(1) In the space provided, sketch the graph of a function f satisfying the properties below. -00 • f is defined and differentiable for all I* 2. • f(0) = 0 . f has vertical asymptotes at I=+2 . f has horizontal asymptote y=0 as 17 00, and y=1 as 1 • f is increasing on (-2,2) . f is decreasing on (-00,-2) and (2,00) • f is concave up on (0, 2) and (2.00) • f is concave down on (-2,0) and (-00,--2)

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Nestor RutherfordLv2

17 Aug 2018

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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

1. (25 pts) The graph of a function f is given in the figure below. The domain of fis (-, +.). Use the graph of fto answer the questions below. yax) y=f(x) -3 - 2 3 4 (I) (1 pt) Find the range of f. (II) Find the following values. (Note: Possible answers include + , -, or "does not exist".) (a) (1 pt) (x) = 5 (b) (1 pt) lim f (x) = 5 (0) (1 pt) Tim f (x) = 5 (d) (1 pt) f(3) = 2 (e) (2 pts) lim f (x) = + 0 x-2- (8) (2 pts) 11m = (x) = 0 (g) (2 pts) lim f (x) = DOES NOT EXIST x+-2 (h) (2 pts) Lim f (x) = 0 (i) (2 pts) lim f (x) = + X +00 (III) (2 pts) Find all vertical asymptotes. x = -2 (IV) (2 pts) Find all horizontal asymptotes. y = 0 (v) (2 pts) Find all slant asymptotes. | y = x + 2 (VI) (4 pts) Determine the intervals of continuity for f. (-00, -2), [-2, 3), and (3, +c)

3. (a) Assume that a function f(x) is concave down on its whole domain, and f(0) = -2 and f'(0) = -1. Circle each statement that must be true: • The function f(2) has no inflection points. • The derivative f'(2) is always negative. • The derivative satisfies f'(3) < -1. • The derivative satisfies f'(-3) < -1. • The function f(x) <-2 for all 1 >0. (b) Sketch a graph of a function f() satisfying all of the following conditions: • The derivative function f'(x) is decreasing when x < 1 and when r > 3. • The derivative function f'(2) is increasing when 1

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