Purdue University Study Guide for MA Credit Exam

Transcription

1 Purdue University Study Guide for MA Credit Exam Students who pass the credit exam will gain credit in MA The credit exam is a two-hour long exam with multiple choice questions. No books or notes are allowed. No formula sheets will be provided or allowed. Any brand of one-line display calculators may be used. No exceptions! 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: 1. Lesson Plan for MA Practice Problems The lesson plan lists the sections of the text that are covered in MA The practice problems provide some preparation for the exam. The current text used for MA16010 as well as the course web page are listed below (A copy of the MA16010 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 MA16010, MA1600 and MA1601. A big portion of the text comes from the book listed below, which contains all the topics covered in MA The following text is also on reserve in the math library. To prepare for the MA16010 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. Good luck! 1

2 Lesson Session Topic MA Applied Calculus I Lesson Plan 1 C. Trigonometric Functions C. & 1.6 Trigonometric, Exponential and Logarithmic Functions 1.6 Exponential and Logarithmic Functions. Finding Limits Graphically and Numerically. Evaluating Limits Analytically 6. Continuity and One-sided Limits 7. Infinite Limits 8.1 The Derivative and the Tangent Line Problem 9. Basic Differentiation Rules 10. Rates of Change 11. Product and Quotient Rules and Higher-order Derivatives 1. Product and Quotient Rules and Higher-order Derivatives 1. The Chain Rule 1. The Chain Rule 1. Implicit Differentiation 16.7 Related Rates 17.7 Related Rates 18.1 Extrema on an Interval 19. Increasing and Decreasing Functions and the First Derivative Test 0. Increasing and Decreasing Functions and the First Derivative Test 1. Concavity and the Second Derivative Test. Concavity and the Second Derivative Test. Limits at Infinity.6 A Summary of Curve Sketching.7 Optimization Problems 6.7 Optimization Problems 7.7 Optimization Problems 8.1 Antiderivatives and Indefinite Integration 9.1 Antiderivatives and Indefinite Integration 0. Area 1. Riemann Sums and Definite Integrals. The Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus.6 Numerical Integration 6. Differential Equations: Growth and Decay 6 6. Differential Equations: Growth and Decay

3 Practice Problems 1. Which function below has a period of π, a maximum of. and a minimum of 19.? A. y = sin 0.x B. y = cos 0.x +. C. y = sin 0.πx. D. y = cos x 19. E. y = cos 0.x +. x. Find the domain of f(x) = x. e x e A. (, ln ) (ln, ) B. (, 1 ln ) (1 ln, ) C. (, ln ) (ln, ) D. (, ln 1 ) (ln1, ) E. (, ln 1 ) ( ln1, ) 1. Find all real solutions of the equation sin(x) =. 7π πn 11π πn A. +, + ; n is an integer π πn π πn B. +, + ; n is an integer 6 6 7π 11π C. + πn, + πn; n is an integer π 11π D. + πn, + πn; n is an integer 9 9 π πn 7π πn E. +, + ; n is an integer Find the limit: x + 8 lim x x + 1 A. B C. D. E.

4 . f(x) = x + : x < 1 x : x 1 Choose the number of correct statements below. I. f is not continuous at x = 1. II. lim f(x) = 1. x 1 + III. lim f(x) = 1. x 1 IV. lim f(x) = lim f(x). x 1 x 1 + A. None of the above statements is true. B. Only one of the above statements is true. C. Only two of the above statements are true. D. Only three of the above statements are true. E. All of the above statements are true. 6. Which of the following function has a non-removable discontinuity at x =? A. y = x + x + x B. y = x + x + C. y = x D. y = x + x x E. y = x 9 7. A ball is thrown straight up from the top of a 6-foot building with an initial velocity of feet per second. Use the position function below for free-falling ob jects and find its velocity after seconds. s(t) = 16t + v 0 t + s 0 A. - ft/sec B. 6 ft/sec C. -16 ft/sec D. 8 ft/sec E. -6 ft/sec 8. Which of following does NOT equal to positive infinity (+ )? 1 A. lim x 0 x 1 B. lim x 1 + x 1 x C. lim x + x 9 x + D. lim x x 1 E. lim x 1 (x 1)

5 x 9. A student used the limit process to find the derivative of f(x) = and his work is shown below. Which of the following statements is true? A. He made a mistake in Line (1). B. He made a mistake in Line (). C. He made a mistake in Line (). D. He made a mistake in Line (). E. He made a mistake in Line (). (x + h) x f ' (x) = lim (1) h 0 h x + xh + h x = lim () h 0 h = xh + h lim h 0 h () = lim (x + h ) h 0 () = x () x + x 10. Find the equation of the tangent line to the graph of g(x) = at x =. 8 A. y = x 0 B. y = x + C. y = x 10 D. y = x 18 E. y = x Find the derivative of y = (sin x + tan x)e x. A. y ' = (cos x + sec x)e x B. y ' = (sin x + cos x + tan x + sec x)e x C. y ' = (sin x + cos x + tan x)e x D. y ' = (sin x + cos x + tan x + sec x)e x E. y ' = (sin x + cos x + tan x + sec x tan x)e x 1. The population P, in thousands, of a small city is given by 0t P (t) = 10 + t + 9 where t is the time in years. What is the rate of change of the population at t = yr? Round your answer to the third decimal place. A thousand per year B..1 thousand per year C thousand per year D..91 thousand per year E..88thousand per year

