00spex2 - A {\fm‘m- 2’2 1 ~* Bfluw‘m‘r « H 2~ r...

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Unformatted text preview: A {\fm‘m- 2’2 1 ~* Bfluw‘m‘r « H 2~ r ‘> (7P4 N a» vzooo Math 222 Prof. Bauman Second Midterm April 26, 2000 YOUR N anew YOUR Instructor’s Name: PROBLEM V 14 _ VII 12 TOTAL 100 - READ AND ANSWER QUESTIONS VERY CAREFULLY. o JUSTIFY YOUR ANSWERS: Answers without supporting work will not re— ceive full credit. 0 SIMPLIFICATION: It is not necessary to simplify your answers. 0 GOOD LUCK! Have a GREAT EXAM! JUSTIFY YOUR ANSWER!!! —ANSWERS WITH LITTLE OR NO SUPPORTING WORK— ' WILL RECEIVE LITTLE OR NO CREDIT I. (18 points) Determine the absolute convergence or conditional convergence or divergence of the following. Show all work and clearly indicate the various tests, integrals, comparison series or theorems used. 00 2 (a) <6 pts) 2 g;— 12:1 JUSTIFY YOUR ANSWER!!! —ANSWERS WITH LITTLE OR NO SUPPORTING WORK— WILL RECEIVE LITTLE OR NO CREDIT II. (14 points) 00 (a) If possible, find the sum 2 6—”. 11:0 1 (b) Show that the sequence an : ( min) n. (Note a sequence is “monotone” always decreasing.) 2 > is monotone for sufliciently large if it is either always increasing or JUSTIFY YOUR ANSWER!!! —ANSWERS WITH LITTLE OR NO SUPPORTING WORK— WILL RECEIVE LITTLE OR NO CREDIT III. (15 points) 00 . 1 , °° 1 (a) The mtegral test shows that E —3— converges smce / —§d$ converges. n_1 n 1 :1: Which of these two is larger? .Iustify your answer. A picture may help. 00 (b) Consider Z an and let 3n : a1+. . .+an. Suppose for an n, 5” : 1+£nrgn). 71:1 00 Show that 2 an converges. n=1 (c) In part (b) what is an? JUSTIFY YOUR ANSWER!!! —ANSWERS WITH LITTLE OR NO SUPPORTING WORK— WILL RECEIVE LITTLE OR N O CREDIT ( ) fl . (a) For which 23 does the series converge absolutely? I oo _ 2 n IV. (15 points) Consider the power series 2 x 71:1 (b) For which (I) does the series converge conditionally? (c) The above series does converge when :r = 1. Is the sum of the series positive or is it negative? Justify your answer. ( Think alternating series.) JUSTIFY YOUR ANSWER!!! —ANSWERS WITH LITTLE OR NO SUPPORTING WORK— WILL RECEIVE LITTLE OR NO CREDIT 1 V. (14 points) The polar graph of the function 7‘ = § + (303(6) (sort of) looks like (c) Write down a definite integral, including limits7 which represents the shaded area. You need not evaluate your integral. w .Mthfihwhw‘..HLHV‘wlilmm.m>:17".1 H, V. ‘ ,, Mum“,“my”, Emu, , N JUSTIFY YOUR ANSWER!!! —ANSWERS WITH LITTLE OR NO SUPPORTING WORK—- WILL RECEIVE LITTLE OR NO CREDIT VI. (12 points) Consider the conic 93:2 — 1823 —16y2 = 35. Graph the above conic. Find and label the center foci and vertices. JUSTIFY YOUR ANSWER!!! —ANSWERS WITH LITTLE OR NO SUPPORTING WORK~ WILL RECEIVE LITTLE OR NO CREDIT VII. ( 12 points) Consider the parametric curve with equations m=t2—t y=t2+t (a) Find the coordinates of the points on the curve where the tangent lines are i) horizontal and ii) vertical. (b) Find the equation of the tangent to the curve at t = 1. ...
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00spex2 - A {\fm‘m- 2’2 1 ~* Bfluw‘m‘r « H 2~ r...

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