MTH 142 Midterm: MTH 142 University of Rochester 2003 Spring Midterm1 solutions

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turquoisegnu128

31 Jan 2019

School

University of Rochester

Department

Mathematics

Course

MTH 142

Professor

All

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Document Summary

February 20, 2003: (15 pts) let f (x) = (a) find the domain of f . 8x2 + 4 (1 x2)3 . x 6= 1, i. e. ( , 1) ( 1, 1) (1, ) (b) find all horizontal and vertical asymptotes of f . There are vertical asymptotes are x = 1 and x = 1, since and lim x 1+ There is a horizontal asymptote at y = 1, since and lim x . 4x (1 x2)2 and f (x) = 8x2 + 4 (1 x2)3 . (c) find the intervals on which f is increasing and the intervals on which f is decreasing. The denominator of f (x) is always positive. Therefore f (x) is increasing if x > 0, and f (x) is decreasing if x < 0. (d) find all local extrema for f . f (x) = 0 only when x = 0.

Related Questions

4. (21 pts) Let g be the function given by ( 6 - X 2 + x if x < 2, x + -2 g(x) = x2 – x-1 if x>2. (a) (6 pts) Use the definition of continuity to determine whether the function g is continuous at 2. Show your work. (b) (3 pts) Evaluate the limit. Justify your answer. Show your work. Write "does not exist" only if the limit does not exist and is not + or -0. lim g(x) = lim -2+ = +00 X 2 + X since the form of the limit is NUM: POS, DENOM: POS (c) (2 pts) Write the equation(s) of all vertical asymptotes of g. Write "none" if appropriate. The result in part (b) implies that the line x= -2 is a vertical asymptote. This is the only vertical asymptote, since the function g is continuous on its domain (d) (2 pts) Determine the largest) intervals of continuity of g. Use interval notation to write your answer. Let g be the function given by 12+ if x < 2, x# -2 2 + x g(x)= x2-x-1 x-1 if x>2. (e) (6 pts) Evaluate each limit. Do not use L'Hôpital's Rule. Write "does not exist" only if the limit does not exist and is not too or -00. Show your work. 6-1 i lim g(x) = lim -00 2 + X = lim -002 + x 0-1 + 1 x -00 x x x -+- 2 +1 0 0 + 11 x2 - x-1 1 ii. lim g(x) = lim x++ V x2 – x-1 X-1 x2 - x-1 X - 1 = lim x +oc x +0 - lim Vam M93001 (f) (2 pts) Write the equation(s) of all horizontal asymptotes of g. Write "none" if appropriate.

MTH 142 Midterm: MTH 142 University of Rochester 2003 Spring Midterm1 solutions

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