final104-2000

final104-2000 - 69.104 Final Exam December 2000 1...

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Unformatted text preview: 69.104 Final Exam December 2000 1 Multiple-Choice Questions Please circle only one answer. 1. [2 marks] Let f ( x ) = x 3 + x . Evaluate f (1). In other words, find the derivative of f at x = 1. (a) f (1) = 0 (b) f (1) = 1 . 5 (c) f (1) = 3 . 5 (d) f (1) = 6 2. [2 marks] Let f ( x ) = ln( x ). Evaluate f (2). In other words, find the second derivative of f at x = 2. (a) f (2) = 1 8 (b) f (2) = 0 (c) f (2) = 1 (d) f (2) = 3 4 3. [2 marks] Let f ( x ) = | x 1 | . Calculate L = lim h f (1 + h ) f (1) h . (a) L = 0 (b) This limit does not exist (c) L = 1 (d) L = 1 4. [2 marks] Evaluate the following limit: L = lim x x sin 1 x . (a) L = 1 (b) L = 4 (c) L = 2 (d) L = 1 5. [2 marks] A differentiable function f has the property that f (1) = 6, f (6) = 2 and f (1) = 3. What is the value of the derivative of f ( f ( x )) at x = 1? (a) 3 (b) 6 (c) x (d) 2 2 69.104 Final Exam December 2000 6. [2 marks] Let f ( x ) = tan(1 + sin x ). Evaluate f ( x ). In other words, find the derivative of f at x . (a) f ( x ) = sin x sec 2 (cos x ) (b) f ( x ) = sec 2 (1 + sin x ) (c) f ( x ) = sec 2 (sin x ) (d) f ( x ) = cos x sec 2 (1 + sin x ) 7. [2 marks] Let y be given implicitly as a differentiable function of x by x 2 cos y + y 2 1 = 0. Then the slope of the tangent line to the curve y = y ( x ) at the point ( x, y ) where x = 0, y = 1 is equal to: (a) 2, (b) + , (c) 1 2 , (d) 0....
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final104-2000 - 69.104 Final Exam December 2000 1...

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