Math 120 Sample questions for Chapters 4&5, with answers Not all questions will necessarily be like the ones on the following pages, but they do give a pretty good idea of the types of questions to expect on the Final Exam on the topics covered in these chapters. Answers (but not complete solutions) follow the set of questions for each of the two chapters. In the sample questions for Chapter 4, do not consider question #s 4,5,6,8,12, leaving 15 questions. The optimization questions (§4.7) are, in general, a bit more difficult then what you can expect, in terms of setting up the function to be maximized or minimized. The first half of the homework questions for §4.7 are more representative of the level of difficulty of questions to expect. In the sample questions for Chapter 5, do not consider question #s 9,11,17,19,20, leaving 15 questions. You will probably need a calculator for question #s 2 & 7 as stated, and certainly for #3, which I am trying to avoid, but there may be on the exam similar questions that are easier to calculate that do not require a calculator. Modifications of these questions that may be done without a calculator are as follows: #2 round each of the given speeds down to the nearest integer #3 0 to π using six approximating rectangles #7 may be done for v(t)=4 t−8. Also, study the seven quizzes and two hour tests, for which you were provided detailed solutions. On the Final Exam, there will be at least two questions similar to those on each of Tests 1 and 2, as well as at least one similar to those on Quiz 7. Stewart - Calculus ET 6e Chapter 4 Form B 1. Find all the critical numbers of the function. g ( x) = 4 x + sin( 4 x) 2. Find the local and absolute extreme values of the function on the given interval. f ( x) = x3 − 6 x 2 + 9 x + 1, [ 2, 4] 3. Find an equation of the line through the point (8, 16) that cuts off the least area from the first quadrant. 4. Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. ª 2 2º f ( x) = sin 3πx , «− , » ¬ 3 3¼ 5. Estimate the absolute maximum value of the function y = x 3x − x 2 to two decimal places on the interval [0, 3]. 6. At 4:00 P.M. a car's speedometer reads 29 mi/ h . At 4:15 it reads 71 mi/h. At some time between 4:00 and 4:15 the acceleration is exactly x mi/ h 2 . Find x. 7. Find f . f ′(t ) = 2t − 3sin t , f (0) = 5 8. Find a cubic function f ( x) = ax 3 − bx 2 + cx − d that has a local maximum value of 112 at 1 and a local minimum value of -1,184 at 7. 9. Find the inflection points for the function. f ( x) = 8 x + 3 − 2 sin x , 0 < x < 3π Stewart - Calculus ET 6e Chapter 4 Form B 10. Find the limit. ex − 2 x → 0 sin 7 x lim 11. Evaluate the limit. lim x →0 1 − cos x x2 + x 12. A company estimates that the marginal cost (in dollars per item) of producing items is 2.75 - 0.002x. If the cost of producing one item is $589 find the cost of producing 100 items. 13. Find the limit. lim (1 − 10 x)1 / x x→0 14. Consider the following problem: A farmer with 800 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? 15. Find the most general antiderivative of the function. 1 1 f ( x) = 8 x 7 − 10 x 9 16. A fence 9 ft tall runs parallel to a tall building at a distance of 7 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth. 17. A particle is moving with the given data. Find the position of the particle. v(t ) = sin t − cos t , s (0) = 0 18. A rectangular beam will be cut from a cylindrical log of radius 10 inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log. Stewart - Calculus ET 6e Chapter 4 Form B 19. A painting in an art gallery has height h = 60 cm and is hung so that its lower edge is a distance d = 17 cm above the eye of an observer (as seen in the figure below). