Math 131 Exam - Fall 2023 - PDF

Math 131 Final Exam Fall 2023 Name: ID: INSTRUCTIONS 1. If your name is printed at the top, verify. 2. Write your name and ID in the boxes. 3...

Math 131 Final Exam Fall 2023 Name: ID: INSTRUCTIONS 1. If your name is printed at the top, verify. 2. Write your name and ID in the boxes. 3. Read the exam instructions on the next page. 4. Write all work on THIS booklet (no scratch paper). 5. Mark your answers clearly. If you erase, erase completely. 6. Turn in your ENTIRE exam booklet. Math 131 Final Exam Page 2 of 19 This exam consists of: 5 true/false questions worth 2 points each. 10 multiple-choice problems worth 5 points each. 4 written problems worth 40 points total. Please make sure you have all 19 problems and no pages are missing. No calculators! For the true/false and multiple choice questions, bubble in your answer. There is only one correct answer unless the problem explicitly says there may be more than one. Show all your work for the written problems. Your ability to make your solution clear will be part of the grade. Furthermore, wrong answers are more likely to receive partial credit if your work is clear. Useful Formulas sin(a + b) = sin(a) cos(b) + cos(a) sin(b) cos(a + b) = cos(a) cos(b) − sin(a) sin(b) π π

sin(θ) = cos 2 − θ , cos(θ) = sin 2 −θ 180 degrees = π radians sin2 x + cos2 x = 1 1 + tan2 x = sec2 x sin(x) cos(x) 1 + cot2 x = csc2 x tan(x) = cos(x) , cot(x) = sin(x) 1 1 sin(x) sec(x) = cos(x) , csc(x) = sin(x) limx→0 x =1 sin(−x) = − sin(x), cos(−x) = cos(x) Exp growth: P = Cekt log(ab) = log(a) + log(b) log(ab ) = b log a logb c loga c = logb a (ab )c = abc √ d dx sin(x) = cos(x), d dx cos(x) = − sin(x) sin(30◦ ) = sin π6 = 21 , sin(45◦ ) = sin π4 = 2 2 Volumes: 34 πr3 (sphere), πr2 h (cylinder), 13 πr2 h (cone) d dx arctan(x) = 1 d 1+x2 , dx arcsin(x) = √ 1 1−x2 Math 131 Final Exam Page 3 of 19 1. The graph of f (x) = sin(x) is shown below: 1 0.5 − π4 π π 3π π 4 2 4 −0.5 True or false: The shaded region on the graph represents a Riemann sum with 3 rectangles approxi- Z π/2 mating sin(x) dx. 0 True False Z 1 √ π 2. True or false: − 1 − x2 dx = −. −1 2 True False Math 131 Final Exam Page 4 of 19 1 3. True or false: The function f (x) = is differentiable at x = 2, but not continuous at x = 2. x−2 True False 4. True or False: If a function f is differentiable everywhere, and f (1) = f (10), then there must be a number a such that 1 ≤ a ≤ 10 and f 0 (a) = 0. True False Math 131 Final Exam Page 5 of 19 5. True or false: The 2003rd derivative of cos(x) is sin(x). True False √ x−3 6. Find lim. x→9 x − 9 0 1 9 1 6 1 ∞ Math 131 Final Exam Page 6 of 19 7. Find the derivative of f (x) = cos(x2 ). f 0 (x) = cos(2x) f 0 (x) = 2x sin(x2 ) f 0 (x) = cos(2x) − sin(x2 ) f 0 (x) = − sin(x2 ) f 0 (x) = −2x sin(x2 ) 8. Find the second derivative of f (x) = x8 ex. f 00 (x) = x9 ex−1 f 00 (x) = 56x6 ex f 00 (x) = (x8 + 16x7 + 56x6 )ex f 00 (x) = (x8 + 8x7 + 56x6 )ex f 00 (x) = x9 (x − 1)ex−2 Math 131 Final Exam Page 7 of 19 1 5 9. Let f (x) = x3 − x2 + 6x − 63. What feature does the graph of f have at x = 2? 3 2 A local minimum A local maximum A critical point that is not a local extremum An inflection point None of the above Z 4 10. Find x2 − 3 dx. 1 9 12 15 18 21 180 Math 131 Final Exam Page 8 of 19 11. The graph of a function f (x) is shown below. −2 −1 1 2 Which of the following could be the graph of f 0 (x)? (Axes may not be to scale in these graphs.) I II III IV V VI −2 −1 1 2 −2 −1 1 2 −2 −1 1 2 I II III −2 −1 1 2 −2 −1 1 2 −2 −1 1 2 IV V VI Math 131 Final Exam Page 9 of 19 12. Find the slope of the tangent line to the ellipse 4x2 + y 2 = 8 at the point (1, −2). 