MATH 142 Final: MATH142 South Carolina 18Final Exam3Solution
Fall 2018 Exam 3
MARK BOX
problem points
0 10
1 10
2 5
3 10
4–16 65=13x5
%100
HAND IN PART
NAME: Solutions
PIN: 17
INSTRUCTIONS
•This exam comes in two parts.
(1) HAND IN PART. Hand in only this part.
(2) STATEMENT OF MULTIPLE CHOICE PROBLEMS. Do not hand in this part.
You can take this part home to learn from and to check your answers once the solutions are posted.
•On Problem 0, fill in the blanks. As you know, if you do not make at least half of the points on
Problem 0, then your score for the entire exam will be whatever you made on Problem 0.
•During this exam, do not leave your seat unless you have permission. If you have a question, raise
your hand. When you finish: turn your exam over, put your pencil down and ✿✿✿✿
raise
✿✿✿✿✿
your
✿✿✿✿✿✿
hand.
•During the exam, the use of unauthorized materials is prohibited. Unauthorized materials include:
books, personal notes, electronic devices, and any device with which you can connect to the
internet. Unauthorized materials (including ✿✿✿
cell✿✿✿✿✿✿✿✿
phones, ✿✿
as
✿✿✿✿✿
well✿✿✿
as✿✿✿✿✿
your✿✿✿✿✿✿✿
watch) must be in a
secured (e.g. zipped up, snapped closed) bag placed completely under your desk or, if you did not
bring such a bag, given to Prof. Girardi to hold during the exam (it will be returned when you
leave the exam). This means no electronic devices (such as cell phones) allowed in your pockets.
•Cheating is grounds for a F in the course.
•At a student’s request, I will project my watch upon the projector screen.
Upon request, you will be given as much (blank) scratch paper as you need.
•The mark box above indicates the problems along with their points.
Check that your copy of the exam has all of the problems.
•This exam covers (from Calculus by Thomas, 13th ed., ET): §10.7–10.10, 11.1, 11.2 .
Honor Code Statement
I understand that it is the responsibility of every member of the Carolina community to uphold and maintain the University of South
Carolina’s Honor Code.
As a Carolinian, I certify that I have neither given nor received unauthorized aid on this exam.
I understand that if it is determined that I used any unauthorized assistance or otherwise violated the University’s Honor Code then
I will receive a failing grade for this course and be referred to the academic Dean and the Office of Academic Integrity for additional
disciplinary actions.
Furthermore, I have not only read but will also follow the instructions on the exam.
Signature :
Prof. Girardi Page 1 of 11 Math 142
Fall 2018 Exam 3
0. Fill-in the boxes.
0A. Power Series. Consider a (formal) power series
h(x) =
∞
X
n=0
an(x−x0)n,(0A)
with radius of convergence R∈[0,∞]. (Here x0∈Ris fixed and {an}∞
n=0 is a fixed sequence of real numbers.)
Without any other further information of {an}∞
n=0, answer the following questions.
The choices for the next 4 boxes are: AC,CC,DIVG, or anything.
AC stands for: is always absolutely convergent.
CC stands for: is always conditionally convergent.
DIVG stands for: is always divergent.
anything stands for: can do anything i.e., there are examples showing that it can AC, CC, or DIVG.
0A.1. At the center x=x0, the power series in (0A) AC .
0A.2. For x∈Rsuch that |x−x0|< R, the power series in (0A) AC .
0A.3. For x∈Rsuch that |x−x0|> R, the power series in (0A) DIVG .
0A.4. If 0 < R < ∞, then for the endpoints x=x0±R, the power series in (0A) anything .
0A.5. Furthermore, if αand βare in the interval (x0−R, x0+R), then (Hint: note the
x=β
x=α
already in there.)
Zx=β
x=α
h(x)dx =
∞
X
n=0
an
n+ 1 (x−x0)n+1
x=β
x=α
.
0B. Taylor/Maclaurin Polynomials and Series.
Let y=f(x) be a function with derivatives of all orders in an interval Icontaining x0.
0B.1. The nth Taylor coefficient of y=f(x) about x0is
cn=f(n)(x0)
n!
0B.2. The Nth-order Taylor polynomial of y=f(x) about x0, in open form (so with ... and without a P-sign), is
PN(x) = f(x0) + f(1)(x0)(x−x0) + f(2)(x0)
2! (x−x0)2+f(3)(x0)
3! (x−x0)3+···+f(N)(x0)
N!(x−x0)N
0B.3. The Taylor series of y=f(x) about x0, in closed form (so, with a P-sign and without ... ), is
P∞(x) =
∞
X
n=0
f(n)(x0)
n!(x−x0)n
Prof. Girardi Page 2 of 11 Math 142
Fall 2018 Exam 3
0C. Parametric Curves. Consider the curve Cparameterized by
x=x(t)
y=y(t)
for a≤t≤b.
0C.1. Express dy
dx in terms of derivatives with respect to t. Answer: dy
dx =
dy
dt
dx
dt
0C.2. The arc length of C, expressed as on integral with respect to t, is
Arc Length = Zt=b
t=asdx
dt 2
+dy
dt 2
dt
1. Commonly Used Taylor Series
Fill in Problem 1’s ✿✿✿✿✿✿
blank boxes with the choices a – ℓ, which are provided below.
You may use a choice more than once or not at all.
A sample question is already done for you.
a.
∞
X
n=0
xnd.
∞
X
n=0
(−1)nx2n
(2n)! g. x∈Rj. (−1,1]
b.
∞
X
n=0
xn
n!e.
∞
X
n=0
(−1)nx2n+1
(2n+ 1)! h. (−1,1) k. [−1,1)
c.
∞
X
n=1
(−1)n+1 xn
nf.
∞
X
n=0
(−1)nx2n+1
2n+ 1 i. [−1,1] ℓ.none of the others
sample. A power series expansion for y=1
1−xis aand is valid precisely when h.
1.1. A power series expansion for y= cos xis dand is valid precisely when g.
1.2. A power series expansion for y= sin xis eand is valid precisely when g.
1.3. A power series expansion for y=exis band is valid precisely when g.
1.4. A power series expansion for y= ln (1 + x) is cand is valid precisely when j.
1.5. A power series expansion for y= tan−1xis fand is valid precisely when i.
Prof. Girardi Page 3 of 11 Math 142
Document Summary
Instructions: this exam comes in two parts. (1) hand in part. Hand in only this part. (2) statement of multiple choice problems. You can take this part home to learn from and to check your answers once the solutions are posted. your hand. When you nish: turn your exam over, put your pencil down and . Problem 0, then your score for the entire exam will be whatever you made on problem 0: on problem 0, ll in the blanks. As you know, if you do not make at least half of the points on: during this exam, do not leave your seat unless you have permission. If you have a question, raise: during the exam, the use of unauthorized materials is prohibited. Unauthorized materials include: books, personal notes, electronic devices, and any device with which you can connect to the internet.
