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Unformatted text preview: Betancourt, Daniel – Exam 2 – Due: Mar 27 2007, 11:00 pm – Inst: Diane Radin 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Determine if lim x →∞ ‡ p x 2 + 2 x · exists, and if it does, find its value. 1. limit doesn’t exist 2. limit = 0 correct 3. limit = √ 2 4. limit = 1 5. limit = √ 3 Explanation: After the rationalization p x 2 + 2 x = ( x 2 + 2) x 2 √ x 2 + 2 + x , we see that p x 2 + 2 x = 2 √ x 2 + 2 + x . On the other hand, lim x →∞ 1 √ x 2 + 2 + x = 0 . Consequently, lim x →∞ ‡ p x 2 + 2 x · exists and limit = 0 . keywords: limit, limit at infinity, square root function, rationalize numerator 002 (part 1 of 1) 10 points Determine if the limit lim x →∞ 1 + 2 x 3 x 3 3 + 5 x 3 exists, and if it does, find its value. 1. limit = 1 3 2. limit = 1 3 3. limit = 3 5 4. limit = 3 5 correct 5. limit does not exist Explanation: Dividing by x 3 in the numerator and de nominator we see that 1 + 2 x 3 x 3 3 + 5 x 3 = 1 x 3 + 2 x 2 3 3 x 3 + 5 . With s = 1 x , therefore, lim x →∞ 1 + 2 x 3 x 3 3 + 5 x 3 = lim s → s 3 + 2 s 2 3 3 s 3 + 5 . Consequently, the limit exists, and limit = 3 5 . keywords: limit, limit at infinity, asymptote 003 (part 1 of 1) 10 points The figure below shows the graphs of three functions: Betancourt, Daniel – Exam 2 – Due: Mar 27 2007, 11:00 pm – Inst: Diane Radin 2 One is the graph of a function f , one is its derivative f , and one is its second derivative f 00 . Identify which graph goes with which function. 1. f : f : f 00 : correct 2. f : f : f 00 : 3. f : f : f 00 : 4. f : f : f 00 : 5. f : f : f 00 : 6. f : f : f 00 : Explanation: Calculus tells us that f (i) has horizontal tangent at ( x ,f ( x )) when f crosses the xaxis, (ii) is increasing when f > 0, and (iii) is decreasing when f < 0, (iv) has a local max at x when f ( x ) = 0 and f 00 ( x ) < 0, (v) has a local min at x when f ( x ) = 0 and f 00 ( x ) > 0, (vi) is concave up when f 00 > 0, (v) and concave down when f 00 < 0. Inspection of the graphs thus shows that f : f : f 00 : . keywords: graph, local minimum, local maxi mum, concavity 004 (part 1 of 1) 10 points Find the derivative of f when f ( x ) = 4 x cos 5 x. 1. f ( x ) = 4 cos 5 x 20 x sin 5 x correct 2. f ( x ) = 4 cos 5 x + 20 x sin 4 x 3. f ( x ) = 4 cos 4 x 4 x sin 5 x 4. f ( x ) = 20 cos 5 x + 5 x sin 5 x 5. f ( x ) = 20 cos 5 x 4 x sin 5 x Explanation: Using the formulas for the derivatives of sine and cosine together with the Chain Rule we see that f ( x ) = (4 x ) cos 5 x + 4 x (cos 5 x ) = 4 cos 5 x 20 x sin 5 x....
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 Fall '09
 Gualdani
 Derivative, Betancourt, Diane Radin, Daniel Exam

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