MATH 201 Midterm, Summer 2015

Math 201—Midterm (Summer 15) B. Shayya and H. Yamani Please write your section number on your booklet. Please answer each problem on the indicated page(s) of the booklet. Any part of your answer not written on the indicated page(s) will not be graded. e Unjustified answers will receive little or no credit. Problem 1 (answer on page 1 of the booklet.) (8 pts each) Which of the following sequences converge, and which diverge? Find the limit of each convergent sequence. 2000 sin(el” — 1) ; = ee aT = Vnl = on 100n + (nl) \ Yn! Problem 2 (answer on pages 2 and 3 of the booklet.) (9 pts each) Which of the following series converge, and which diverge? When possible, find the sum of the series. Na > GS + cr) ii) s Cyn eae (Hi) ¥ (1 — cos =a) n=1 n=1 roblem 3 (answer on page 4 of the booklet.) (ik pts) Find the interval of convergence of the power series RS Se Inn ee eae . 4 Problem 4 (answer on page 5 of the booklet.) \@ (7 pts) Use ASET to estimate In(1.01) with an error of magnitude less than 10~°. Does your estimate tend to be an overestimate or an underestimate? SeQ7 pts) Use Taylor’s theorem to estimate the error resulting from the approximation l+oz or |x| < 0.01. ii) (7 pts) Use Taylor’s theorem to estimate the error resulting from the approximation Vi+2¢ 21+ § for |z| < 0.01. roblem 5 (answer on page 6 of the booklet.) a ts) Does 3 2 xy lim (x,y) (0,0) 2? + y4 exist? Why or why not? (b) (7 pts) Does . xy lim (x,y) (0,0) 20 + y exist? Why or why not? (c) (7 pts) Suppose f(z, y, z) is a differentiable function of three variables with felt,y,2) = 3a? +y? + 2ez, fyleysz)=20y, and f.(z,y,2) = 2%. Let s=*, y=rtns, z=r+s’, and w=f(z,y,2). Find Ow/dr and w/Qs at the point (r,s) = (2,1). Problem 6 (answer on page 7 of the booklet. ) (12 pts) Match each of the following equations with the surface it defines. pee —2 Jo=2—y? 242 =1 22 + dy? — 42? = a? 4 Az” ae . teen Oy? + 22 = 16 Qe 224227 = be @ SS +4 +3? = 36 ®) 2? +4? +42 =10