009C Sample Final 2 - Revision history

MathAdmin: Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.''' '''Click on the
2017-03-12T16:45:46Z

Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.''' '''Click on the <span class..."

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'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.'''

'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''
<div class="noautonum">TOC</div>

== [[009C_Sample Final 2,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] ==
<span class="exam">Test if the following sequences converge or diverge. Also find the limit of each convergent sequence.

<span class="exam">(a)  <math style="vertical-align: -16px">a_n=\frac{\ln(n)}{\ln(n+1)}</math>

<span class="exam">(b)  <math style="vertical-align: -15px">a_n=\bigg(\frac{n}{n+1}\bigg)^n</math>

== [[009C_Sample Final 2,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] ==
<span class="exam"> For each of the following series, find the sum if it converges. If it diverges, explain why.

<span class="exam">(a)  <math style="vertical-align: -14px">4-2+1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots</math>

<span class="exam">(b)  <math>\sum_{n=1}^{\infty} \frac{1}{(2n-1)(2n+1)}</math>

== [[009C_Sample Final 2,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] ==
<span class="exam">Determine if the following series converges or diverges. Please give your reason(s).

<span class="exam">(a)  <math>\sum_{n=0}^{\infty} \frac{n!}{(2n)!}</math>

<span class="exam">(b)  <math>\sum_{n=0}^{\infty} (-1)^n \frac{1}{n+1}</math>

== [[009C_Sample Final 2,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] ==
<span class="exam">(a) Find the radius of convergence for the power series

::<math>\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.</math>

<span class="exam">(b) Find the interval of convergence of the above series.

== [[009C_Sample Final 2,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] ==
<span class="exam"> Find the Taylor Polynomials of order 0, 1, 2, 3 generated by  <math style="vertical-align: -5px">f(x)=\cos(x)</math>  at  <math style="vertical-align: -14px">x=\frac{\pi}{4}.</math>

== [[009C_Sample Final 2,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] ==
<span class="exam">(a) Express the indefinite integral  <math style="vertical-align: -13px">\int \sin(x^2)~dx</math>  as a power series.

<span class="exam">(b) Express the definite integral  <math style="vertical-align: -14px">\int_0^1 \sin(x^2)~dx</math>  as a number series.

== [[009C_Sample Final 2,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] ==

<span class="exam">(a) Consider the function  <math style="vertical-align: -16px">f(x)=\bigg(1-\frac{1}{2}x\bigg)^{-2}.</math>  Find the first three terms of its Binomial Series.

<span class="exam">(b) Find its radius of convergence.

== [[009C_Sample Final 2,_Problem_8|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] ==

<span class="exam">Find  <math>n</math>  such that the Maclaurin polynomial of degree  <math>n</math>  of  <math style="vertical-align: -5px">f(x)=\cos(x)</math>  approximates  <math style="vertical-align: -13px">\cos \frac{\pi}{3}</math>  within 0.0001 of the actual value.

== [[009C_Sample Final 2,_Problem_9|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] ==

<span class="exam">A curve is given in polar coordinates by
::<span class="exam"><math>r=\sin(2\theta).</math>

<span class="exam">(a) Sketch the curve.

<span class="exam">(b) Compute  <math style="vertical-align: -14px">y'=\frac{dy}{dx}.</math>

<span class="exam">(c) Compute  <math style="vertical-align: -14px">y''=\frac{d^2y}{dx^2}.</math>

== [[009C_Sample Final 2,_Problem_10|<span class="biglink"><span style="font-size:80%"> Problem 10 </span>]] ==

<span class="exam">Find the length of the curve given by
::<span class="exam"><math>x=t^2</math>
::<span class="exam"><math>y=t^3</math>
::<span class="exam"><math>0\leq t \leq 2</math>