Exam 2 Spring, 1999

Exam 2 Spring, 1999
Midterm Exam 2
Math 116
April 20, 1999
Name:
Social Security Number:
Section:
Instructor’s Name:
Before starting to work, make sure that you have a complete exam: there are 4 numbered pages in this Web
version.
The point value of each question appears after its statement. Budget your time accordingly: about five
minutes for each ten points. That will leave time to check your work or to attempt the bonus question.
The maximum score for averaging purposes is 100, although up to 110 points can be earned if the bonus
is done correctly.
For full credit, solution methods must be complete, logical and understandable, and must involve only
techniques and results developed thus far in this course or Math 115. Answers must be clearly labeled,
must give the information asked for, and must follow logically from earlier work. Be sure to read each
question carefully!
Show all work clearly in the space provided. Work done outside a question’s space can be considered only
if there are clear and explicit directions to it within the workspace. Mark out (or fully erase) any work that
you do not want graded.
The absolute deadline for handing in the exam is 55 minutes after the exam begins.
Do not write anything on this cover page below the following solid line.
1.
6.
2.
7.
3.
8.
4.
9.
5.
(Bonus)
TOTAL SCORE:
1
1. ( 5 pts.) A function f defined on the interval [−1.5, 3] has the following tabulated values:
−1.5
1.2
x
f (x)
3
Estimate
−1.5
−0.75
0.8
0
0.5
0.75
0.2
1.5
−0.2
2.25
0.3
3
0.6
f (x) d x by the trapezoidal rule.
2. (5 pts.) Suppose that F is an antiderivative of f , F(5) = 7 and lim F(x) = 5.
x→∞
∞
Compute
f (x) d x.
5
3. (10 pts., 5 pts.each) In each of the following, determine whether the sequence {an } converges, where
ln(n 2 )
a) an =
n
b) an = (−1)n
n
n+1
4. Multiple Choice: (25 pts., 5 pts.each) In each of the following, you are given two questions: Indicate
your choices by writing clear capital block letters in the space provided. For the first choice, indicate
whether the series converges absolutely, converges conditionally or diverges. For the second, indicate all
tests that you need to use in order to determine the convergence you indicated in the first part.
a)
∞
tan−1 n where tan−1 x is the arctangent function.
n=1
i.
(A) Absolutely Convergent
(B) Conditionally Convergent
(C) Divergent
Choice:
ii. (A) Geometric Series
(B) Integral Test or p-Series
(D) Basic or Limit Comparison Test
(C) Test for Divergence
(E) Alternating Series Test
(F) Ratio Test
Choice:
b)
i.
∞
1
n=2
n [ln(n)]2
(A) Absolutely Convergent
(B) Conditionally Convergent
(C) Divergent
Choice:
ii. (A) Geometric Series
(B) Integral Test or p-Series
(D) Basic or Limit Comparison Test
(C) Test for Divergence
(E) Alternating Series Test
(F) Ratio Test
Choice:
2
c)
∞
(−1)n
n=1
i.
1
n · ln n
(A) Absolutely Convergent
(B) Conditionally Convergent
(C) Divergent
Choice:
ii. (A) Geometric Series
(B) Integral Test or p-Series
(D) Basic or Limit Comparison Test
(C) Test for Divergence
(E) Alternating Series Test
(F) Ratio Test
Choice:
d)
∞
n=1
i.
n2
1
+ 2n
(A) Absolutely Convergent
(B) Conditionally Convergent
(C) Divergent
Choice:
ii. (A) Geometric Series
(B) Integral Test or p-Series
(D) Basic or Limit Comparison Test
(C) Test for Divergence
(E) Alternating Series Test
(F) Ratio Test
Choice:
e)
∞
n=1
i.
an where an =
n!
3 · 5 · 7 · · · (2n + 1)
(A) Absolutely Convergent
(B) Conditionally Convergent
(C) Divergent
Choice:
ii. (A) Geometric Series
(B) Integral Test or p-Series
(D) Basic or Limit Comparison Test
(C) Test for Divergence
(E) Alternating Series Test
(F) Ratio Test
Choice:
∞
(−1)n
5. ( 5pts.) How many terms in
must be used to approximate
n2
n=1
the sum to within .001. Show all work.
6. (15 pts., 5 pts.each) Consider the series
∞
(x − 2)n
.
n
n=1
a) Determine the radius of convergence.
b) What is the interval of convergence? Explain you answer carefully!
c) For which real numbers x is the series absolutely convergent?
3
7. Multiple Choice: (5 pts.)
If f (x) = −2x + 4x 3 − 6x 5 + · · · + (−1)n 2nx 2n−1 + · · · for x ∈ (−1, 1), then f (0) =
(A) 0
(B) 24
(D) −2
(C) 12
(E) −24
Choice:
8. The error function erf is important in statistics, engineering and science.
Its definition is
2
erf (x) = √
π
x
e−t dt
2
0
for any real number x.
(a) (10 pts.) Use the Maclaurin series for e x (Taylor series about x = 0) to find the Maclaurin series for
the function erf (x).
(b) (5 pts.) For which real numbers x does the series in (a) converge to erf (x)? How do you know that?
9. (10 pts., 5 pts.each) Consider the curve parameterized by
x = 3 cos θ
where θ ∈ [0, 2π ]
y = 2 sin θ
a) Find
dy
dx
b) Find the equation of the tangent line to the curve at θ =
π
.
6
Bonus: (10 pts.) Suppose that a and b are real numbers and that c0 , c1 , c2 , . . . , cn , . . . is a sequence of
strictly positive real numbers. Suppose further that the Ratio Test was successfully used to compute the
∞
cn2 (x − a)n , and then the endpoints were tested, resulting in
radius of convergence of the power series
n=0
the interval of convergence
(− 19 , 79 ].
find the radius of convergence of the power series
∞
n=0
4
cn3 (x − a)n .
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