MATH 125g Study Guide - Midterm Guide: Inverse Function

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viridianrhinoceros717

4 Aug 2020

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You must show your work to obtain full credit. Points may be deducted if you do not justify your nal answer. Please indicate clearly whenever you continue your work on the back of the page. The exam is worth a total of 200 points. [24 points] in each case, evaluate the limit if it exists, including . If no limit exists, explain why. (a) lim x 3 (b) lim x 0 (c) lim x 1 (d) lim x x2 4x + 3 x2 + 4x 21 sin x. 0: [30 points] evaluate the following integrals. 0 (a) z 1 (b) z 3 (c) z (3x 1)4 dx. 0 |x2 1| dx cos x (1 + sin x)2 dx (d) z e2x 1 + ex dx (e) z. 1 dx: [16 points] (i) give the de nition of: f is di erentiable at a. (ii) use the de nition to show that the function g(x) = x2 sin(1/x)

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[ -2 e* if x< 0 3. (24 pts) Let g(x) = 1 X+6 if x20. [x - 3 (I) Evaluate the limit. (Note: Possible answers include + or - .) [You may not use a table of values, a graph, or L'Hospitals's Rule to justify your answer. ] SHOW YOUR WORK. (a) (3 pts) lim g(x) = ; (b) (3 pts) lim g(x) = x 10 x-10+ (c) (3 pts) lim g(x) = ; (d) (3 pts) lim g(x) = 14 = * 3+ (e) (3 pts) lim g(x) = ; (f) (3 pts) lim g(x) = X + (II) (6 pts) Using the definition of continuity, show that gis continuous at 0. SHOW YOUR WORK.

3. (21 pts) Evaluate the limit or say that the limit does not exist. If a limit does not exist, explain why. Show your work. You may NOT use a table of values, a graph, or L'Hospitals's Rule to justify your answer. (Note: Possible answers include + or -..) (a) (7 pts) x2 - 2x - 3 - lim x+3 Vx+1 - 2 x2 - 2x - 3 Vx+ 1 + 2 lim x+3 Vx+1 -2 Vx+1 +2 (x - 3) (x + 1) (Vx+ = lim x+3 x + 1 - 4 = lim x+3 (x - 3) (x + 1) (Vx+1 +2). -- x-3 = lin (x + 1) (Vx+1 +2) = (3 + 2) (V3+1 +2) = = 4 (V4 +2) = 16 (b) (7 pts) lim xsin (In x) X >0+ NOTE: We cannot apply the Product Law, since lim sin (1n x) DOES NOT EXIST. On the other hand, -1s sin (in x) s 1 and , therefore, -xs x sin (in x) sx, (since x > 0). Since lim X = lim (-X) = 0, x0+ X0 the inequality above implies that lim xsin (in x) = 0, by the Squeeze Theorem . x0+ EXPLANATION : Since lim ln x = -0, x →0+ sin (lnx) oscillates between 1 and -1 as x + 0*, and, therefore lim sin (In x) DOES NOT EXIST. x >0+ (c) (7 pts) in x+2- (lnx) +In 20 -= -00 (x - 2) 70-

"Find the domain" for #23 please and thank you.

MATH 125g Study Guide - Midterm Guide: Inverse Function

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