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!

a_{n}={\frac {\ln n}{n}}

Problem 2

Consider the infinite series $\sum _{n=2}^{\infty }2{\bigg (}{\frac {1}{2^{n}}}-{\frac {1}{2^{n+1}}}{\bigg )}.$

(a) Find an expression for the $n$ th partial sum $s_{n}$ of the series.

(b) Compute $\lim _{n\rightarrow \infty }s_{n}.$

Problem 3

Determine whether the following series converges absolutely,

conditionally or whether it diverges.

Be sure to justify your answers!

\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!

\sum _{n=1}^{\infty }{\frac {1}{n^{2}3^{n}}}

Problem 5

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

(a) $\sum _{n=0}^{\infty }{\sqrt {n}}x^{n}$

(b) $\sum _{n=0}^{\infty }(-1)^{n}{\frac {(x-3)^{n}}{2n+1}}$

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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"> Find the following limits:
+<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!
-::<span class="exam">a) Find <math>\lim _{x\rightarrow 2} g(x),</math> provided that <math>\lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]=5</math>
+::<math>a_n=\frac{\ln n}{n}</math>
-::<span class="exam">b) Find <math>\lim _{x\rightarrow 0} \frac{\sin(4x)}{5x} </math>
-::<span class="exam">c) Evaluate <math>\lim _{x\rightarrow -3^+} \frac{x}{x^2-9} </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 following function <math> f:</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>
-::::::<math>f(x) = \left\{
-     \begin{array}{lr}
-       x^2 &  \text{if }x < 1\\
-      \sqrt{x} & \text{if }x \geq 1
-     \end{array}
-   \right.
-</math>
-::<span class="exam">a) Find <math> \lim_{x\rightarrow 1^-} f(x).</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">b) Find <math> \lim_{x\rightarrow 1^+} f(x).</math>
-::<span class="exam">c) Find <math> \lim_{x\rightarrow 1} f(x).</math>
+<span class="exam">(b) Compute &nbsp;<math style="vertical-align: -11px">\lim_{n\rightarrow \infty} s_n.</math>
-::<span class="exam">d) Is <math>f</math> continuous at <math>x=1?</math> Briefly explain.
 == [[009C_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam"> Let <math>y=\sqrt{3x-5}.</math>
+<span class="exam"> Determine whether the following series converges absolutely,
-::<span class="exam">a) Use the definition of the derivative to compute <math>\frac{dy}{dx}</math> for <math>y=\sqrt{3x-5}.</math>
+<span class="exam"> conditionally or whether it diverges.
-::<span class="exam">b) Find the equation of the tangent line to <math>y=\sqrt{3x-5}</math> at <math>(2,1).</math>
+<span class="exam"> Be sure to justify your answers!
+::<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>