pastexams - MATH 1013 3.0 APPLIED CALCULUS I Please tick...

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1 MATH 1013 3.0 Please tick one section APPLIED CALCULUS I Location SECTION A: Prof. Szeto ACW006 SECTION B: Prof. Taylor ACW109 SECTION C: Prof. Rost ACW206 SECTION D: Prof. Hu ACW206 NAME: STUDENT #: Final Exam: Saturday Dec 16, 2006, 10:00-13:00 Aids allowed: notes on one sheet only (two sides permitted) of letter size paper plus a non-graphing calculator. ANSWER ALL (13) QUESTIONS. All questions carry equal marks. In all questions, unless otherwise stated, it is essential to briefly explain your reasoning and to provide details of the intermediate steps taken in reaching your answers. Answers are to be written in this question book. Do not disassemble this book. Each question starts on a new page, followed by a blank page for continuation where appropriate. 3 blank pages are appended at the end for rough work ONLY. Do not remove any pages. Question Mark /10 1 2 3 4 5 6 7 8 9 10 11 12 13 Total
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2 Question 1 Clearly circle True or False. No need to explain. a) True / False , 2 ! , ln 1, exp(1) are all irrational numbers. b) True / False | | x x + is nonnegative for all real number x . c) True / False By L x f c x = ! ) ( lim we mean that, for every 0 > , there is a corresponding 0 > , such that " < # $ < # < | ) ( | | | 0 L x f c x . d) True / False If ) ( c f does not exist, then ) ( x f is not continuous at c x = . Space Question 2 Find each of the following limits or indicate that it does not exist. Show intermediate steps. If the limit does not exist, explain why not. a) 6 10 3 lim 2 2 2 ! + ! + " x x x x x b) ) 3 2 5 2 ( lim 2 2 ! ! + " # x x x c) ) 2 7 1 4 ( lim 2 3 3 x x x ! + ! " d) [ ] | | lim 0 x x ! (Recall that [ ] | | x is the greatest integer less than or equal to x ). Space Question 3 a) Let x x x f sin ) ( = , 0 ! x . How should f be defined at 0 = x in order to make it continuous there? b) Show that the equation 0 10 2006 12 16 = ! + x x has a solution in ). 1 , 0 ( Space
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3 Question 4 Determine the derivative with respect to x for the following functions, a) f(x) = (tan 2x) 3 b) g(x) = (x 2 - 2)/(x 2 + 1) c) USING the limit definition of the derivative show that D x (x -1/2 ) = -(1/2)x -3/2 Space Question 5 A circular cylindrical oil drum of length 1m is lying on its side. It contains diesel fuel but has a small leak at the base so that fuel is leaking at a rate proportional to the depth of fuel remaining in the drum (i.e. dV/dt = - k a h, where V is volume of fuel in m 3 , h is the depth of fluid (in meters) in the drum, a is the drum radius (in meters), and k is a positive constant with units of m/s). Using the notation in the diagram, a) Obtain expressions for the depth h and volume V of the fuel remaining in the drum in terms of the angle θ and the radius a. These should be valid for 0 θ π . b) Determine the rate at which h is changing (i.e. dh/dt) as a function of a, k and θ . c) Rearrange to give dh/dt as a function of a, k and h. Space top of fuel θ h a
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4 Question 6 Find the global maximum and minimum values of the function f(x) = x - sin 2x on the interval [0, π /2], as well as local max and min values, if they exist.
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This note was uploaded on 11/05/2011 for the course CHEM 1500 taught by Professor Hameedmmirza during the Spring '11 term at York University.

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pastexams - MATH 1013 3.0 APPLIED CALCULUS I Please tick...

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