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MATH 1310.03F A Exam 2 Thursday, November 9, 2000

NAME:

Student Number:tex2html_wrap_inline253 No.tex2html_wrap_inline255Marks tex2html_wrap_inline257


Instructions:

1. You have 50 minutes for this exam. You are not allowed to use any calculators.

2. This exam contains 3 questions and has a total of 100 marks.

3. Show all of your work. Your work must justify the answer that you give.

1. (30 Marks)Evaluate each of the following indefinite integrals.
displaymath267

Solution. (a)
eqnarray36

(b) 1. Factorizing the polynomial tex2html_wrap_inline273, we have
displaymath275

2. Get partial fraction decomposition.

Let tex2html_wrap_inline277.

Multipling the above equation by tex2html_wrap_inline279, we obtain
displaymath281

Taking x=1 in the above equation, we obtain A=1/2.

Taking x=0 in the above equation, we obtain C=-1/2.

Taking x=2 in the above equation, we obtain 5A+2B+C=1. This implies B=-1/2. Hence, we have
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3. Evaluate the indefinite integral.
eqnarray76

2. (35 Marks)Determine whether each of the following integrals converges or diverges. If it converges, find its value.
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Solution. (a) tex2html_wrap_inline305.

(b) Let tex2html_wrap_inline309. Then x=1 is a singularity. Let tex2html_wrap_inline313. Then
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We consider the following cases:

(i) When 1-p>0, tex2html_wrap_inline321 and tex2html_wrap_inline323 converges.

(ii) When 1-p<0, tex2html_wrap_inline329 and tex2html_wrap_inline323 diverges.

(iii) When p=1, tex2html_wrap_inline337. Hence, tex2html_wrap_inline339. In this case, tex2html_wrap_inline341 diverges.

tex2html_wrap_inline343 (35 Marks) Evaluate each of the following limits.


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Solution. (a) tex2html_wrap_inline349.

(b)
eqnarray189

Hence, tex2html_wrap_inline353.




Kunquan Lan
Thu Nov 9 10:46:22 EST 2000