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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (g)

2012-05-05T15:25:55Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (g)/Hint 1

2012-05-05T15:24:56Z

IsabellGraf:

Recall the taylor series of <math> \cos(x)\ </math> and use just the first two terms for a simplification of the integrand.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (g)/Hint 1

2012-05-05T15:24:14Z

IsabellGraf: Created page with "Recall the taylor series of <math> \cos(x)\ </math> and use just the first two terms for a simplification."

Recall the taylor series of <math> \cos(x)\ </math> and use just the first two terms for a simplification.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)

2012-05-05T15:20:46Z

IsabellGraf:

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Solution 1

2012-05-05T15:19:29Z

IsabellGraf:

Recall that

:<math>
\log(1-x) = -x -\frac{x^2}2 -\frac{x^3}3 - \cdots,
</math>

therefore

:<math>
x\log(1-x)= -x^2 - \frac{x^3}2 - \frac{x^4} 3-\cdots
</math>

The coefficient of ''x''<sup>3</sup> term is '''-1/2'''.

Answer '''D'''.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Solution 1

2012-05-05T15:19:05Z

IsabellGraf:

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Hint 1

2012-05-05T15:18:10Z

IsabellGraf: Created page with "Recall the taylor series for <math> \log(1-x) \ </math> and multiply it with <math>\ x</math>."

Recall the taylor series for <math> \log(1-x) \ </math> and multiply it with <math>\ x</math>.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (e)

2012-05-05T15:14:01Z

IsabellGraf:

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (e)/Hint 2

2012-05-05T15:09:25Z

IsabellGraf: Created page with "Then use integration by parts another time."

Then use integration by parts another time.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (e)/Hint 1

2012-05-05T15:09:02Z

IsabellGraf: Created page with "Use integration by parts."

Use integration by parts.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)

2012-05-05T15:06:04Z

IsabellGraf:

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)/Hint 2

2012-05-05T15:04:40Z

IsabellGraf: Created page with "Then use the substitution <math>\ u=\cos(x). </math>"

Then use the substitution

<math>\ u=\cos(x). </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)/Hint 1

2012-05-05T15:04:12Z

IsabellGraf:

Use the formula

<math>\ \sin^2(x) + \cos^2(x)=1.</math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)/Hint 1

2012-05-05T15:03:57Z

IsabellGraf: Created page with "Use the formula <math> \sin^2(x) + \cos^2(x)=1.</math>"

Use the formula

<math> \sin^2(x) + \cos^2(x)=1.</math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (c)

2012-05-05T14:55:16Z

IsabellGraf:

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (c)/Hint 1

2012-05-05T14:54:47Z

IsabellGraf: Created page with "First use the integral comparison test. Then use the integral ''p'' test."

First use the integral comparison test.

Then use the integral ''p'' test.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)

2012-05-05T14:43:13Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)/Hint 2

2012-05-05T14:42:47Z

IsabellGraf: Created page with "If you do not know how to use the integral ''p'' test, estimate the integral with a lower bound first."

If you do not know how to use the integral ''p'' test, estimate the integral with a lower bound first.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)/Hint 1

2012-05-05T14:38:07Z

IsabellGraf: Created page with "Use the integral ''p'' test."

Use the integral ''p'' test.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)/Solution 1

2012-05-05T14:36:49Z

IsabellGraf:

The difference between the highest exponents is 3-2 = 1. By the intergral ''p'' test (with ''p'' = 1) this integral diverges.

More precisely we estimate the integral with

<math> \int_1^\infty\frac{x^2}{x^3 + 2x + 5}dx > \int_1^\infty\frac{x^2}{x^3 + 2x^3 + 5x^3}dx = \frac18\int_1^\infty\frac1xdx </math>

Here we see exactly how to use the integral ''p'' test. We also can continue with the anti-derivative

<math> \frac18\int_1^\infty\frac1xdx = \frac18\lim_{\alpha\rightarrow\infty}[\text{ln}(x)]_1^\alpha = \frac18\lim_{\alpha\rightarrow\infty}\text{ln}(\alpha)=\infty. </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)/Solution 1

2012-05-05T14:31:14Z

IsabellGraf:

User:IsabellGraf

2012-05-05T14:22:15Z

IsabellGraf: Created page with "Visiting scholar from Augsburg, Germany"

Visiting scholar from Augsburg, Germany

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (a)

2012-05-05T14:19:41Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (a)/Hint 2

2012-05-05T14:19:12Z

IsabellGraf: Created page with "If you do not see a possibility to use the series ''p'' test, try to estimate the sum downstairs."

If you do not see a possibility to use the series ''p'' test, try to estimate the sum downstairs.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (a)/Hint 1

2012-05-05T14:17:52Z

IsabellGraf: Created page with "Use the series ''p'' test."

Use the series ''p'' test.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (a)/Solution 1

2012-05-05T14:16:14Z

IsabellGraf:

The difference between the highest exponents is 1.1-0.1 = 1. By the series ''p'' test (with ''p'' = 1) this series '''diverges'''.

More precisely we can prove

<math> \sum_{n=1}^\infty\frac{n^{0.1}}{n^{0.99} + n^{1.1}+1} > \sum_{n=1}^\infty \frac{n^{0.1}}{n^{1.1} + n^{1.1} + n^{1.1}} = \sum_{n=1}^\infty\frac{n^{0.1}}{3n^{1.1}} = \frac13\sum_{n=1}^\infty\frac1n=\infty</math>

The last sum diverges because of the series ''p'' test with ''p=1''.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (a)/Solution 1

2012-05-05T14:15:25Z

IsabellGraf:

The difference between the highest exponents is 1.1-0.1 = 1. By the series ''p'' test (with ''p'' = 1) this series '''diverges'''.