2 4 6 8 12 Z x2 dy 13. Use the Fundamental Theorem of Calculus to find if y = cos(t2 )dt. dx 1 4 cos(x ) −2x sin(x4 ) 2x cos(x4 ) −2x sin(x2 ) cos(x4 ) x2 cos(x2 ) Math 131 Final Exam Page 10 of 19 14. In a particularly difficult Webwork problem, you are trying to calculate 1 − cos(x) lim. x→0 x + x2 You keep getting the answer wrong. Your work goes through the following steps: (1) Note that 1 − cos(x) and x + x2 are both 0 when x = 0. (2) Apply L’Hopital’s Rule: 1 − cos(x) sin(x) lim 2 = lim x→0 x+x x→0 1 + 2x (3) Apply L’Hopital’s Rule again: sin(x) cos(x) lim = lim x→0 1 + 2x x→0 2 (4) This is not an indeterminate form, so plug in x = 0: cos(x) cos(0) lim = x→0 2 2 (5) Finally, cos(0) = 1, so 1 − cos(x) 1 lim 2 =. x→0 x+x 2 In which step did you make a mistake? (1) (2) (3) (4) (5) Math 131 Final Exam Page 11 of 19 15. In subsection, you and your group are working on the following problem: A child throws a glass marble into the water. The function h(t) = −t3 + 5t2 − 8t + 4 models the height of the marble above the water level as a function of time (in seconds), and negative values of h correspond to the marble being underwater. We start observing the marble as soon as the child throws it, and stop after 3 seconds. Find the time when the position of the marble is farthest below the water. A member of your group presents the following solution, broken into steps: (1) We are looking for the absolute minimum of the function h(t) on the interval [0, 3]. (2) The derivative of the height is equal to h0 (t) = −3t2 + 10t − 8. (3) The derivative is never undefined, so to find the critical points, we solve the equation h0 (t) = 0. 4 (4) The solutions to −3t2 + 10t − 8 = 0 are t = and t = 2, so these are the critical points. 3 (5) Finally, to see when the height is thelowest,  we compare the value at the these two values of t, 4 4 and find that since h(2) = 0 and h = − , the time when the height is the lowest is at 3 27 4 t=. 3 At which step did your classmate make a mistake? (1) (2) (3) (4) (5) Math 131 Final Exam Page 12 of 19 Written Problem. You will be graded on the readability and reasoning of your work. Put all work inside the boxes, and leave a note if additional work you want graded is on one of the extra pages. All graded work must be done in this exam booklet. 16. Evaluate the following indefinite integrals: Z 1 (a) (3 points) dx x4 Z (b) (3 points) 4x2 − sec2 (x) dx Problem 16 continues on next page! Math 131 Final Exam Page 13 of 19 √ Z (c) (4 points) 9(ex − sin(x)) + 5 x dx Math 131 Final Exam Page 14 of 19 Written Problem. You will be graded on the readability and reasoning of your work. Put all work inside the boxes, and leave a note if additional work you want graded is on one of the extra pages. All graded work must be done in this exam booklet. 17. Let g(x) = f (x2 + 1), where f is a function with f (5) = 4 and f 0 (5) = 7. (a) (6 points) Find g 0 (2). (b) (4 points) Find the equation of the tangent line to the graph of g(x) at x = 2. Math 131 Final Exam Page 15 of 19 Written Problem. You will be graded on the readability and reasoning of your work. Put all work inside the boxes, and leave a note if additional work you want graded is on one of the extra pages. All graded work must be done in this exam booklet. 18. Suppose you know that f 000 (x) = cos(x), f (0) = 3, f 0 (0) = 2, and f 00 (0) = 1. (a) (3 points) Find f 00 (x). (b) (3 points) Find f 0 (x). (c) (4 points) Find f (x). Math 131 Final Exam Page 16 of 19 Written Problem. You will be graded on the readability and reasoning of your work. Put all work inside the boxes, and leave a note if additional work you want graded is on one of the extra pages. All graded work must be done in this exam booklet. 19. A cylinder has volume 16π cubic inches. What is the minimum possible value of its surface area, and what radius should it have in order to achieve this minimum? (Note: A cylinder with radius r and height h has surface area 2πr2 + 2πrh.) Math 131 Final Exam Page 17 of 19 This page left blank for additional scratch work. If you want anything on this page graded, you must indicate that on the page containing the relevant free-response problem. Math 131 Final Exam Page 18 of 19 This page left blank for additional scratch work. If you want anything on this page graded, you must indicate that on the page containing the relevant free-response problem. Math 131 Final Exam Page 19 of 19 This page left blank for additional scratch work. If you want anything on this page graded, you must indicate that on the page containing the relevant free-response problem.