Final Exam Fall 22 Solutions

Math 1551 Version A Name (Print): Fall 2022 Final Exam Canvas email: December 8, 2022 Time Limit: 170 Minutes GT ID: By signing here, you agree to abide by the Georgia Tech Honor Code : I commit to uphold the ideals of honor and integrity by refusing to betray the trust bestowed upon me as a member of the Georgia Tech Community. Sign Your Name: This exam contains 33 problems, all multiple choice. The exam is double-sided. Check to see if any pages are missing. Enter all requested information on the top of this page. You may not use your books, notes, or any calculator on this exam. The following rules also apply: Do not cut apart, tear apart, or in any other way remove any pages from this exam packet. Mark your exam version on your bubble sheet. Failure to do so may lead to a score of 0. Mark all of your multiple choice answers on the separate bubble sheet you were provided. You may write as much as you like, including indicating your final answers, on this exam packet. However, nothing written in this packet for the multiple choice problems will be graded under any circumstances. So, be very careful when filling your answers in on the bubble sheet to make sure you fill the bubble you intended. Only fill in a single bubble for each question on your bubble sheet; any question with multiple bubbles filled will be marked incorrect. Do not make any marks within bubbles that you do not want to select as your answer or those problems will be marked incorrect. No partial credit will be given on any multiple choice problem. Each multiple choice problem is worth the same number of points. You must submit this entire exam packet when turning in your bubble sheet. Failure to do so will result in a score of 0 on the final exam .

The graphs of two functions f and g are depicted below. Answer questions 1-3 based on these two graphs: 1. lim x → 0 f ( x ) g ( x ) : A. - 2 B. DNE C. 0 D. 3 E. -∞ 2. lim x → 4 g ( x ) f ( x ) 2 : A. DNE B. 2 C. ∞ D. 1 / 4 E. 1 / 2 3. lim x →- 6 g ( f ( x )): A. DNE B. - 1 C. 8 D. 6 E. 21 4. Suppose f ( x ) = sin(2 x ). Select the expression that is equivalent to the limit definition of the derivative f 0 ( π 6 ) . A. lim h → π 6 sin ( 2( π 6 + h ) ) - sin π 6 h - π 6 B. lim x → 0 sin (2 x ) - sin π 6 x C. lim x → π 6 sin ( x ) - sin π 3 x - π 6 D. lim h → 0 sin ( 2( π 6 + h ) ) - sin π 3 h E. lim x → π 6 = sin (2 x ) - π 3 x - π 6

10. lim x →∞ 2 x + 7 x + 8 3 2 x - 9 = A. 0 B. 1 C. ∞ D. 2 3 11. Find the points on the curve x 2 4 - y 2 9 = 1 where the tangent line to the curve is horizontal or vertical. A. No vertical tangent line; Horizontal tangent line at (2 , 0) and ( - 2 , 0) B. No horizontal tangent line; Vertical tangents line at (2 , 0) and ( - 2 , 0) C. No vertical tangent line; Horizontal tangent line at (0 , 0) D. No horizontal tangent line; Vertical tangent line at (4 , √ 27) E. Curve does not have horizontal or vertical tangent lines 12. Use Linear Approximation to estimate √ 17. Is your answer an overestimate or underestimate of the exact value? A. L (17) = 4 + 1 8 (17 - 16) = 4 . 125, underestimate B. L (17) = 4 + 1 8 (17 - 16) = 4 . 125, overestimate C. L (17) = 4 + 1 4 (17 - 16) = 4 . 2, overestimate D. L (17) = 4 - 1 8 (17 - 16) = 3 . 875, overestimate E. L (17) = 4 - 1 8 (17 - 16) = 3 . 875, underestimate 13. Consider the function f ( x ) = (2 x + 1) sin( x ) . Find f 0 ( x ). A. f 0 ( x ) = cos( x ) ln(2 x + 1) + 2 sin( x ) 2 x + 1 B. f 0 ( x ) = (2 x + 1) sin( x ) cos( x ) ln(2 x + 1) + 2 sin( x ) 2 x + 1 C. f 0 ( x ) = sin( x )(2 x + 1) sin( x ) - 1 D. f 0 ( x ) = (2 x + 1) sin( x ) cos( x ) 2 x + 1 E. f 0 ( x ) = 2(2 x + 1) sin( x ) ln | sin( x ) |

