Difference between revisions of "009C Sample Midterm 1"

Latest revision as of 18:45, 26 February 2017

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

Click on the boxed problem numbers to go to a solution.

Problem 1

Does the following sequence converge or diverge?

If the sequence converges, also find the limit of the sequence.

Be sure to jusify your answers!

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n=\frac{\ln n}{n}}

Problem 2

Consider the infinite series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=2}^\infty 2\bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg).}

(a) Find an expression for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th partial sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_n} of the series.

(b) Compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow \infty} s_n.}

Problem 3

Determine whether the following series converges absolutely,

conditionally or whether it diverges.

Be sure to justify your answers!

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n}}

Problem 4

Determine the convergence or divergence of the following series.

Be sure to justify your answers!

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \frac{1}{n^23^n}}

Problem 5

Find the radius of convergence and interval of convergence of the series.

(a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^\infty \sqrt{n}x^n}

(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^\infty (-1)^n \frac{(x-3)^n}{2n+1}}

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-'''This is a sample, and is meant to represent the material usually covered in Math 9C for the midterm. An actual test may or may not be similar. Click on the''' '''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
+'''This is a sample, and is meant to represent the material usually covered in Math 9C for the midterm. An actual test may or may not be similar.'''
+'''Click on the''' '''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
 <div class="noautonum">__TOC__</div>
 == [[009C_Sample Midterm 1,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
-<span class="exam"> Does the following sequence converge or diverge? If the sequence converges, also find the limit of the sequence.
+<span class="exam"> Does the following sequence converge or diverge?
+<span class="exam"> If the sequence converges, also find the limit of the sequence.
 <span class="exam"> Be sure to jusify your answers!
-::::::<math>a_n=\frac{\ln n}{n}</math>
+::<math>a_n=\frac{\ln n}{n}</math>
 == [[009C_Sample Midterm 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-<span class="exam"> Consider the infinite series <math>\sum_{n=2}^\infty 2\bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg).</math>
+<span class="exam"> Consider the infinite series &nbsp;<math>\sum_{n=2}^\infty 2\bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg).</math>
+<span class="exam">(a) Find an expression for the &nbsp;<math style="vertical-align: 0px">n</math>th partial sum &nbsp;<math style="vertical-align: -3px">s_n</math>&nbsp; of the series.
-::<span class="exam">a) Find an expression for the <math>n</math>th partial sum <math>s_n</math> of the series.
+<span class="exam">(b) Compute &nbsp;<math style="vertical-align: -11px">\lim_{n\rightarrow \infty} s_n.</math>
-::<span class="exam">b) Compute <math>\lim_{n\rightarrow \infty} s_n.</math>
 == [[009C_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam"> Determine whether the following series converges absolutely, conditionally or whether it diverges.
+<span class="exam"> Determine whether the following series converges absolutely,
+<span class="exam"> conditionally or whether it diverges.
 <span class="exam"> Be sure to justify your answers!
-::::::<math>\sum_{n=1}^\infty \frac{(-1)^n}{n}</math>
+::<math>\sum_{n=1}^\infty \frac{(-1)^n}{n}</math>
 == [[009C_Sample Midterm 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> Find the derivatives of the following functions. Do not simplify.
+<span class="exam"> Determine the convergence or divergence of the following series.
+<span class="exam"> Be sure to justify your answers!
-::<span class="exam">a) <math>f(x)=\sqrt{x}(x^2+2)</math>
+::<math>\sum_{n=1}^\infty \frac{1}{n^23^n}</math>
-::<span class="exam">b) <math>g(x)=\frac{x+3}{x^{\frac{3}{2}}+2}</math> where <math>x>0</math>
-::<span class="exam">c) <math>h(x)=\frac{e^{-5x^3}}{\sqrt{x^2+1}}</math>
 == [[009C_Sample Midterm 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam"> The displacement from equilibrium of an object in harmonic motion on the end of a spring is:
+<span class="exam"> Find the radius of convergence and interval of convergence of the series.
-::::::<span class="exam"><math>y=\frac{1}{3}\cos(12t)-\frac{1}{4}\sin(12t)</math>
+<span class="exam">(a) &nbsp;<math>\sum_{n=0}^\infty \sqrt{n}x^n</math>
-<span class="exam">where <math>y</math> is measured in feet and <math>t</math> is the time in seconds. Determine the position and velocity of the object when <math>t=\frac{\pi}{8}.</math>
+<span class="exam">(b) &nbsp;<math>\sum_{n=0}^\infty (-1)^n \frac{(x-3)^n}{2n+1}</math>