MATH 201 Midterm, Summer 2017
American University of Beirut · Calculus and Analytical Geometry III · 2 pages · Prof. Bassam Shayya
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Math 201—Midterm (Summer 17)
B. Shayya and H. Yamani
e Please write your section number on your booklet.
e Please answer each problem on the indicated page(s)
not written on the indicated page(s) will not be graded.
¢ Unjustified answers will receive little or no credit.
of the booklet. Any part of your answer
Problem 1 (answer on page 1 of the booklet.)
(8 pts each) Which of the following sequences converge, and which diverg
e? Find the limit of each
convergent sequence.
m—2y\r 5 sinn\"/(—1)" 1yn
= bp = n(e%/" — 1) =(1- va 5)
an (Ga ) = l(e @, 1 = - +5
Problem 2 (answer on page 2 of the booklet.)
(8 pts each) Which of the followin
sum of the series.
oa Le inn)? —— 1
(i) ae (ii) ieee (iii) Dede (a
§ series converge, and which diverge? When possible, find the
Problem 3 (answer on page 3 of the book et.)
(15 pts) Find the interval of convergence of the power series )>°°., —1_(x + 4)”.
n=2 ninn
Problem 4 (answer on page 4 of the book et.)
(10 pts) State and prove the n-th term test.
Problem 6 (answer on page 5 of the booklet.)
(11 pts) Find the tangent plane and normal line of the cone z = —/x? + y? at the point (3, 4, —5).
Problem 6 (answer on page 6 of the booklet.)
(i) (8 pts) Use the fact that sinz = eeo(—1)" Soy for all x € R to find a power series expan-
sion for the function f(x) = cosz about the point z = 0. Also, find the Taylor polynomial p(x)
generated by f at the point z = 0. ;
(ii) (8 pts) Use ASET to estimate cos(0.1) with an error of magnitude less than 1078. Does your
estimate tend to be an over-estimate or an under-estimate?
(ili) (8 pts) Let p4(x) be the Taylor polynomial from part (i). Use Taylor’s theorem to prove that
cosz < pa(x) for allO<a<r7.
Problem 7 (answer on page 7 of the booklet.)
(12 pts) Match each of the following equations with the surface on the back of this page that it
defines.
1. 2? 4+2277=8 20 2742? —y=1
3. 9y? +2? = 16 4. 9x? + 4y? + 22? = 36
5. raz? y? OS a
BS apa Gh @ Sor =e
9. 27+ 4? + 42? = 10 10. z= —42?—y?
11. 224 4y? — 42? =4 D2? = yt iz?
Your answer should be written in the form number — letter.