MATH 222 Final: MATH 222 UW Madison Final 2017 Solutions
MATH 222 (Lectures 2,3, and 4) Fall 2017
Final Exam Solutions
Circle your TA’s name from the following list.
Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park
James Hanson Julia Lindberg Mao Li Polly Yu Qiao He
Tejas Bhojraj Weitong Wang Yu Sun Zihao Zheng
Problem 1 Problem 2 Problem 3 Problem 4 Problem 5
Score
Problem 6 Problem 7 Problem 8 Problem 9 Problem 10
Score
Instructions
•On Problems 1 and 2 only the answer will be graded. On all other problems, you must
show your work and we will grade the work and your justification.
•All problems are worth approximately 10 points.
•No calculators, books, or notes (except for those notes on your 3 inch by 5 inch notecard.)
•Please simplify any formula involving a trigonometric function and an inverse trigonometric
function. For example, please write cos(arcsin x) = √1−x2.
•Final answers should not involve functions applied to either infinity or applied to a point
outside of their domain. For instance, arctan(∞) and ln(0) and √−5 will not be accepted
as a final answer. In a question involving a limit, we will accept a final answer of “∞” as
synonymous with “The limit does not exist”.
•Please simplify any binomial expression b
k.
•The formulas page is the LAST PAGE of this exam.
1. For each statement below, CIRCLE true or false. You do not need to show your work.
(a) (b) (c) (d) (e)
True False True False True False True False True False
(a) The series P∞
k=0 k!
(2k)! converges.
(b) The series P∞
k=0 3+e−k
√kconverges.
(c) The integral R∞
3
√x+x2
x3+x5dx converges.
(d) If sin θ=x+2
25 then sec θ=25
√25−(x+2)2.
(e) (1 + x3)1/2−1 is o(x2).
Solution:
(a) True (ratio test).
(b) False.
(c) True (limit comparison to R∞
3
x2
x5dx).
(d) False (should be a 252under the square root sign).
(e) True.
2. On this page only the answer will be graded.
(a) Compute R1
(x+1)(2x+1) dx
Solution: We write 1
(x+1)(2x+1) =1
x+1 +B
2x+1 =A(2x+1)+B(x+1)
(x+1)(2x+1) . We thus have
1 = A(2x+ 1) + B(x+ 1) = (2A+B)x+ (A+B).
This yields B= 2 and A=−1. So we compute
Z1
(x+ 1)(2x+ 1)dx =Z−1
x+ 1 +2
2x+ 1dx
=−ln |x+ 1|+ ln |2x+ 1|+C
So the answer is −ln |x+ 1|+ ln |2x+ 1|+C.
(b) Compute T4(1
1−x3·ex4)
Solution Since T∞1
1−z=P∞
k=0 zkand since 03= 0, we can use the substitution
method to get T∞1
1−x3= 1 + x3+o(x4). Similarly since T∞ez=P∞
k=0 zk
k!and 04= 0,
we can use the substitution method to get ex4= 1 + x4+o(x4). We then have that
T∞(1
1−x3·ex4) = (1 + x3+o(x4))(1 + x4+o(x4)) + 1 + x3+x4+o(x4).
The answer is thus T4(1
1−x3·ex4) = 1 + x3+x4.
(c) Compute
1
1
0
×
1
−2
3
.
Solution:
3
−3
−3
.
Document Summary
Math 222 (lectures 2,3, and 4) fall 2017. Circle your ta"s name from the following list. Problem 1 problem 2 problem 3 problem 4 problem 5. Problem 6 problem 7 problem 8 problem 9 problem 10. Instructions: on problems 1 and 2 only the answer will be graded. For example, please write cos(arcsin x) = 1 x2: final answers should not involve functions applied to either in nity or applied to a point outside of their domain. For instance, arctan( ) and ln(0) and 5 will not be accepted as a nal answer. You do not need to show your work. (a) (b) (c) (d) (e) K converges. (a) the series p k=0 (b) the series p k=0 (c) the integral r 3 (d) if sin = x+2. 25 then sec = (e) (1 + x3)1/2 1 is o(x2). Solution: we write (x+1)(2x+1) = 1 x+1 + b.
