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Midterm2009 - is discontinuous at x = 0 3 Use the deFnition...

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Calculus I - Fall 2009 Midterm Exam, October 14, 2009 In the following problems you are required to show all your work and provide the necessary explana- tions everywhere to get full credit. 1. For each of the following, decide if the limit exists. If it does, find the limit. If it does not, decide also if the limit is , -∞ , or neither. (a) lim x →− 2 x 3 - 4 x 2 + x - 1 (b) lim x 3 x 2 + 2 x - 15 x 2 - x - 6 (c) lim x 3 x - 3 x - 3
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(d) lim x 0 ( x 4 - 5 x ) cos x tan x (e) lim x →∞ cos x (f) lim x →∞ 8 x 3 - x + 2 1 - 4 x 3 (g) lim x →− 1 1 x + 1 (h) lim x →− 1 - 1 x + 1
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2. Use limits to explain why the function f ( x ) = sin x if x 0 cos x if x > 0 is discontinuous at
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Unformatted text preview: is discontinuous at x = 0 . 3. Use the deFnition of the derivative to Fnd f ( x ) = 1 x . 4. Find an equation of the tangent and normal lines to the curve f ( x ) = x sin x at the point p π 6 , π 12 P . 5. Let f ( x ) = x √ x + x 5 . Find f ′ ( x ) and f ′′ ( x ) . 6. Let f ( x ) = x 2-1 sec x + 3 . Find f ′ ( x ) . 7. Match the graph of each function in (a) - (d) with the graph of its derivative in I-IV. (a) (b) (c) (d) (I) (II) (III) (IV) ANSWER: (a) (b) (c) (d)...
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