Name Student Number Section Number Instructor Math Winter 005 Key Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems 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/), etc. must be simplified for full credit. For administrative use only: / M.C. / 0 /7 /7 /7 /7 /7 5 /7 6 /7 7 /7 8 /7 Total /00
Math Winter 005 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 The integral π/ 0 cos(x) dx equals sin x π/ 0 = 0 sin x cos x dx equals cos x + C dx (c) The integral 0 + x equals tan x = π 0 The radius of convergence of n x n is n=0 (e) The first three lowest order terms of the power series of ( + x) / may be written as + ( ) ( ) x + x + = +! x 8 x + (f) For what values of p does the following improper integral converge? dx p + < 0 p > x p (g) Indicate which convergence test one could use in determining the convergence/ divergence of i. ii. iii. n n limit comparison with /n, integral test, comparison test with /n n n + comparison test/ limit comparison with /n ( ) n ratio test, alternating series test, comparison test with /n! n! n= n= n= (h) State the nth term of the MacLaurin series for i. e x n! xn ii. x x n (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 [, ]. ȳ = [f(x)] dx/ f(x) dx (j) A focus of the hyperbola (x ) (y ) = is [0, ] or [, ]
Problems through are multiple choice. Each multiple choice problem is worth points. In the grid below fill in the square corresponding to each correct answer. 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. Find 6 + x + x ( + x) ( + x ) dx (a) ln + arctan ln + arctan 7 π (e) ln + 8 π ln arctan ln + 8 π arctan (f) 0 (c) ln + 8 π ln arctan (g) π ln + arctan ln π (h) None of the above Answer: ln + 8 π ln arctan. The base of a solid is an elliptical region bounded by x + y =, and cross sections perpendicular to the y axis are squares. Find the volume of the solid. (a) 0 (e) 0 (i) None of the above 6 (f) 8 (c) (g) 8 6 (h) 7 Answer: 6
. Find the length of the graph of y = x / x/ for x [, 6]. (a) 7 + 6 (e) 6 (c) 6 + 7 7 (f) 6 + 7 7 5 (g) 7 + 6 6 + 85 6 + 7 7 (h) None of the above Answer: 6 x cos x 5. Find lim x 0 x (a) (e) (i) None of the above 6 (f) (c) (g) 6 (h) 0 Answer: 6. e t sin (t) dt (a) 7 e ( cos + sin ) (e) 6 e ( cos + sin ) 7 e ( cos + sin ) (f) 7 e ( cos + sin ) (c) 7 e ( cos + sin ) (g) The integral does not converge. 7 e ( cos + sin ) (h) None of the above Answer: 7 e ( cos + sin ) 7. Find the power series expansion for the function sin x (or arcsin x) expanded about 0. ( ) k ( ) k (a) k + xk+ (e) (k + )! xk+ (i) None of the above ( ) / ( ) k x k+ k k + xk+ (f) (k + )! (c) ( ) k x k+ (g) ( ) k x k k + k! ( ) / ( ) k x k+ (h) ( ) k x k+ k (k + )! (k + )!
Answer: ( ) ( ) k x k+. k+ k 8. Identify the equation that best goes with the following graph in rectangular coordinates. (a) (x ) + (y ) = (e) y = x (i) None of the above (y ) (x ) = (f) (x ) (y ) = (c) x = y (g) (x ) (y ) = (x ) + (y ) = (h) Answer: (x ) + (y ) = (y ) (x ) =. Which of the following integrals represents the surface area of the surface generated by revolving the curve y = e x, 0 x, about the line y =. (a) 0 π(ex ) + e x dx (f) 0 π(ex ) dx (c) 0 π(ex + ) + e x dx (g) 0 π(ex ) + e x dx (h) 0 π(ex + ) dx 0 π(ex ) dx 0 π(ex ) + e x dx (i) None of the above (e) 0 π(ex ) + e x dx Answer: 0 π(ex + ) + 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 a x dx. Here a, b are positive constants. Let ax = b sin t, then a dx = b cos t dt, so b a x dx = b b sin t b cos t dt a = b cos t dt a = b ( + cos t) dt a = (t b + ) a sin t + C ( = b sin ax a b + ax ( ax b b ) ) + C = b ax a sin b + x b a x + C. Find the area of the region bounded by the curve x = y y and the line y = x. The curve and the line intersect when x = x ( x) or at x = 0, y = 0 and x =, y =. Area of region A is given by A = 0 (y y ) y dy = y y =.. Find the volume of the solid generated by revolving the region enclosed by y = and y = (x ) + about the y-axis. The two curves intersect at (, ) and (, ). Using the shell method, volume V is given by V = so πx( ((x ) +)) dx = π 0 [ ( x)(8 6x+x ) dx = 6π x + ] x x V = π. Determine the values of p for which the integral answer. x (ln x) p dx converges. Justify your 5
Let u = ln x, then du = dx and so the integral may be written as x x (ln x) p dx = ln u du p The above improper integral is convergent if p >, thus the integral for p >.. (a) Find a Maclaurin series which represents the function sin x x when x > 0. sin x Hence calculate lim. x 0+ x (c) Find the interval of convergence of this power series. From the Maclaurin series for the sine function, x (ln x) p dx converges sin t = t! t + 5! t5 + = n=0 ( ) n (n + )! tn+ sin x = ( x x x! ( x) + ) 5! ( x) 5 + =! ( x) + 5! ( x) + =! x + 5! x + ( ) n = (n + )! xn n=0 Hence sin x lim x 0+ x = lim x 0+! x + 5! x + = Interval of convergence: from ratio test, the series is (absolutely) convergent if ( ) n+ ((n+)+)! xn+ < ( ) n (n+)! xn as n. So x (n + )(n + ) < x < (n + )(n + ) so the radius of convergence is infinite. 6
5. Find the Taylor polynomial of degree for f (x) = x + x + x + which is centered at. Applying Taylor series expansion at x =, and noting that the series terminates after the cubic term, Now so f(x) = f() + f ()(x ) + f ()! (x ) + f () (x )! f(x) = x + x + x + f() = 0 f (x) = x + 6x + f () = 0 f (x) = x + 6 f () = 0 f (x) = f () = f(x) = f() + f ()(x ) + f () (x ) + f () (x )!! = 0 + 0(x ) + 0! (x ) + (x )! = 0 + 0 (x ) + 5 (x ) + (x ) 6. Compute I = x ln x dx and determine a reduction formula for I n = x(ln x) n dx, n > Integrating by parts, with u = ln x, dv = x dx, I = (ln x) x x x dx = x ln x x + C For n >, with u = (ln x) n, dv = x dx, I n = (ln x) n x x n(ln x)n x dx = x (ln x) n n x(ln x) n dx and thus I n = x (ln x) n n I n. ( 7. Sketch the closed curve r = 7 cos θ π ) and determine the area enclosed by the curve. The equation represents a circle with radius= 7/, passing the origin and center along y = x. 7
8. (a) For which values of x does k= What is the sum of this series? The series obviously converges when x = 0. t k Consider the series S(t) = k. Now k= k ( ex ) k converge? S (t) = t k = k= t k For t <, S (t) = and so the series converges. t Hence the series k ( ex ) k converges when x = ln or when k= e x < < e x < x < ln. Also for t <, since S(0) = 0. Consequently, S (t) = t S(t) = dt t = ln( t) k= k ( ex ) k = S( e x ) = ln(e x ) = x. 8