Purdue University Study Guide for MA Credit Exam

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1 Purdue University Study Guide for MA 60 Credit Exam Students who pass the credit exam will gain credit in MA60. The credit exam is a twohour long exam with 5 multiple choice questions. No books or notes are allowed. A formula sheets (included with this study guide) will be provided. Any brand of one-line display calculators may be used. A calculator with exponential, logarithmic and basic trigonometric functions will be needed for the exam. No any other electronic devices may be used. To prepare for the exam, you are being provided:. Lesson Plan for MA60. Practice Problems The lesson plan lists the sections of the text that are covered in MA60. The practice problems provide some preparation for the exam. The current text used for MA60 as well as the course web page are listed below (A copy of the MA60 text is on reserve in the math library.) Most of the material covered on the credit exam can be studied from any calculus textbook. Textbook: Edition: Authors: Applied Calculus and Differential Equations Purdue Custom First Edition Larson, Edwards, Zill and Wright The url for the course web page is: The book listed above is a custom made, loose leaf text for MA600, MA600 and MA60. A big portion of the text comes from the book listed below, which contains all the topics covered in MA60. The following text is also on reserve in the math library. To prepare for the MA60 credit exam, you may use either of these two texts. The section numbers listed on the lesson plan on the next page apply to both texts. Textbook: Edition: Authors: Calculus of a Single Variable Sixth Edition Larson and Edwards When you are ready for the examination, obtain the proper form from your academic advisor. Follow the instructions on the form.

2 MA 60 Lesson Plan Lesson Topic -3 Integration by substituion and integration by parts -5 Integration of rational functions 6-7 Volumes of solids of revolution 8 Centroids 9- Work and Fluid pressure 3 Improper integrals Infinite series 5 Maclaurin and Taylor series, operations with series 6-7 Computations with series; applications 8-9 Fourier series 0- First-order linear differential equations; separable differential equations First-order linear differential equations 3- Applications of first-order linear differential equations 5-6 Higher-order homogeneous differential equations 7-9 Nonhomogeneous differential equations 30-3 Introduction and basic properties of the Laplace transform 33-3 Solutions of linear differential equations by Laplace transforms

3 The Formula Page may be used. It will be attached to the credit exam.. Evaluate x + dx (A) (x + ) 3/ + C (B) (x + ) 3/ + C (C) (x + ) / + C (D) (x + ) / + C 3 3 (E) None of these x dx. Evaluate x (A) x ln x + C (B) x + C (C) ln x + C (D) x + C (E) None of these 3x + 3. Evaluate dx x + x (A) 6 ln x + 5 ln x + + C (B) 3 ln(x ) + ln x + C (C) 3 ln x + x + C (D) ln x ln x + + C (E) ln x + ln x + + C 3. Evaluate x ln x dx (Give your answer correct to decimal places.) (A).9 (B).50 (C) 0. (D).0 (E).7 5. Find the area of the region bounded by the graph of y = sin x, the x-axis, and the lines π x = 0 and x =. 3 (A) (B) (C) 0 (D) (E) 6. Find the area of the region bounded by the curves x + y = 0 and x y 8 = (A) (B) (C) 6 (D) (E) Calculate the volume generated by revolving the region bounded by y = x, the x-axis and x = about the y-axis. (Express your answer as a definite integral.) (A) π x dx (B) π x dx (C) π ( x) x dx (D) π x dx (E) π x 3/ dx Calculate the volume generated by revolving the area bounded by y = x, the y-axis and y = about the x-axis. (Express your answer as a definite integral.) (A) π 0 ( x) dx (B) π 0 ( x) dx (C) π 0 ( x) dx (D) π 0 ( x x) dx (E) π 0 x dx 9. Calculate the centroid of a quarter circle of radius r. r r r r r r r (A) x =, ȳ = (B) x =, ȳ = (C) x =, ȳ = 0 (D) x =, ȳ = (E) 3π 3π 3 3 π π π r r x =, ȳ = 3π 3π 0. Calculate x (the x-coordinate of the centroid) of the region given in problem 7. Note 6 the area of the region is square units (A) x = (B) x = (C) x = (D) x = (E) x =

4 . Find the work done in pumping the water out of the top of a cylindrical tank 5 ft in radius and 0 ft high, if the tank is initially half full of water, which weighs 6. lb/ft 3. (A) 93, 750π ft-lb (B) 58, 550π ft-lb (C) 7, 800π ft-lb (D) 5, 600π ft-lb (E) None of these. A spring of natural length ft requires a force of 6 lb to stretch it by ft. Find the work done in stretching it by 6 ft. (A) 5 ft-lb (B) 08 ft-lb (C) 6 ft-lb (D) 36 ft-lb (E) ft-lb 3. A vertical rectangular floodgate on a dam is 5 ft long and ft deep. Find the force on the floodgate if its upper edge is 3 ft below the water surface. (The weight density of water is 6. lb/ft 3.) Give your answer correct to the nearest integer. (A) 76 lb (B) 3900 lb (C) 8 lb (D) 60 lb (E) 00 lb. A horizontal tank with vertical circular ends is filled with oil. If the radius of each end is m, find the force on one end of the tank. (Assume w is the weight of the oil.) Express your answer as a definite integral. (Hint: Assume that the origin is at the center of one of the circular ends.) (A) w y 0 y dy (B) w w ( y) dy (E) None of these y dy (C) w ( y) y dy (D) 5. Evaluate the improper integral x dx (A) It is divergent (B) ln ln 5 (C) ln 5 ln (D) ln 3 ln 5 (E) ln 5 ln n 6. = n= 5 n 5 5 (A) It is divergent (B) (C) (D) (E) 7. Find the first three non-zero terms of the Maclaurin series of f(x) = + 3x. (A) f(x) = + 3 x 9 x (B) f(x) = + + 3x ( + 3x) (C) f(x) = + x x 8 8 (D) f(x) = x 9 ( + 3x) (E) f(x) = + 3 x 9 x Using the Maclaurin series ln( + x) = x x + x 3 x + x 5, find the minimum 3 5 number of terms required to calculate ln(.3) so that the error is (A) (B) 3 (C) (D) 5 (E) 6 9. Find the first three non-zero terms in the Taylor series for f(x) = sin x in powers of (x π ). 8 (A) f(x) = + (x π ) (x π ) (B) f(x) = (x π ) 3 (x π ) + (x π ) (C) f(x) = (x π ) (x π ) 3 + (x π ) 5 (D) f(x) = + (x π ) (x π ) 8 3! 8 5! (E) None of these Approximate cos x dx using three terms of the appropriate Maclaurin series. Give 0 your answer correct to decimal places. (A) (B) (C) (D) (E) 0.863