6 1. If h(t) = sin(t) + cos(t), find h () (t). A. sin(t) cos(t) B. sin(t) + cos(t) C. 7 sin(t) 7 cos(t) D. 7 sin(t) + 7 cos(t) E. 7 sin(t) + 7 cos(t) ( x ) 1. Given f(x) =. Find f ' (1). x + 1 A. 7 B. 9 C. 1 D. 1 6 E. 1. A spherical balloon is inflated with gas at a rate of cubic centimeters per minute. How fast is the radius of the balloon changing at the instant the radius is centimeters? The volume V of a sphere with a radius r is V = πr. A. 6π centimeters per minute B. π centimeters per minute C. 16π centimeters per minute 6π D. centimeters per minute E. π centimeters per minute 16. A toy rocket is launched from a platform on earth and flies straight up into the air. Its height after launch is given by: s(t) = t + t + t + 16, where s is measured in meters, and t is in seconds. Find the velocity when the acceleration is 18 m/s. A. m/s B. m/s C. 16 m/s D. 8 m/s E. 1 m/s 6

7 17. According to a joint study conducted by Oxnard s Environmental Management Department and a state government agency, the concentration of CO in the air due to automobile exhaust t yr from now is given by C(t) = 10(0.t + t + 6) parts per billion. Find the rate at which the level of CO is changing 0 years from now. Round your answer to the nearest integer. A. 9 parts per billion per year B. 11 parts per billion per year C. 1 parts per billion per year D. 19 parts per billion per year E. parts per billion per year dy 18. Find by implicit differentiation. dx dy y A. = dx x e y dy y B. = dx 1 ye y dy y C. = ye y y dx x dy 1 + xy D. = dx xye y dy xy y E. = dx x xye y ln (xy) + x = e y 19. An airplane flies at an altitude of y = miles towards a point directly over an observer (see figure). The speed of the plane is 00 miles per hour. Find the rate at which the π angle of elevation θ is changing when the angle is. A. 7 rad/hour B. rad/hour 8 1 C. rad/hour D. 7 rad/hour E. 0 rad/hour 7

8 x 0. Find the critical numbers of y = x e. A. x =, 1 B. x = 0, C. x = 0, 1 D. x =, E. x =, 0 1. Given the function and its derivative, 8x f(x) =, x + 8x + f ' (x) =. (x + ) The y values of the absolute maximum and the absolute minimum of f(x) over the closed interval [ 1, ] are respectively: A. 8 B. 8 and 8 and C. and 8 D. 8 and E. and. Find the open interval where g(t) is increasing. A. (, 0) B. (0, ) C. (, ) D. (, ) E. (0, ) 1 g(t) = t + t 8

9 . The graph of the first derivative of a function f(x) is shown below. Which of the following statements are true? (I) f(x) has critical numbers. (II) On (, ), f(x) is increasing. (III) On (0, ), f(x) is decreasing. (IV) A relative maximum occurs at x = 0. A. I and II are true. B. I and III are true. C. I and IV are true. D. II and III are true. E. III and IV are true.. The position function s(t) = t t + t describes the motion of a particle along a line for t 0. Choose the correct statement below. A. The particle is always moving in a positive direction. B. The particle is always moving in a negative direction. 1 C. The particle changes from a negative direction to a positive direction at t =. D. The particle changes from a negative direction to a positive direction at t = 1. E. The particle changes from a negative direction to a positive direction at t =.. Find the open interval where f(x) = 1 x + x is concave downward. A. (, 0) B. (, ) C. (, ) D. (, 0) E. (, ) 9

10 6. Find the inflection point of y = x + x. A. (, ) B. (, 0) C. (0, 0) D. ( 1, 0) E. ( 1, ) 7. lim f(x) = is true for which of the following functions? x x + x A. f(x) = x + 7 B. f(x) = + x x + 9 C. f(x) = x + x + 6 x x D. f(x) = x + E. f(x) = x + x x + 8. Choose the correct statement regarding the asymptotes of f(x). f(x) = x x + 6 x + 1 A. Horizontal Asymptote: y = 1; Vertical Asymptote: x = 1; Slant Asymptote: None B. Horizontal Asymptote: y = 0; Vertical Asymptote: x = 1; Slant Asymptote: None C. Horizontal Asymptote: None; Vertical Asymptote: x = 1; Slant Asymptote: None D. Horizontal Asymptote: y = 1; Vertical Asymptote: x = 1; Slant Asymptote: y = x E. Horizontal Asymptote: None; Vertical Asymptote: x = 1; Slant Asymptote: y = x 9. A manufacturer has determined that the total cost C of operating a factory is C(x) = 1.x + x where x is the number of units produced. Which of the following statements is true regarding the average cost? A. The minimum average cost is 19 B. The maximum average cost is 19 C. The minimum average cost is D. The maximum average cost is E. The minimum average cost is 00 10

