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# SampleProblems - AUBG Spring 2009 MAT 103c Calculus I...

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AUBG, Spring 2009, MAT 103c: Calculus I Midterm Exam Sample Problems Problem 1. Express the limit as a derivative and evaluate. (a) lim x 1 x 17 - 1 x - 1 (b) lim h 0 4 16 + h - 2 h (c) lim x π/ 3 cos x - 0 . 5 x - π/ 3 Answers. (a) 17 (b) 1 32 (c) - 3 2 Problem 2. If f is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x 2 f ( x ) (b) y = f ( x ) x 2 (c) y = x 2 f ( x ) Answers. (a) y 0 = 2 xf ( x ) + x 2 f 0 ( x ) (b) y 0 = f 0 ( x ) x 2 - 2 xf ( x ) x 4 (c) y 0 = 2 xf ( x ) - x 2 f 0 ( x ) f ( x ) 2 Problem 3. Calculate y 0 . (a) y = x + 1 3 x 4 (b) y = sin mx x (c) y = 2 x p x 2 + 1 (d) y = x 1 - x 2 (e) y = cos(tan x ) (f) y = sin(cos x ) (g) y = sin(tan p 1 + x 3 ) (h) y = (1 - x - 1 ) - 1 Answers. (a) y 0 = 1 2 x - 4 3 x 2 3 x (b) y 0 = mx cos mx - sin mx x 2 (c) y 0 = 2(1 + 2 x 2 ) x 2 + 1 (d) y 0 = 1 + x 2 (1 - x 2 ) 2 (e) y 0 = - sin(tan x ) cos 2 x (f) y 0 = - cos(cos x ) sin x (g) y 0 = 3 x 2 cos(tan 1 + x 3 ) 2 1 + x 3 cos 2 1 + x 3 (h) y 0 = - 1 ( x - 1) 2 Problem 4. Find equations of both lines through the point (2 , - 3) that are tangent to the parabola y = x 2 + x .

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