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle subtended at his eye by the painting?) 20. A steel pipe is being carried down a hallway 15 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 9 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner? ANSWER KEY Stewart - Calculus ET 6e Chapter 4 Form B 1. π (2n + 1) 4 2. Max ( 4,5 ) , Min ( 3,1) 3. y = −2 x + 32 4. 1 −1 1 −1 , , , 6 6 2 2 5. 2.92 6. x = 168 7. f (t ) = t 2 + 3cos t + 2 8. 12 x 3 − 144 x 2 + 252 x − 8 9. (π , 8π + 3) , (2π , 16π + 3) 10. 1 7 11. lim 1 − cos x x →0 x2 + x =0 12. $851.25 13. e −10 14. 16000 15. 7 x 8 / 7 − 9 x10 / 9 + C 16. 22.57 17. s (t ) = 1 − cos t − sin t 18. 20 / 3 in., 20 2 / 3 in. 19. 36.18 20. 33.58 Stewart - Calculus ET 6e Chapter 5 Form A 1. Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be left endpoints. 2. The speed of a runner increased steadily during the first three seconds of a race. Her speed at halfsecond intervals is given in the table. Find a lower estimate for the distance that she traveled during these three seconds. t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0 (ft/s) 0 3.7 8.3 13.2 15.1 15.7 16.2 3. Approximate the area under the curve y = sin x from 0 to π / 4 using ten approximating rectangles of equal widths and right endpoints. 4. Evaluate the Riemann sum for f ( y ) = 9 − y 2 , 0 ≤ y ≤ 2 with four subintervals, taking the sample points to be right endpoints. 5. If f ( x) = x − 2, 1 ≤ x ≤ 6 , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints. 6. Evaluate the definite integral. π /8 x 2 sin x dx 5 + x6 −π / 8 ³ 7. The velocity function (in meters per second) is given for a particle moving along a line. Find the distance traveled by the particle during the given time interval. v(t ) = 4t − 3, 0 ≤ t ≤ 6 8. If ³ 6 0 9. f ( x)dx = 10 and ³ 4 f ( x)dx = 7 , find 0 ³ 6 f ( x )dx . 4 Find the area of the region that lies under the given curve. y = 5x + 2 , 0 ≤ x ≤ 1 Round the result to the nearest thousandth. Stewart - Calculus ET 6e Chapter 5 Form A 10. Evaluate. ³ 1 0 d arctan x (e )dx dx 11. Evaluate the indefinite integral. 4 + 6x ³ dx 6 + 4 x + 3x 2 12. Evaluate the integral by interpreting it in terms of areas. 3 ³ (1 + 4 x) dx 1 13. Express the sum as a single integral in the form b ³ f ( z) dz a 8 11 2 8 ³ f ( z ) dz + ³ f ( z) dz 14. Find the derivative of the function. F ( x) = x t2 0 1+ t3 ³ dt 15. Evaluate the integral if it exists. ³ 2 § 1− x · ¨ x ¸ dx © ¹ 16. Evaluate the integral. π /4 ³ sin(t ) dt π /6 17. Evaluate the integral if it exists. ³ cos(ln x) dx x Stewart - Calculus ET 6e Chapter 5 Form A 18. Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate § dx 130 ·¸ of production of these calculators after t weeks is = 4,500¨¨1 − calculators per week. 2 ¸ dt © (t + 9) ¹ Production approaches 4,500 per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques. Find the number of calculators produced from the beginning of the third week to the end of the fourth week. Round the answer to the nearest integer. 19. The marginal cost of manufacturing x yards of a certain fabric is C ′( x) = 3 − 0.01x + 0.000006x 2 (in dollars per yard). Find the increase in cost if the production level is raised from 500 yards to 3,000 yards. 20. Evaluate the integral if it exists. 1 ex ³ 1+ e 0 2x dx ANSWER KEY Stewart – Calculus ET 6e Chapter 5 Form A 1. 8 2. 28 3. 0.32 4. 14.25 5. -0.857 6. 0 7. 54 8. 3 9. 2.092 10. eπ / 4 − 1 11. 2 6 + 4 x + 3 x 2 + C 12. 18 11 13. ³ f ( z) dz 2 14. x2 1 + x3 15. x − 2 ln x − 1 +C x 16. 0.158919 17. sin(ln x) + C 18. 818 19. 17500 20. arctan(e) − π 4