MAT 1322 Study Guide - Final Guide: Improper Integral, Partial Fraction Decomposition, Summation

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turquoisegnu128

24 Oct 2018

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Total = 53 marks: (4 marks) determine whether each of the following improper integrals is convergent or divergent. If it is convergent, find its value; if it is divergent, use appropriate method to justify your conclusion: (a) 3 lim ln b dx lim b ln 3. Since x3 x2 in the interval [0, 1], The cross sections of the solid perpendicular to the x-axis are semicircles with diameter on the x-y plane. The area of the cross section is a(x) = p (1 - 15: (4 marks) find the length of the arc y = The length of the arc is l = Find the work (in joules) needed to lift the stone from the ground: (5 marks) a monument of the shape of a truncated circular cone is built with stone of density. A layer of the monument at height h of thickness dh weights dw = dgp (3 -

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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-

CALCULUS 2: Please solve numbers 9 and 10, if you can solve 11 and 12 also and fit it in the answer board. That would be great. Please show step by step solution.

CALCULUS 2: Please solve numbers 4 and 5 all parts. Show step by step solution please.

MAT 1322 Study Guide - Final Guide: Improper Integral, Partial Fraction Decomposition, Summation

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