MathAdmin: /* Problem 10 */

2016-05-13T20:21:43Z

Problem 10

2016-05-13T17:59:04Z

Problem 10

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
2016-04-19T01:03:34Z

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  boxed problem numbers  to go to a solution.'''
<div class="noautonum">TOC</div>

== [[009C_Sample Final 1,_Problem_1| Problem 1 ]] ==
Compute

::a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math>

::b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math>

== [[009C_Sample Final 1,_Problem_2| Problem 2 ]] ==
 Find the sum of the following series:

::a) <math>\sum_{n=0}^{\infty} (-2)^ne^{-n}</math>

::b) <math>\sum_{n=1}^{\infty} \bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg)</math>

== [[009C_Sample Final 1,_Problem_3| Problem 3 ]] ==
Determine whether the following series converges or diverges.

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

== [[009C_Sample Final 1,_Problem_4| Problem 4 ]] ==
 Find the interval of convergence of the following series.

::::::<math>\sum_{n=0}^{\infty} (-1)^n \frac{(x+2)^n}{n^2}</math>

== [[009C_Sample Final 1,_Problem_5| Problem 5 ]] ==
 Let

::::::<math>f(x)=\sum_{n=1}^{\infty} nx^n</math>

::a) Find the radius of convergence of the power series.

::b) Determine the interval of convergence of the power series.

::c) Obtain an explicit formula for the function <math style="vertical-align: -5px">f(x)</math>.

== [[009C_Sample Final 1,_Problem_6| Problem 6 ]] ==
 Find the Taylor polynomial of degree 4 of <math style="vertical-align: -5px">f(x)=\cos^2x</math> at <math>a=\frac{\pi}{4}</math>.

== [[009C_Sample Final 1,_Problem_7| Problem 7 ]] ==

A curve is given in polar coordinates by
::::::<math>r=1+\sin\theta</math>

::a) Sketch the curve.

::b) Compute <math style="vertical-align: -12px">y'=\frac{dy}{dx}</math>.

::c) Compute <math style="vertical-align: -12px">y''=\frac{d^2y}{dx^2}</math>.

== [[009C_Sample Final 1,_Problem_8| Problem 8 ]] ==

A curve is given in polar coordinates by
::::::<math>r=1+\sin 2\theta</math>
::::::<math>0\leq \theta \leq 2\pi</math>

::a) Sketch the curve.

::b) Find the area enclosed by the curve.

== [[009C_Sample Final 1,_Problem_9| Problem 9 ]] ==

A curve is given in polar coordinates by
::::::<math>r=\theta</math>
::::::<math>0\leq \theta \leq 2\pi</math>

Find the length of the curve.

== [[009C_Sample Final 1,_Problem_10| Problem 10 ]] ==

A curve is given in polar parametrically by
::::::<math>x(t)=3\sin t</math>
::::::<math>y(t)=4\cos t</math>
::::::<math>0\leq t \leq 2\pi</math>

::a) Sketch the curve.

::b) Compute the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math>.

← Older revision		Revision as of 20:21, 13 May 2016
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	::<span class="exam">b) Compute the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math>.		::<span class="exam">b) Compute the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math>.
		+
		+	'''Contributions to this page were made by [[Contributors\|Kayla Murray]]'''

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 == [[009C_Sample Final 1,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
-<span class="exam">A curve is given in polar parametrically by
+<span class="exam">A curve is given parametrically by
 ::::::<span class="exam"><math>x(t)=3\sin t</math>
 ::::::<span class="exam"><math>y(t)=4\cos t</math>