Consider the following function for questions 14 and 15: f ( x ) =            2 x - 3 , x < - 2 - 7 , x = - 2 x 2 - 11 , - 2 < x ≤ 5 x - 3 , x > 5 14. Classify the discontinuity at the point x = - 2. A. Jump B. Removable C. Wild-Oscillatory D. Not a discontinuity E. Vertical asymptote 15. Classify the discontinuity at the point x = 5. A. Wild-Oscillatory B. Removable C. Vertical asymptote D. Jump E. Not a discontinuity 16. An ice cream cone factory must determine the dimensions for their new cone. The cone must have a fixed volume of 2 cubic units and will, as the name implies, be shaped like a hollow cone. The factory wants to use the smallest amount of ingredients per cone as possible, which is equivalent to minimizing the surface area of the cone. Given that the volume of a cone of radius r and height h is V = 1 3 πr 2 h , and the surface area of such a cone is A = πr √ h 2 + r 2 , which expression below should the company minimize in order to design their new cone? A. πr r 36 π 2 r 4 + r 2 B. 1 3 πr 2 r 4 π 2 r 2 - r 2 C. πr r 36 π 2 r 3 - r 2 D. πr 2 r 4 πr + r 2 E. None of the above 17. lim x → 3 x 2 - 5 x + 6 x 2 - 9 = A. 1 6 B. 3 C. ∞ D. 0 E. Does Not Exist

24. Which one of the following choices correctly characterizes the existence of a global maximum of the function f ( x ) = x 2 + 5 x - 2 on the closed interval [0 , 4]? A. f ( x ) does not have a global maximum on the interval [0 , 4] B. f ( x ) must have a global maximum on [0 , 4] because this is a closed interval, and the global maximum must be located at either a critical point or one of the two endpoints of the interval. C. All functions have a global maximum, so f ( x ) must have one as well. D. If f ( x ) is discontinuous anywhere on [0 , 4], then it will definitely not have a global maximum, otherwise it will. E. None of the above. 25. Which one of the following functions has a point where it is continuous but NOT differentiable? A. f ( x ) = 1 x - 1 B. f ( x ) = | x | C. f ( x ) = x 2 D. f ( x ) = ( - 1 , x < 0 1 , x ≥ 0 E. None of the above 26. Amy is trying to find the global minimum of a function f ( x ) on an open interval (0 , 10). She has correctly identified that f ( x ) has a critical point located at x = 3. Which one of the following additional statements is definitely true? A. If f ( x ) is continuous on the open interval and the critical point at x = 3 is the only local minimum on the open interval, then f (3) is the global minimum. B. If the critical point at x = 3 is the only critical point of f ( x ) on the open interval and it is a local minimum, then f (3) is the global minimum. C. If f 0 (1) < 0 and f 0 (5) > 0, then f (3) is the global minimum. D. If f 00 (3) > 0, then f (3) is the global minimum. E. None of the above.

Consider the function y = x 3 + 3 x 2 - 9 x + 12 when answering problems 27 and 28. 27. For what values of x is the function increasing? A. The function is never increasing. B. - 3 < x < 1 C. x > 3 D. x > - 1 E. x < - 3 , x > 1 28. For what values of x is the function concave up? A. The function is never concave up. B. x > 3 C. x < 3 D. x < - 1 E. x > - 1 29. Z 2 x 2 + x 4 dx = A. 2 x 3 + 4 x 3 + C B. 2 3 x 3 + 4 x 3 + C C. 2 x 3 + x 5 + C D. 2 3 x 3 + 1 5 x 5 + C

33. Which of the following functions satisfy the conditions of the Mean Value Theorem on the given interval? A. f ( x ) = 1 x ( x + 2) on x ∈ [ - 3 , - 1] B. f ( x ) = | x | on x ∈ [ - 2 , 2] C. f ( x ) = 1 x on x ∈ [ - 1 , 1] D. f ( x ) = x 1 3 on x ∈ [ - 1 , 1] E. f ( x ) = 1 x on x ∈ [1 , 2] EXTRA CREDIT: Use the space below to answer the extra credit problem. Think about one problem/topic you have worked on in this class this semester (from a homework, quiz, worksheet, exam, etc) that you struggled to understand and solve, and explain how the struggle itself was valuable. In the context of this question, describe the struggle and how you overcame the struggle. You should discuss whether struggling built aspects of character in you (e.g. endurance, self-confidence, competence to solve new problems) and how these virtues might benefit you in later ventures. Please note that your answer should include some form of self-reflection in order to attain the 3 % points. Your answer should be no longer than 3-4 lines.