MAC 2311 Midterm: MAC2311 S97 Test 2

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turquoisegnu128

31 Jan 2019

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Instructions: do not write your answers on this paper, use separate answer sheets. To receive credit, you must show all work. You are not required to simplify your answers. (1) (10 pts. ) Find a value for a that makes the function continuous f (x) =(sin x. A cos x if x 0, if x > 0. (3) (10 pts. ) Verify that f (x) = x3 + 5x2 3x 1 has a root between 0 and 1. (4) (10 pts. ) Analyze the global behavior of each rational function f de ned below. If the graph has a horizontal asymptote, give its equation y = c. if the graph has no horizontal asymptote, nd cxk such that f (x) cxk for |x| large. (a) f (x) = 7x8 x3 + x + 1 (b) f (x) = x7 3x4 + 1. This is the graph of y = f (x): y. Sketch the graph of y = f (x). (6) (15 pts. )

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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.

4. (21 pts) Let g be the function given by ( 6 - X 2 + x if x < 2, x + -2 g(x) = x2 – x-1 if x>2. (a) (6 pts) Use the definition of continuity to determine whether the function g is continuous at 2. Show your work. (b) (3 pts) Evaluate the limit. Justify your answer. Show your work. Write "does not exist" only if the limit does not exist and is not + or -0. lim g(x) = lim -2+ = +00 X 2 + X since the form of the limit is NUM: POS, DENOM: POS (c) (2 pts) Write the equation(s) of all vertical asymptotes of g. Write "none" if appropriate. The result in part (b) implies that the line x= -2 is a vertical asymptote. This is the only vertical asymptote, since the function g is continuous on its domain (d) (2 pts) Determine the largest) intervals of continuity of g. Use interval notation to write your answer. Let g be the function given by 12+ if x < 2, x# -2 2 + x g(x)= x2-x-1 x-1 if x>2. (e) (6 pts) Evaluate each limit. Do not use L'Hôpital's Rule. Write "does not exist" only if the limit does not exist and is not too or -00. Show your work. 6-1 i lim g(x) = lim -00 2 + X = lim -002 + x 0-1 + 1 x -00 x x x -+- 2 +1 0 0 + 11 x2 - x-1 1 ii. lim g(x) = lim x++ V x2 – x-1 X-1 x2 - x-1 X - 1 = lim x +oc x +0 - lim Vam M93001 (f) (2 pts) Write the equation(s) of all horizontal asymptotes of g. Write "none" if appropriate.

These topics are related with 1st year university Calculus Math. Could you help to calculate the Question ( 9 ) for solution steps? Thanks lots.

MAC 2311 Midterm: MAC2311 S97 Test 2

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