MATH 201 Exam 2, Fall 2015

Math 201-Exam 2 (Fall 15) Prof. B. Shaya and Dr. H. Y; i b e Please write your section number on your oak niet e Please place your student ID card ot oa ' front of you. » Please answer each problem on the indicat" Page(s) of the booklet not written on the indicated page(s) will not De graded. et. Any part of your answer e Unjustified answers will receive little or DO Credit, Problem 1 (answer on pages 1 and 2 of the booklet.) (a) (8 pts) Does 4 im 2 (zy)7.0) 328 + yA exist? Why or why not? (b) (8 pts) What about 6 Wi se a (zy) -+(00) 328 + yf * (c) (8 pts) What about : Ho lim 2.9 (zy)+(0,0) 32 + y Problem 2 (answer on page 3 of the booklet.) Suppose f(x,y, 2) is a differentiable function of three variables such that f(1, 2,3) = f(,1,-4)=4, VfU,1,-4)=i+j+k, Vf(8,3,-4) =3i-j4+k. Let z= 5r+3s, y=2r+s, z= -2(r?+s*), and w= f(x,y, 2). (i) (12 pts) Find w/dr and Ow/ds at the point (r,s) = (1, 1), (ii) (12 pts) Estimate f(1.01, 1.02, —3.98). / Problem 3 (answer on page 4 of the booklet.) Consider the function f(x) = e?. (a) (12 pts) Use Taylor's theorem to find a power series expansion for {() about the point # = 0. (b) (12 pts) Find the Taylor polynomial p3(z) generated by f at z = 0. Then use Taylor’s theorem to estimate the error resulting from the approximation e~°! ~ p3(—0.1). Conclude that e®! < 10/9. Problem 4 (answer on pages 5 and 6 of the booklet.) Let C be the two-sided cone 2? = 2° + y’. (i) (15 pts) Find the tangent plane and normal line of C at the point (3,4, 5). (ii) (10 pts) Let S be the set of all points (x,y, 2) in C such that the normal line of C at (x,y, z) is perpendicular to the vector v = i+jt V2k. Prove that S lies on a line L. (iii) (3 pts) Find p arametric equations for the line L from part (ii).