009C Sample Final 3 - 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
2017-03-19T18:34:16Z

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 3,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] ==
<span class="exam"> Which of the following sequences  <math style="vertical-align: -5px">(a_n)_{n\ge 1}</math>  converges? Which diverges? Give reasons for your answers!

<span class="exam">(a)  <math style="vertical-align: -15px">a_n=\bigg(1+\frac{1}{2n}\bigg)^n</math>

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

== [[009C_Sample Final 3,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] ==
<span class="exam"> Consider the series

::<math>\sum_{n=2}^\infty \frac{(-1)^n}{\sqrt{n}}.</math>

<span class="exam">(a) Test if the series converges absolutely. Give reasons for your answer.

<span class="exam">(b) Test if the series converges conditionally. Give reasons for your answer.

== [[009C_Sample Final 3,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] ==
<span class="exam">Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.

::<math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math>

== [[009C_Sample Final 3,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] ==
<span class="exam"> Determine if the following series converges or diverges. Please give your reason(s).

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

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

== [[009C_Sample Final 3,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] ==
<span class="exam"> Consider the function

::<math>f(x)=e^{-\frac{1}{3}x}</math>

<span class="exam">(a) Find a formula for the  <math>n</math>th derivative  <math style="vertical-align: -5px">f^{(n)}(x)</math>  of  <math style="vertical-align: -5px">f</math>  and then find  <math style="vertical-align: -5px">f'(3).</math>

<span class="exam">(b) Find the Taylor series for  <math style="vertical-align: -5px">f(x)</math>  at  <math style="vertical-align: -5px">x_0=3,</math>  i.e. write  <math style="vertical-align: -5px">f(x)</math>  in the form

::<math>f(x)=\sum_{n=0}^\infty a_n(x-3)^n.</math>

== [[009C_Sample Final 3,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] ==
<span class="exam"> Consider the power series

::<math>\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}</math>

<span class="exam">(a) Find the radius of convergence of the above power series.

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

<span class="exam">(c) Find the closed formula for the function  <math style="vertical-align: -5px">f(x)</math>  to which the power series converges.

<span class="exam">(d) Does the series

::<math>\sum_{n=0}^\infty \frac{1}{(n+1)3^{n+1}}</math>

<span class="exam">converge?

== [[009C_Sample Final 3,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] ==

<span class="exam">A curve is given in polar coordinates by

::<math>r=1+\cos^2(2\theta)</math>

<span class="exam">(a) Show that the point with Cartesian coordinates  <math style="vertical-align: -15px">(x,y)=\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg)</math>  belongs to the curve.

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

<span class="exam">(c) In Cartesian coordinates, find the equation of the tangent line at  <math style="vertical-align: -15px">\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg).</math>

== [[009C_Sample Final 3,_Problem_8|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] ==

<span class="exam">A curve is given in polar coordinates by  <math style="vertical-align: -2px">r=4+3\sin \theta</math>
::<math>0\leq \theta \leq 2\pi</math>

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

<span class="exam">(b) Find the area enclosed by the curve.

== [[009C_Sample Final 3,_Problem_9|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] ==

<span class="exam">A wheel of radius 1 rolls along a straight line, say the  <math>x</math>-axis. A point  <math style="vertical-align: 0px">P</math>  is located halfway between the center of the wheel and the rim. As the wheel rolls,  <math style="vertical-align: 0px">P</math>  traces a curve. Find parametric equations for the curve.

== [[009C_Sample Final 3,_Problem_10|<span class="biglink"><span style="font-size:80%"> Problem 10 </span>]] ==

<span class="exam">A curve is described parametrically by
::<span class="exam"><math>x=t^2</math>
::<span class="exam"><math>y=t^3-t</math>

<span class="exam">(a) Sketch the curve for  <math style="vertical-align: -2px">-2\le t \le 2.</math>

<span class="exam">(b) Find the equation of the tangent line to the curve at the origin.