Name Student Number Section Number Instructor Math Winter 2005 Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems 2 through are multiple choice questions. For problems 0 through 8 give the best answer and justify it with suitable reasons and/or relevant work. Work on scratch paper will not be graded. Do not show your work for problem. Please write neatly. Notes, books, and calculators are not allowed. Expressions such as ln(), e 0, sin(π/2), etc. must be simplified for full credit. For administrative use only: / M.C. /2 0 /7 /7 2 /7 /7 /7 5 /7 6 /7 7 /7 8 /7 Total /00
Math Winter 2005 Departmental Final Exam Part I: Short Answer and Multiple Choice Questions Do not show your work for problem.. Fill in the blanks with the correct answer. (a) The integral (b) The integral (c) The integral π/2 0 0 cos(2x) dx equals sin x cos 2 x dx equals dx + x 2 equals The radius of convergence of 2 n x n is n=0 (e) The first three lowest order terms of the power series of ( + x) /2 may be written as (f) For what values of p does the following improper integral converge? dx x p (g) Indicate which convergence test one could use to determine the convergence/ divergence of i. ii. iii. n n n n + ( ) n n=2 n= n= n! (h) State the nth term of the MacLaurin series for i. e x ii. x (i) Express in terms of a quotient of integrals the y coordinate of the centroid of the region below y = f(x) with f(x) > 0 for all x over [ 2, 2]. (j) A focus of the hyperbola (x 2)2 (y )2 = is
Problems 2 through are multiple choice. Each multiple choice problem is worth points. In the grid below fill in the square corresponding to each correct answer. 2 A B C D E F G H I A B C D E F G H I A B C D E F G H I 5 A B C D E F G H I 6 A B C D E F G H I 7 A B C D E F G H I 8 A B C D E F G H I A B C D E F G H I 2. Find 2 6 + x 2 + x (2 + x) ( + x 2 ) dx (a) 2 ln 2 + arctan ln + 2 arctan 7 2 π (e) 2 ln 2 + 8 π ln 2 arctan 2 (b) ln 2 + 8 π 2 arctan 2 (f) 0 (c) ln 2 + 8 π ln 2 arctan 2 (g) π 2 ln 2 + arctan 2 ln π (h) None of the above. The base of a solid is an elliptical region on the xy-plane enclosed by x2 + y2 =, and cross sections perpendicular to the y axis are squares. Find the volume of the solid. (a) 20 (e) 0 (i) None of the above (b) 6 (f) 8 (c) 2 (g) 8 6 (h) 72 2
. Find the length of the graph of y = x /2 x/2 for x [, 6]. (a) 7 + 2 6 (e) 6 (b) (c) 6 + 7 7 2 (f) 6 + 7 7 5 (g) 7 + 6 6 + 85 2 6 + 7 7 (h) None of the above 2 x 2 2 cos x 5. Find lim x 0 x Hint: You might want to consider using power series to do this. (a) (e) (i) None of the above (b) 6 (f) 2 (c) (g) 6 (h) 0 6. e t sin (t) dt (a) 7 e ( cos 2 + sin 2) (e) 6 e ( cos 2 + sin 2) (b) 7 e ( cos 2 + sin 2) (f) 7 e ( cos 2 + sin 2) (c) 7 e ( cos 2 + sin 2) (g) The integral does not converge. 7 e ( cos 2 2 + sin 2) (h) None of the above 7. Find the power series expansion for the function sin x or arcsin x expanded about 0. Hint: (a) (b) (c) You might want to write the function in the form x something dt. 0 ( ) k ( ) k 2k + x2k+ (e) (k + )! xk+ (i) None of the above ( ) /2 ( ) k x 2k+ k 2k + x2k+ (f) (2k + )! ( ) k x k+ (g) ( ) k x k k + k! ( ) /2 ( ) k x 2k+ (h) ( ) k x 2k+ k (2k + )! (2k + )!
8. Identify the equation that best goes with the following graph in rectangular coordinates. (a) (x ) 2 + (y 2)2 = (e) y = x 2 (i) None of the above (b) (y ) 2 (x 2)2 = (f) (x ) 2 (y 2)2 = (c) x = y 2 2 (g) (x 2) 2 (y )2 = (x 2) 2 + (y )2 = (h) (y 2) 2 (x )2 =. Which of the following integrals represents the surface area of the surface generated by revolving the curve y = e 2x, 0 x, about the line y = 2. (a) 0 2π(e2x 2) + e x dx (f) 0 2π(e2x 2) dx (b) (c) 0 2π(e2x + 2) + e x dx (g) 0 2π(e2x ) + e x dx (h) 0 2π(e2x + 2) dx 0 2π(e2x ) dx (e) 0 2π(e2x ) + e x dx (i) None of the above 0 2π(e2x 2) + e x dx The answers to the multiple choice MUST be entered on the grid on the previous page. Otherwise, you will not receive credit.
Part II: Written Solutions For problems 0 8, write your answers in the space provided. Neatly show your work for full credit. 0. Find a formula for b 2 a 2 x 2 dx. Here a, b are positive constants.. Find the area of the region bounded by the curve x = y y 2 and the line y = x. 5
2. Find the volume of the solid generated by revolving the region enclosed by y = and y = (x ) 2 + about the y-axis.. Determine the values of p for which the integral answer. 2 x (ln x) p dx converges. Justify your 6
. (a) Find a Maclaurin series which represents the function sin x x when x > 0. sin x (b) Hence calculate lim. x 0+ x (c) Find the interval of convergence of this power series. 5. Find the Taylor polynomial of degree for f (x) = x + x 2 + 2x + which is centered at. 7
6. Compute I = x ln x dx and determine a reduction formula for I n = x(ln x) n dx, n >. ( 7. Sketch the closed curve r = 7 cos θ π ) and determine the area enclosed by the curve. 8
8. (a) For which values of x does k= k ( ex ) k converge? (b) What is the sum of this series? End