MATH100 Study Guide - Midterm Guide: Mean Value Theorem

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turquoisegnu128

24 Oct 2018

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University of Alberta

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Mathematics

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MATH100

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MATH100 Full Course Notes

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4 x2 (cid:0) 4(cid:19) and (b) lim x!0(cid:18)j3x (cid:0) 1j (cid:0) j3x + 1j x (cid:19) : Solution for (a): the (cid:147)test(cid:148)gives 1 x!2(cid:18) 1 (a) lim x (cid:0) 2 (cid:0) x2 (cid:0) 4(cid:19) = lim (cid:19) = lim x!2. 4 lim x (cid:0) 2 (cid:0) x!2(cid:18) 1 x!0(cid:18)j3x (cid:0) 1j (cid:0) j3x + 1j lim x. 0 = ? so further work is needed. 0 (cid:0) 1 x + 2 (cid:0) 4 x2 (cid:0) 4. 0 = ? so further work is needed. x (cid:0) 2 x2 (cid:0) 4. 4 x!0(cid:18)(cid:0) (3x (cid:0) 1) (cid:0) (3x + 1) x (cid:19) = lim x!0 (cid:0)6x x. Problem 2: determine the horizontal and vertical asymptotes for. For the vertical asymptotes we need to solve px2 (cid:0) 2x (cid:0) x = 0 ()px2 (cid:0) 2x = x () x2 (cid:0) 2x = x2 =) x = 0:

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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)

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

MATH100 Study Guide - Midterm Guide: Mean Value Theorem

MATH100 Full Course Notes

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