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practice-problems

practice-problems - University of Engineering and...

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University of Engineering and Technology Lahore Department of Electrical Engineering Midterm Practice Problems Calculus I January 5, 2010 Total marks: 30 Time 90 mins Instructions Give brief and concise answers. Do not needlessly fill pages. 1. Compute lim x →∞ 1 x + 1 ln x using the Sandwich theorem and other limit the- orems. 2. Let F ( x ) = x 2 - 1 | x - 1 | . Find (a) lim x 1 + F ( x ) (b) lim x 1 - F ( x ). Does the limit lim x 1 F ( x ) exist? 3. If lim x a [ f + g ] ( x ) = 2 and lim x a [ f - g ] ( x ) = 1, what is lim x a [ fg ] ( x )? 4. Show that when p > 0, lim x →∞ ln x x p = 0 5. Find a point on the parabola y = 2 x 2 that is closest to the point (1 , 4) . 6. Consider the function w defined by w ( x ) = f ( g ( h ( x ))) + k ( x ) where f , g , h and k are functions which are differentiable everywhere on the real line. Compute w 0 ( x ). 7. Compute the limit (Hint: Write d dx e x as a limit) lim x 1 e x - e x 2 - 1 8. Construct a function f , defined on all of R with the following property: For each M > 0, there exists a δ > 0 such that f ( x ) M for all x with 0 < | x | < δ . Show that this property implies that lim

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