Review Session for Final Exam
April 17, 2015
All Problems in Final Exam in 2014 Spring
9
1. (a) Let Q : −x + 5y − 3z = 2 and R : 3x − y − 4z = 0 be two planes. Determined if Q and R are parallel,
5
orthogonal, or identical.
(b) The volume of a right circular cone of radius x and height y is V (x, y) =
V (x, y) = π.
πx2 y
. Graph the level curve
3
(c) Let f (x, y) = sin(xy). Find
∂2f
.
∂x∂y
(d) Use sigma notation to write the midpoint Riemann sum for f (x) = x8 on [5, 15] with n = 50. Do not
evaluate the midpoint Riemann sum.
Z
(e) Evaluate
5
f (x) dx, where f (x) =
1
3, if x ≤ 3,
x, if x ≥ 3.
(f) If
f 0 (1)
= 2 and
f 0 (2)
Z
= 3, find
1
2
f 0 (x)f 00 (x) dx.
Z
(g) Evaluate
cos−1 (y) dy.
Z
(h) Evaluate
cos3 x sin4 x dx.
Z
(i) Evaluate
√
dx
.
3 − 2x − x2
Z
(j) Evaluate
x − 13
dx.
x2 − x − 6
(k) The random variable X has probability density function
(
f (x) =
0,
if x < 1,
3 −5
x 2 , if x ≥ 1.
2
Find the expected value E(X) of the random variable X.
"
n−1 #
∞ n
X
1
2
(l) Evaluate
+ −
.
3
5
n=1
(m) Let k be a constant. Find the value of k such that f (x) = 1 + k|x| is a probability density function on
−1 ≤ x ≤ 1.
Z
(n) Let h(s) be a continuous function with h(10) = 2. If f (x, y) =
xy
h(s) ds, find fx (2, 5).
1
2. (a) Determine whether the series
∞
X
k=1
√
3
k4 + 1
√
converges.
k5 + 9
(b) Find the radius of convergence of the power series
∞
X
k=0
xk
10k+1 (k
+ 1)!
.
(c) Let
∞
X
n=0
bn .
bn xn be the Maclaruin series for f (x) =
∞
X
3
1
3
1
−
, i.e.,
bn xn =
−
. Find
x + 1 2x − 1
x + 1 2x − 1
n=0
3. (a) Use the method of Lagrange multipliers to find the maximum and minimum values of (x + 1)2 + (y − 2)2
on the circle x2 + y 2 = 125. A solution that does not use the method of Lagrange multipliers will receive
no credit, even if the answer is correct.
(b) Find the point on the circle x2 + y 2 = 125 that has minimum distance from the point (−1, 2).
1
4. Let T (x, y) = y 3 + x2 − 2xy + 6x − 6y.
9
(a) Find all critical points of T (x, y) and classify each as a local maximum, local minimum or saddle point.
(b) Find the maximum and minimum values of T (x, y) on the region R =
1 2
(x, y) : y − 3 ≤ x ≤ 0 .
3
5. An endowment is an investment account in which the balance ideally remains constant and withdrawals are
made on the interest earned by the account. Such an account may be modeled by the initial value problem
B 0 (t) = aB − m for t > 0, with B(0) = B0 . The constant a reflects the annual interest rate, m is the annual
rate of withdrawal, and B0 is the initial balance in the account.
(a) Solve the initial value problem with a = 0.02 and B(0) = B0 = $30, 000. Note that your answer depends
on the constant m.
(b) If a = 0.02 and B(0) = B0 = $30, 000, what is the annual withdraw rate m that ensures a constant
balance in the account?
6. (a) Evaluate the sum of the convergent series
∞
X
1
.
k
π k!
k=1
(b) Assume that the series
∞
X
nan − 2n + 1
n=1
n+1
converges, where an > 0 for all n ≥ 1. Is the following series
− ln a1 +
∞
X
n=1
ln
an
an+1
convergent or divergent. If your answer is YES, evaluate the sum of the series − ln a1 +
∞
X
n=1
ln
an
.
an+1
Selected Problems in Sample Exam for Midterm Exam 1
7. The function f (x, y) obeys
f (x, y) + sin(f (x, y)) = 2x + 4xy,
Find fx (0, 0) and fxy (0, 0).
and f (0, 0) = 0.
Selected Problems in Midterm Exam 1
8. Let Q and R be two planes given by the equations:
Q : 2x − y = 0,
R : y + z = 0.
Let P be the plane which passes through the point (1, 0, 0) and is orthogonal to both Q and R. Find the
equation of the plane P .
Selected Problems in Sample Exam for Midterm Exam 2
9. Evaluate lim
n→∞
n
X
6(k − 1)2
k=1
n3
r
1+2
(k − 1)3
.
n3
Selected Problems in Midterm Exam 2
Z
10. Evaluate
0
1
x2
dx.
( 4 − x2 )3
√
Z
11. Evaluate
sin4 x sec2 x dx.
Selected Problems in Sample Exam for Final Exam
12. Let

1
 1
f (x) = lim  +
n→∞
3 2+ 1+

x−1 3
n
+ ··· +
 x − 1
,
3  
n
(n−1)(x−1)
1
2+ 1+
n
where x ≥ 1. Find the equation of the tangent line to the graph y = f 0 (x) at x = 2.
Z
13. Compute the following indefinite integral
sin(ln x) dx.
Selected Problems in Final Exam in 2013 Spring
14. Fill in the blanks with right, left, or midpoint; an interval; and a value of n.
3
X
k=0
f (1.5 + k) · 1 is a
Riemann sum for f on the interval [
,
] with n =
.
15. Find the interval of convergence of the following series:
∞
X
(x + 1)2k
(a)
.
k 2 9k
k=1
(b)
∞
X
k=1
ak (x − 1)k , where ak > 0 for k = 1, 2, · · · , and
∞ X
a1
ak+1
ak
= .
−
ak+1 ak+2
a2
k=1
16. Let f (0) = 1, f (2) = 3 and
f 0 (2)
Z
= 4. Calculate
0
4
√
f 00 ( x) dx.
Selected Problems in Final Exam in 2012 Spring
17. Consider the function

if x < 0,
 a,
k tan−1 (x), if 0 ≤ x ≤ 1,
F (x) =

b,
if x > 1.
Find the values of a, k and b for which F is a valid cumulative distribution function of a continuous random
variable X, and the probability density function of X.
18. Find the limit, if it exists, of the sequence {ak }, where
ak =
k! sin3 k
.
(k + 1)!
Selected Problems in Quiz 3
Z
19. Use the (right) Riemann sum to compute
1
(2x + 1) dx.
0