More precisely we can prove

<math> \sum_{n=1}^\infty\frac{n^{0.1}}{n^{0.99} + n^{1.1}+1} > \sum_{n=1}^\infty \frac{n^{0.1}}{n^{1.1} + n^{1.1} + n^{1.1}} = \sum_{n=1}^\infty\frac{n^{0.1}}{3n^{1.1}} = \frac13\sum_{n=1}^\infty\frac1n=\infty</math>
The last sum diverges because of the series ''p'' test with ''p=1''.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (g)

2012-04-26T00:09:40Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (g)/Statement

2012-04-26T00:09:20Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

Which of the following is the closest approximation to
the value of the integral <math> \int_0^{1} \frac{1-\cos x} x dx </math>? (Hint: use Taylor series)

A. <math>\ 1 </math>

B. <math> \frac14</math>

C. <math>\frac12 </math>

D. <math> \frac56</math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (g)/Statement

2012-04-26T00:08:54Z

IsabellGraf: Created page with "Answer the following multiple choice questions. Check your answer very carefully. Your answer will be marked right or wrong (work will not be considered for this problem). W..."

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

Which of the following is closest approximation to
the value of the integral <math> \int_0^{1} \frac{1-\cos x} x dx </math>? (Hint: use Taylor series)

A. <math>\ 1 </math>

B. <math> \frac14</math>

C. <math>\frac12 </math>

D. <math> \frac56</math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)

2012-04-26T00:07:08Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Statement

2012-04-26T00:06:47Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

In the Taylor series of <math>\ x\log(1-x) </math> (about the point <math>\ x=0</math>), the coefficient
of the <math>\ x^3 </math> term is

A. <math> \frac13 </math>

B. <math> -\frac13 </math>

C. <math>\ \frac 12 </math>

D. <math> -\frac12 </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Statement

2012-04-26T00:06:35Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

In the Taylor series of <math>\ x\log(1-x) </math> (about the point <math>\ x=0</math>), the coefficient
of the <math>\ x^3 </math> term is

A. <math>: \frac13 </math>

B. <math> -\frac13 </math>

C. <math>\ \frac 12 </math>

D. <math> -\frac12 </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Statement

2012-04-26T00:06:19Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

In the Taylor series of <math>\ x\log(1-x) </math> (about the point <math>\ x=0</math>), the coefficient
of the <math>\ x^3 </math> term is

A. <math>\ \frac13 </math>

B. <math> -\frac13 </math>

C. <math>\ \frac 12 </math>

D. <math> -\frac12 </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Statement

2012-04-26T00:05:49Z

IsabellGraf:

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Statement

2012-04-26T00:05:02Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

In the Taylor series of <math>\ x\log(1-x) </math> (about the point <math>\ x=0</math>), the coefficient
of the <math>\ x^3 </math> term is

A. <math>\ 1 </math>

B. <math> \frac14 </math>

C. <math> \frac 12 </math>

D. <math> \frac56 </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (f)/Statement

2012-04-26T00:04:49Z

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

In the Taylor series of <math>\ x\log(1-x) </math> (about the point <math>\ x=0</math>), the coefficient
of the <math>\ x^3 </math> term is

A. <math> 1 </math>

B. <math> \frac14 </math>

C. <math> \frac 12 </math>

D. <math> \frac56 </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (e)

2012-04-26T00:02:08Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (e)/Statement

2012-04-26T00:01:51Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

The value of the integral <math> \int_0^{\infty} x^2 e^{-x} dx </math> is

A. <math> e-1 </math>

B. <math> 2e-1 </math>

C. <math> 0.5e </math>

D. <math> 1 </math>

E. <math> 2 </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (e)/Statement

2012-04-26T00:01:21Z

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)

2012-04-25T23:58:49Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)/Statement

2012-04-25T23:58:24Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

The value of the integral <math> \int_0^\pi \sin^3 x dx </math> is

A. <math> 0\ </math>

B. <math> 1\ </math>

C. <math> \frac{\pi}{2}\ </math>

D. <math> \pi\ </math>

E. <math> \frac43.\ </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)/Statement

2012-04-25T23:57:35Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

The value of the integral <math> \int_0^\pi \sin^3 x dx </math> is

A. <math> 0,\ </math> B. <math> 1,\ </math> C. <math> \frac{\pi}{2},\ </math> D. <math> \pi,\ </math> E. <math> \frac43.\ </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (d)/Statement

2012-04-25T23:56:34Z

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

The value of the integral <math> \int_0^\pi \sin^3 x dx </math> is

A. <math> 0\ </math>

B. <math> 1\ </math>

C. <math> \frac{\pi}{2}\ </math>

D. <math> \pi\ </math>

E. <math> \frac43\ </math>

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (c)

2012-04-25T23:54:30Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (c)/Statement

2012-04-25T23:53:54Z

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

Does the integral <math> \int_1^{\infty} \frac{1+\sin x} {x^{2}} dx </math> converge?

A. Converge

B. Diverge.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)

2012-04-25T23:53:04Z

IsabellGraf:

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Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)/Statement

2012-04-25T23:52:46Z

IsabellGraf:

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

Does the integral <math> \int_1^{\infty} \frac {x^2}{x^3+2x+5} dx </math> converge?

A. Converge.

B. Diverge.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)/Statement

2012-04-25T23:52:33Z

Answer the following multiple choice questions. Check
your answer very carefully. Your answer will be marked right or
wrong (work will not be considered for this problem).

Does the integral <math> \int_1^{\infty} \frac {x^2}{x^3+2x+5} dx </math> converge?

A. Converge.

B.Diverge.

Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (a)

2012-04-25T23:51:12Z

IsabellGraf:

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