MAT134Y5 Study Guide - Final Guide: Asymptote

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silvermouse444

20 Jan 2019

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Mat134y5y - calculus for life sciences - utm - 2012-2013. Final exam - final answers - version a. Part a: c, d, d, b, e. Part b: a, a, b, c, d, a, d, c, e, a, d, e, a, c, b. 1. (a) y = 0 is the only horizontal asymptote. No vertical asymptotes. (b) in ection points are (cid:18) 1. , e 1/2(cid:19) : the dimensions of the rectangle of largest area are 6 for the base, and 18 for the height. 3 ln|x| + ln(x2 + 1) + c: the interval of convergence is [0 , 4) .

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3. (18 pts) MULTIPLE CHOICE ! ! ! A table of values for f(x) and f'(x) is shown below. Suppose that f is a one-to-one function and f (x) is its inverse. f' x 1 3 4 f (x) 3 4 6 (x) 4 5 3 CIRCLE THE CORRECT ANSWER IN EACH PART. (I) Evaluate f-1 (f (x)) at x = 3. (a) 1 ; (b) 3; (c) 6; (a) 4; (e) DOES NOT EXIST; (£) NONE OF THE PREVIOUS ANSWERS. (11) Evaluate e f (f (x)) at x = 3. (a) 6 ; (b) 25; (c) 5; (d) 15; (e) DOES NOT EXIST ; () NONE OF THE PREVIOUS ANSWERS. (111) Evaluate e In (f (x)) at x = 3. (e) DOES NOT EXIST; (f) NONE OF THE PREVIOUS ANSWERS. (IV) Evaluate f-1 (x) at x = 3. (a) 4; (a) 4 ; (b) 1; (b) 1 ; (com () 5; (d) 5; (e) DOES NOT EXIST; () NONE OF THE PREVIOUS ANSWERS. (v) Evaluate -? (x) at x = 3. (a) 1 ; (b) 4; (e) 5; () NONE OF THE PREVIOUS ANSWERS. (VI) Find the average rate of chnage of f over the interval (1, 3). (a) 2 ; (b)1; (c) ; (a) 5; (e) DOES NOT EXIST; () NONE OF THE PREVIOUS ANSWERS.

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.

MAT134Y5 Study Guide - Final Guide: Asymptote

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