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31 Oct 2018

3 Consider the function f (a) = * + 1 Its first and second derivatives are given by f'(x) = - by fee_*2 +1 (2x2 - 1) and and I" (2) - 20 (x2 + 3) (221) B. Find the intervals where the function is increasing, decreasing. Write DNE if they don't exist.

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1 Nov 2018
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Long Problems. In questions 2 - 6, show your work. No credit will be given for the answer without the correct accompanying work. [14] 2. Consider the function f(3) = -2 Its first and second derivatives are given by 82 f' (t) =(22 – 4)2 8(3.+ 4) (x2 - 4 f"(c) = 1 + 4) (a) Find all z such that f'(x) = 0 or f'(x) does not exist.

4. For the function 22 - 1 f(0) = 72-4 determine all of the following if they are present: (i) critical points (where f'(x) = 0 or f'(x) does not exist), local maxima and minima, intervals where f(x) is increasing or decreasing; (ii) inflection points and intervals where f(2) is concave upward or downward; (iii) asymptotes (horizontal, vertical). Sketch the graph of y = f(c), giving the (x,y) coordinates for all of the points of interest above. Please make your sketch big enough to see clearly all features of interest. 6(3.x2 + 4) You may use, without demonstrating it, the fact that f"(x) = ? (x2 – 4)3

1. (18 pts) The graph of a function f, defined on (-0, +o), is given in the figure below. - - y = f(x) - - - - - - N - - 4 A -2-1 144 -1 - - - - - - - -- - - -- Use the graph of f to answer questions (a)-(f) below. (a) Determine the value or write "does not exist". i. f(-3) = -2 ii. f(0) = 1 (b) Determine the limit or write "does not exist" (DNE). Write "does not exist" only if a limit does not exist and is not too or -0. i. lim f(x) = 0 iv. lim f(x) = +00 X- 3 x2 ii. lim f(x) = -4 v. lim f(x) = -0 X- 0- x -00 iii. lim f(x) = DNE vi. lim f(x) = -2 x0 x++00 (C) MULTIPLE CHOICE! Circle ONLY one answer! Find a number a for which the following statement is true (the function f is shown above): STATEMENT: The function f is continuous at a, and lim f(x) = 0. i. a=-3 iii. a=0 V. a=3 ii. a = -2 iv. a = 2 vi. Such a number does not exist. (d) Determine the largest intervals of continuity of f. Use interval notation to write your answer. (-0, -3), (-3,0), (0, 2), (2, +00) (e) Write the equation(s) of all vertical asymptotes of f. Write "none" if appropriate. x= 2 (f) Write the equation(s) of all horizontal asymptotes of f. Write "none" if appropriate. y = -2

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