MATH 2413.0012 – Spring 2015 Exam II Retake, Version 1 Due Wednesday, 15 April 2015 at the START of class Instructions: 1) There are nine(9) problems/questions on this exam, and each problem/question is a free response question worth ten (10) points. The maximum score on this exam is 90�90. 2) For each free response question you are expected to provide one and only one neatly written solution. Show all appropriate work and calculations necessary to produce your answer. If there is work that you do not wish to have graded then please mark it as such. 3) Answers should be given as exact values or in analytical form unless otherwise stated. In other words, do not use decimal approximations unless the problem is an applications problem, or it is otherwise indicated in the instructions. 4) This is a take home exam. You may use your textbook and course notes to complete these problems. Any attempt at collusion or use of a private tutor or public tutoring services will be considered cheating. No one may assist you or work this exam for you. 5) Scientific calculators are allowed, however, calculators that have built in computer algebra systems (e.g., TI-89) are not allowed. Any attempt to utilize such a calculator or a cell phone, iPad, iPod, tablet, computer, or similar device will be considered cheating. 6) You must work the version of the exam assigned to you by the first letter of your last name. If you work the other version, you will not receive credit. 7) By signing the exam below you are agreeing to be bound by the regulations concerning academic honesty and academic integrity as set forth in HCCS protocol. Your exam will not be graded and you will receive no credit unless a signature is provided. Name: ________________________________________ I.D. Number: ________________________________________ Signature: ________________________________________ 1) Find the first derivative for each of the following functions: a) 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥 2 ) cos 2 (𝑥𝑥) b) 𝑔𝑔(𝑥𝑥) = 1 + tan�√𝑥𝑥� 2) Find a simplified expression for the second derivative of the following function. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2⁄5 (𝑥𝑥 − 10)3⁄5 3) Find the global/absolute extrema (maxima and minima), and their locations as ordered pairs for the function 𝑓𝑓(𝑥𝑥) = −9𝑥𝑥 5 + 65𝑥𝑥 3 − 60𝑥𝑥 + 5 over the interval −1 ≤ 𝑥𝑥 ≤ 3. 4) Determine where the function 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 + 3) sin 𝑥𝑥 + cos 𝑥𝑥 is increasing and/or decreasing over the interval �− 7𝜋𝜋�4 , 5𝜋𝜋�4�. 5) Determine if the function 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥) cos(𝑥𝑥) satisfies the Mean Value Theorem on the interval �− 𝜋𝜋�4 , 𝜋𝜋�4�. If so, find all values of 𝑥𝑥 such that − 𝜋𝜋�4 < 𝑥𝑥 < 𝜋𝜋�4 and 𝑓𝑓 ′ (𝑥𝑥) 𝜋𝜋 𝜋𝜋 𝑓𝑓 � 4� − 𝑓𝑓 �− 4� . = 𝜋𝜋 𝜋𝜋 − �− � 4 4 6) Find an equation for the line tangent to the curve (𝑥𝑥 − 1)(𝑥𝑥 2 + 𝑦𝑦 2 ) = 2𝑥𝑥 2 at the point (2,2). 7) Show that the function 𝑓𝑓(𝑥𝑥) = 1 − sin 𝑥𝑥 − cos2 𝑥𝑥 satisfies the hypotheses of Rolle’s Theorem on the interval −𝜋𝜋 ≤ 𝑥𝑥 ≤ 𝜋𝜋. Find all values of 𝑥𝑥 in the interval (−𝜋𝜋, 𝜋𝜋) such that 𝑓𝑓 ′ (𝑥𝑥) = 0. 8) Does the function 𝐿𝐿(𝑥𝑥) = 2 + sin 𝑥𝑥 + cos 𝑥𝑥 , 𝑥𝑥 2 − 𝑥𝑥 − 20 have an absolute/global maximum on the interval [−3,3]? Carefully explain why or why not. 9) A bowling ball is dropped from a balloon tethered in place at an altitude of 1600 𝑓𝑓𝑓𝑓. The bowling ball’s height (measured in feet) above ground 𝑡𝑡 seconds after it is dropped is given by 𝑠𝑠(𝑡𝑡) = −16𝑡𝑡 2 + 1600. A camera is positions at a lateral distance of 500 𝑓𝑓𝑓𝑓. from the base of the tether. a) How fast is the distance between the camera and the bowling ball changing when the ball is at an altitude of 1200 𝑓𝑓𝑓𝑓.? b) If the camera is kept focused on the falling ball, how fast is the camera’s angle of elevation changing at this same instant?