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Calculus II Instructor: Elizabeth Maltais
MAT1322E – Test 2 – Monday, March 4, 2019
- Clearly write your name and student number on this test, and sign it below to confirm that you have read, understood and agreed to follow these instructions: / This is a 75-minute closed-book test. No notes. No calculators. The exam consists of 6 questions on 6 pages (including this cover page). ⇧ Each question is worth 2 points. . maximum points possible = 12 points. ⌥ Read all questions carefully and be sure to follow the instructions for the individual problems. ⇤ All questions are *long-answer. To receive full marks, your solution must be cor- rect, complete, and show all relevant details. *Some pages include a multiple-choice question. You will not earn any points for circling the correct response without having properly justified your choice. J You must use proper mathematical notation and terminology. Make sure that your notation is consistent with the notation used in class. N For additional work space, you may use the backs of pages. Do not use any of your own scrap paper. Cellular phones, unauthorized electronic devices or course notes are not allowed during this test. Phones and devices must be turned o↵ and put away in your bag. Do not keep them in your possession such as in your pockets. If caught with such a device or document, academic fraud allegations may be filed which may result in you obtaining zero for this test. † (^) By signing below, you acknowledge that you have read, understood, and will comply with the above instructions.
Family Name: Student Number:
First Name: †^ Signature:
Version A (^) SOLUTIONS
1. Consider the sequence {a n }^1 n=1 where a n = ln(3n 2 2) ln(n 2 + 1). page 2 of 6
Circle the most appropriate response regarding this sequence. Show your work!
A. converges to 3 B. diverges to 1 C. converges to ln(3) D. converges to ln(5)
E. converges to e 5 F. diverges to 1 G. converges to e 3 H. none of the previous answers
2. Consider the sequence {a n }^1 n=1 whose first few terms are
a 1 =
, a 2 =
, a 3 =
, a 4 =
Find an expression for a (^) n and compute the sum S of the series
X^1
n=
an. Show your work!
Circle the most appropriate response:
A. a (^) n = ( 1) n ^1 2 n 3 n^ and S =
B. a (^) n = ( 1) n^2 n ^1 3 n^ and S =
C. a (^) n = 2 n 3 n^ and S =
D. a (^) n = ( 1) n ^1 2 n ^1 3 n^ and S = 1
E. a (^) n = ( 1) n ^1 2 n ^1 3 n^ and S =
F. none of the previous answers
0
lnirnoan
= nltfmaoln( 3M^ - 2) - buff ti (^) )
= figments:^ )
= hinting shift)
= In (3)
for nzl^ , an =f2L
= HEIDI
£7an^
= II If^ }^ )^
" (^) is a geometric series^ with^ first^ term^ a^ = and (^) common ratio^ r =^ - Zz since I^ rt^ = f- If L^1
it converges to
Fr
= Iz (^) ,
= 1¥ = ÷
= Is '
Z
's
0
- Consider the series
X^1
n=
( 1) n^
2 n^ cos(n) 4 n^ n 2
. page 4 of 6
Of the following words, circle all those words (if any) which accurately describe this series.
geometric alternating absolutely conditionally convergent divergent
For each word you circled, you must briefly justify why it describes the series in the space below.
X X^ O (^) x a (^) x
an =fDkos#
4h -^ h
- (^) since Cosby 's (^) sign changes in a (^) non - alternating pattern (^) , this is not (^) an (^) alternating series.
geometric
.^ series^ either^ because^ there^ is^ not^ a^ common^ ratio^ between^ consecutive^ terms
00 Consider (^) Elan I instead^. N (^) =/
o ⇐^ EM "If^ f-
Eiht
" E
Ey
use Limit^ Comparison^ Test^ on^ this^ series
with bn^ = In
his = figs 4¥.
= thing 4 = I
441 - Ya )
¥0 as nooo
Since limit (^) exists and (^) equals a (^) positive constant (^) 6=17 (^) , We (^) Know (^) that (^) £ yf÷ behaves^ like^ £7^ In^ by^ virtue^ of the (^) Limit (^) comparison Test^.
£7 In^ is^ a^ geometric^ series^
with common ratio^ r^ =L
hence it's^ convergent.^ :^. fi yn¥na^ is^ convergent.
By the^ Comparison^ Test^ ,^ we^ see^ that^ Elam^ is^ convergent.
% Ean^ is^ absolutely convergent ( hence^ convergent)
- Consider the series
X^1
n=
n 3 page 5 of 6
We want to estimate its sum with the sum of its first n terms. How many terms to we need to add so that the error in our estimate R (^) n is guaranteed to be less than 0.01?
A. 7 B. 8 C. 15 D. 14 E. 1 F. 11 G. 10
H. none of the above
Show your work!
Let Fk (^) ) =^ ¥3.^ Then^ f^ is^ positive , continuous^ and^ decreasing for^ all^ x^ >^ o
⇒ Integral Test & Remainder Estimate^ Theorem^ apply.
Rn E^ Sno (^) # DX
Sf # dx^ = find fntxsdx
= this - tax
= figs - ate - tant
= (^) O t I 2h 2
To have (^) Rn L (^) 0.01 it suffices to find n such^ that (^) Sno (^) ¥ d (^) x L^ 0.
⇒ solve for n in the inequality
2¥ 20.
=) (^) 2¥ L (^) ¥
⇒ (^) 10oz L h
⇒ 50 Ln^ 2
⇒ FO L^ h
since 72=49 and^ 82=64^ , it^ follows^ that (^) we need n > 8
( since (^) we need n^ >^ Bo^ >^ 549=7 and^ n^ must^ be^ an^ integer)