11 0. f(x) is a polynomial and f ' () = 0, f ' () = 0 '' '' f '' () = 0, f (x) < 0 on (, ) and f (x) > 0 on (, ) Which of the following statements are true? I. (, f ()) is an inflection point of f(x). II. (, f ()) is an inflection point of f(x). III. f(x) has a relative maximum at x =. IV. f(x) has a relative minimum at x =. A. Only I and III are true. B. Only I and IV are true. C. Only II and III are true. D. Only I, II and IV are true. E. Only II, III and IV are true. sin x cos x 1. dx = sin x + cos x A. + C sin x cos x B. + C sin x + cos x C. + C sin x cos x D. + C sin x + cos x E. + C. An evergreen nursery usually sells a certain shrub after years of growth and shaping. The growth rate during those years is approximated by dh = 1.t + 8, dt where t is the time in years and h is the height in centimeters. The seedlings are 1 centimeters tall when planted. How tall are the shrubs when they are sold? A. 9 cm B. 6 cm C. 7. cm D. 71. cm E. 9. cm 11

12 . A company s marketing department has determined that if their product is sold at the price of p dollars per unit, they can sell q = p units. Each unit costs $ 10 to make. What is the maximum profit that the company can make? A. 600 dollars B. 800 dollars C. 980 dollars D dollars E. 100 dollars. A particle is moving on a straight line with an initial velocity of 10 ft/sec and an acceleration of a(t) = t +, where t is time in seconds and a(t) is in ft/sec. What is its velocity after 9 seconds? A. 90 ft/sec B. 10 ft/sec C. 6 ft/sec D. 1 ft/sec E. ft/sec. A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 160 m of wire at your disposal, what is the largest area you can enclose? A m B. 00 m C. 600 m D. 800 m E. 000 m 6. A rectangular box with square base and top is to be constructed using sturdy metal. The volume is to be 16 m. The material used for the sides costs $ per square meter, and the material used for the top and bottom costs $1 per square meter. What is the least amount of money that can be spent to construct the box? A. $0 B. $ C. $96 D. $16 E. $160 1

13 7. Use left endpoints and rectangles to approximate the area of the region between the graph of y = x + x and the x-axis over the interval [, 10]. A. 0 B. 80 C. 78 D. 8 E. 88 u 8. Evaluate du. u 1 A. u + C u 10 B. + + C u u 1 C. u + u + C 1 D. + C u u E. u 10 u + C 9. Which of the following definite integral represents the area of the shaded region? A. 7 dx 0 B. dx 0 C. 7x dx 0 D. x dx 0 E. (x + ) dx 0 1

14 0. The growth rate of the population of a city is P ' (t) = 100(10 t), where t is time in years. How does the population change from t = 0 to t =? A. The population increases by 600 B. The population decreases by 600 C. The population increases by 00 D. The population decreases by 00 E. The population increases by The velocity function, in feet per second, is given for a particle moving along a straight line, v(t) = 8t + 0, where t is time in seconds. Find the total distance that the particle travels from t = 0 to t = 8. A. ft B. 6 ft C. 16 ft D. 7 ft E. 100 ft. Given f(x) dx = a, f(x) dx = b and g(x) dx = a, evaluate 1 1 A. b a B. b a C. a + b D. a + b E. a + b [f(x) g(x)] dx. Find the area of the region bounded by the graphs of the following equations. π y = cos x, y = 0, x = 0 and x =. 6 A. 1 1 B. C. D. π E. 1

15 . At 1:00 P.M., oil begins leaking from a tank at a rate of + 0.8t gallons per hour, where t = 0 corresponds to 1:00 P.M. How much oil is lost from :00 P.M. to 6:00 P.M.? A.. gallons B..0 gallons C. 9.6 gallons D gallons E. 0.8 gallons 6. Use the Trapezoidal Rule to approximate x + dx with n =. 0 1 A. T = ( + + 6) 1 B. T = ( ) 1 C. T = ( ) D. T = E. T = The rate of change of a population P is proportional to P. If P = 100 when t = 0 and P = 900 when t =, what is P ()? A B. 00 C. 600 D E The radioactive isotope 6 Ra has a half-life of 199 years. If there are 100 grams initially, how much is there after 000 years? Round to the third decimal place. A. 0.7 grams B. 1. grams C..1 grams D. 8.7 grams E..0 grams 8. An archaeologist measures that an artifact has only 9% of its initial 1 C remaining. Given that the half-life of 1 C is 70 years, about how old is the artifact? A. 9 years B. 10 years C. 987 years D. 77 years E. 697 years 1

16 Answers to Practice Problems 1. E. B. A. A. C 6. E 7. A 8. D 9. D 10. B 11. D 1. C 1. C 1. A 1. A 16. D 17. C 18. E 19. D 0. E 1. C. B. E. D. A 6. E 7. D 8. E 9. C 0. E 1. D. D. B. C. B 6. C 7. B 8. E 9. C 0. D 1. C. A. B. D. D 6. D 7. E 8. D 16