5 . If f is a periodic function of period π and 0 π x < 0 f(t) = 0 x π π 0 < x π calculate the first three non-zero terms of the Fourier series for f(x). (A) π +cos x+sin x (B) + cos x sin x (C) cos x + cos x (D) + cos x + sin x (E) None π π π π π π of these. ". Find the particular solution of the differential equation y + y = x where y = when x x = x 7 x 5 x 7 x 7 (A) y = + (B) y = + (C) y = + (D) y = + (E) None of 3 3x x these. 3. Find the general solution of the differential equation y dx + (x + ) dy = 0 3 (A) (x+ ) 3 + y = C (B) + = C (C) ln x+ + ln y = C (D) (x+ ) + y = C 3 3 x+ y (E) x + y = C. Find the equation of the curve for which the slope at any point (x, y) is x + y and which passes through the point (0, ). x x (A) y = e x (B) y = + e x (C) y = x + (D) y = ex x (E) y = e x + " 5. Find the particular solution of the differential equation y "" + y " 6y = 0 where y = 0 and y = when x = 0. 3x 3x x 3x (A) y = (e + 3e x ) (B) y = (e + 3e ) (C) y = (e + e x ) (D) 5 5 y = (e 3x + e x ) (E) None of these. "" 6. Find the general solution of the differential equation y y " + y = 0. (+ 3)x/ ( 3)x/ (A) y = c e + c e (B) y = e x [c sin ( 3x/) + c cos ( 3x/)] (C) y = e x [c sin ( 3x) + c cos ( 3x)] (D) y = e x/ [c sin ( 3x/) + c cos ( 3x/)] (E) None of these. 7. Find the general solution of the differential equation y "" + 8y " + 6y = 0. x x x x x x (A) y = c e + c xe (B) y = c e + c xe (C) y = c e + c e (D) y = c sin x + c cos x (E) y = c e x + c e x d x 8. An ob ject moves with simple harmonic motion according to the equation + 6x = 0. dt dx Find the displacement x as a function of t if x = and = 3 when t = 0. dt (A) x = sin 8t + 3 cos 8t (B) x = 3 sin 8t + cos 8t (C) x = 3 sin 6t + cos 6t (D) x = 8 sin 8t + cos 8t (E) x = 8 sin 8t + cos 8t 9. Find the Laplace transform of e 3t sin t. 5

6 8 8 8 (A) (B) (C) (D) (E) (s 3) + 6 (s + 3) + 6 (s 3) + 6 (s + 3)(s + 6) (s + 3) + 6 s 30. Find the inverse Laplace transform of. s + 3s t e t t t ) + e + e t t (A) (e (B) (e t ) (C) (e ) (D) (e + e t ) (E) None of these. "" " 3. Find the Laplace transform of the expression y 3y +y, where y(0) = and y " (0) =. In the following choices the Laplace transform of y(x) is denoted by Y (s). (A) (s 3s + )Y (s) (B) s Y (s) + s (C) (s 3s + )Y (s) + s (D) (s 3s + )Y (s) + s + (E) (s 3s + )Y (s) + s 5 " t 3. Find the Laplace transform of the solution of the differential equation y + y = e, y(0) =. (A) (B) + (C) + (D) + (E) (s + ) s + s + (s + ) s (s ) (s ) 33. Use Laplace transforms to solve the differential equation y "" + 9y = 3t, y(0) =, y " (0) =. (A) y = t sin 3t + cos 3t (B) y = t 0 sin 3t + cos 3t (C) y = cos 3t sin 3t (D) y = cos 3t 3 sin 3t (E) None of these. 3. Use Laplace transforms to solve the differential equation y "" y " + y = e t, y(0) = 0, y " (0) = 0. t (A) y = t e (B) y = t t t t t e (C) y = e (D) y = t e (E) y = te t s 35. If f(s) =, which of the following is the partial fraction expansion of f(s)? (s ) (s + ) In the following choices K, K, and K 3 are constants. K K K 3 K K K K K 3 s (A) + + (B) + (C) + + (D) s s s + (s ) s + s (s ) s + K K K 3 K K + + (E) + s (s ) s + s s + Answers. B. D 3. E. A 5. B 6. B 7. E 8. B 9. E 0. D. B. A 3. D. C 5. E 6. D 7. E 8. C 9. A 0. B. D. C 3. B. D 5. A 6. D 7. A 8. D 9. B 30. B 3. E 3. C 33. A 3. C 35. D 6