MATH 124 Midterm: MATH 124 UW 124 Winter 13 Sol1 v1

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turquoisegnu128

31 Jan 2019

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$MATH 124 Full Course Notes$

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Math 124, sections e and f, winter 2013, solutions to midterm i: compute the following limits and derivatives. (a) (b) (c) + 3x2 e2x(cid:19) = dx (cid:18)5 x dx (cid:0)x cos(cid:0)x2(cid:1)(cid:1) = cos(cid:0)x2(cid:1) 2x2 sin(cid:0)x2(cid:1) Note that when x < 0, we have x = . 7 x2 + 6x 2e2x x2 q1 + 4. 5 x2 3x x2 4x + 3 (d) lim x 3. = lim x 3: answer the following questions about the function (x 3)(x 1) Yes, and lim x 0 f (x) = lim x 0+ 2 lim x 0+ f (x) = lim x 0 x + 4 x + 8. = f (0). (d) (2 points) is f di erentiable at x = 0? d dx (cid:18) 1 x + 2(cid:19) = . 1 (x + 2)2 at x = 0 the value is -1/4. d dx (cid:18) x + 4 x + 8(cid:19) =

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4. (16 pts) MULTIPLE CHOICE! CIRCLE THE CORRECT ANSWER IN EACH PART. (I) ( 4 pts) Complete the statement of the Intermediate Value Theorem: Suppose f is continuous on the interval [a, b] and L is a number between f (a) and f (b). Then there is at least one number c in (a, b) such that (a) C = L; (b) f(c) = 0; (c) f(c) = L; (a) f' (c) = L; (e) C = 0; (f) f (a) < c < f (b); (g) NONE OF THE PREVIOUS ANSWERS. (II) (4 pts) The figure shows the graph of a function f. At what point (points) c does the conclusion of the Intermediate Value Theorem hold for f (x) on the interval [1, 5] and L = 2. --- - - - -1 - - - 2 - - I- - - (a) c = 1; (b) C = 0; (c) C = 2; (a) c = 3; (e) C = 4; (f) C = 5; (g) c= -1. (III) (4 pts) Given that cos x s f (x) s et, for all x > 0 evaluate lim f (x) x+0+ (a) lim f (x) = 0; (b) lim f (x) = 1; (c) lim f (x) = e; x0+ X0+ (d) lim f(x) DOES NOT EXIST. WE DON'T HAVE ENOUGH INFORMATION TO ANSWER THE QUESTION *40+ (IV) (4 pts) Given the function 1 f (x) = , find its inverse, f+(x) x -2 (a) f (x) = x -2; (b) f«+ (x) = x+2 ; (c) fº+ (x) = -2; (f) f-+ (x) -2; (e) f (x) DOES NOT EXIST.

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

Answer all 3 questions and show all work

1) Find Lim Square rootof 1+x -1 / x

x> 0

2) Find Lim f(x) if, f (x)= X+2 X<-1

x^2+2x+3 X>-1

3) Find Lim x^2 * sin 1/x

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MATH 124 Midterm: MATH 124 UW 124 Winter 13 Sol1 v1

MATH 124 Full Course Notes

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