2003 Fall - Fall_2003 math 125

2003 Fall - Fall_2003 math 125 - Math 125 - Final(Common) -...

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Math 125 - Final(Common) - Fall 2003 1. [20 points] Find the limits, if they exist, of the following expressions (you may not use L’Hospital’s rule). 1a. lim x 0 x cot x 1b. lim x →∞ ( x - x ) 1c. lim x 0 1 2+ x - 1 2 x . 2. [20 points] Consider f ( x ) = x 2 - x | x - 1 | if x 6 = 1 2 if x = 1 . 2a. Sketch the graph of f ( x ). 2b. Find the numbers at which f ( x ) is discontinuous . If none, write NONE. You must justify your answer. 3. [15 points] If a snowball melts so that its surface area ( S = 4 πr 2 ) decreases at a rate of 1 cm 2 /min , find the rate at which the radius decreases when the radius is 10 cm . 4a. [10 points] Find the linear approximation to f ( x ) = e x near x = 0. 4b. [5 points] Sketch the graph of f and its linear approximation in the x interval of [ - 1 , 1]. 5. Find the derivatives of the following functions: 5a. [5 points] f ( x ) = x 3 + 1 x 2 + 5 5b. [5 points] f ( x ) = 3 e 2 x 2 +1 5c. [5 points] f ( x ) = ln (1 + x 2
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2003 Fall - Fall_2003 math 125 - Math 125 - Final(Common) -...

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