∞
X
2k (x − 3)k
k=1
k+1
.
(a) Find the radius of convergence of this series.
(a)
(b) Find the interval of convergence of this series.
(b)
∞
X
38. Find the value of x such that
ekx = 1.
k=1
38.
39. Find the power series of f (x) =
4
centered at 0.
4 + x2
39.
40. Find the power series of f (x) =
1
centered at 1.
4 − 2x
40.
41. Find the function represented by the series
k
∞ 2
X
x −1
3
k=0
.
41.
42. Find the function represented by the series
∞
X
x2k
k=1
4k
.
42.
43. Consider the power series
∞
X
(−1)k 32k+1 x3k+1 .
k=0
(a) Find the radius of convergence of this series.
(a)
(b) Evaluate the sum of this series.
(b)
44. Is the series
∞
X
sin(k 2 + 1)
√
convergent or divergent?
k3 − 1
k=1
44.
45. Find the Taylor series of e2x centered at 1.
45.
46. Find the Maclaurin series of log3 (x + 1).
46.
47. Identify the functions represented by the power series
∞
X
xk
k=1
47.
k
.
48. Identify the functions represented by the power series
∞
X
k(k − 1)xk
3k
k=2
.
48.
49. Identify the functions represented by the power series
∞
X
k=2
xk
.
k(k − 1)
49.
50. Consider differential equation y 0 (x) = 2xy with initial condition y(0) = 1, find the power series solution to the
this differential equation.
50.
51. (a) Use the integral test to show that the series
∞
X
k=2
(b) Use the comparison test to show that the series
1
is convergent.
k 2 (k + 1)
∞
X
k=2
(c) Evaluate the series
∞
X
k=2
1
is convergent.
+ 1)
k 2 (k
1
.
+ 1)
k 2 (k
∞
X
1
π2
Hint:
=
and the result of the partial fraction decomposition in Problem (a).
k2
6
k=1
52. Assume that the series
∞
X
n=1
∞
X
nan − 2n + 1
n=1
ln
an
an+1
n+1
converges, where an > 0 for all n ≥ 1. Is the series − ln a1 +
convergent or divergent. If it’s convergent, evaluate the sum of the series − ln a1 +
∞
X
n=1
ln
an
.
an+1