review 4s11

# review 4s11 - MAC 2311 Test Four Review Spring 2011 Exam...

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MAC 2311 Test Four Review, Spring 2011 Exam covers Lectures 25 – 34, except lecture 29 Online Calculus Text, Chapter 4 sec. 24 and 26 – 30, plus Chapter 5, sec. 31 – 33 omit Chapter 4, sec. 25 and 29 1. On which interval(s) is f ( x ) = 1 2 x - 1 x 2 increasing and decreasing? Find all local extrema. 2. Find each interval on which the graph of f ( x ) = ( x 2 + 1) e x is both increasing and concave down. Find each inflection point. 3. Suppose that f ( x ) has horizontal tangent lines at x = - 2, x = 1 and x = 5. If f 00 ( x ) > 0 on intervals ( -∞ , 0) and (2 , ) and f 00 ( x ) < 0 on the interval (0 , 2), find the x -valuse at which f ( x ) has relative extrema. Find the x -value of each inflection point of f ( x ). Assume that f and each of its derivatives are continuous on ( -∞ , ). 4. Evaluate the following limits. Use L’Hospital’s Rule if it applies. a. lim x 0 x - ln( x + 1) 1 - sec 2 x b. lim x 0 cos 2 x e x/ 3 + 1 c. lim x 1 - ln(1 - x 2 ) ln(1 - x ) 2 d. lim x 1 (2 - x ) tan( πx/ 2) e. lim x →∞ ( e x + x ) 1 x f. lim x 0 (cot x - csc x ) 5. Find each horizontal asymptote of f ( x ) = x 2 e - x . Use L’Hospital’s Rule if necessary. Then sketch the graph of f ( x ), showing all local extrema and inflection points.

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