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<h2 class="hd hd-2 unit-title">Introduction to Fourier Decomposition</h2>
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Any general waveform can be expressed as a linear combination of normal modes, as we have stated many times before! With this in mind, we look at how to express a general waveform as the sum of normal modes, which is something called "Fourier Decomposition." </p>
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<h2 class="hd hd-2 unit-title">L15v1: General Solution to Wave Equation and Fourier decomposition</h2>
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<h2 class="hd hd-2 unit-title">L15Q1: Fourier Decomposition [WITH SIMULATION]</h2>
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Fourier Series Simulation
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The general solution that describes an arbitrary waveform on a string is a summation over all normal modes, given by: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\psi (x,t) = \sum _{m} A_{m}\sin (\omega _{m}t + \beta _{m})\sin (k_{m}x + \alpha _{m})[/mathjax]</td>
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Note the following:<br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;each mode corresponds to a specific frequency [mathjaxinline]\omega _{m}[/mathjaxinline] <br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;each mode has a specific amplitude [mathjaxinline]A_{m}[/mathjaxinline] <br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;the general waveform will be a superposition of all normal modes <br/></p>
<p>
Below are some examples of waveforms (black-dashed lines) that are each approximated by a summation of [mathjaxinline]N=20[/mathjaxinline] normal modes (cyan lines). This summation is called a <i class="itshape">Fourier series</i>&#8212;<b class="bfseries">keep scrolling for the question on this page!</b> </p>
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<p>
The above plots show that some functions are better approximated than others, when comparing Fourier series with the same number of terms. What spatial characteristic is the MOST "responsible" for a poor approximation? OR, put another way, what characteristic requires the most Fourier terms to approximate well? </p>
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<text> a) symmetry</text>
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<text> b) asymmetry</text>
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<text> c) an abrupt change</text>
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<text> d) horizontal lines</text>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=fourier_decomposition_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/15189fff9166c334a8d3c90967d9082a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_fourier_decomposition_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L15Q2: Fourier Decomposition II</h2>
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Fourier Decomposition Examples - part a
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<p>
Once again, the general solution that describes an arbitrary waveform on a string is a summation over all possible normal modes: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\psi (x,t) = \sum _{m} A_{m}\sin \left(\omega _{m}t + \beta _{m}\right)\sin \left(k_{m}x + \alpha _{m}\right)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
We have discussed how to find [mathjaxinline]k_{m}[/mathjaxinline] and [mathjaxinline]\alpha _{m}[/mathjaxinline] from boundary conditions, and now we know how to solve for [mathjaxinline]A_{m}[/mathjaxinline] and [mathjaxinline]\beta _{m}[/mathjaxinline] from initial conditions. </p>
<p>
The initial conditions that we use are:<br/>&#8195;&#8195;&#8195;1. &#8194;[mathjaxinline]\frac{\partial }{\partial t}{\psi }(x,t=0)=0[/mathjaxinline] <br/>&#8195;&#8195;&#8195;2. &#8194;[mathjaxinline]\psi (x,t=0)[/mathjaxinline] is known <br/></p>
<p>
The first initial condition will always give [mathjaxinline]\cos \left(\beta _{m}\right)=0[/mathjaxinline], and therefore [mathjaxinline]\beta _{m}=\frac{\pi }{2}[/mathjaxinline]. </p>
<p>
The second initial condition is for [mathjaxinline]A_{m}[/mathjaxinline], which is calculated using <i class="itshape">Fourier decomposition</i>. In particular, for a function which is non-zero only within [mathjaxinline]0 \lt x \lt L[/mathjaxinline], the following is used:<br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;if [mathjaxinline]\alpha _{m}=0[/mathjaxinline], [mathjaxinline]A_{m}=\frac{2}{L}\int _{-\infty }^{\infty }\psi (x,0)\sin \left(k_{m}x\right)dx[/mathjaxinline] <br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;if [mathjaxinline]\alpha _{m}=\frac{\pi }{2}[/mathjaxinline], [mathjaxinline]A_{m}=\frac{2}{L}\int _{-\infty }^{\infty }\psi (x,0)\cos \left(k_{m}x\right)dx[/mathjaxinline] <br/></p>
<p>
The factors of [mathjaxinline]2/L[/mathjaxinline] arise from the fact that the average value of both [mathjaxinline]\sin ^2[/mathjaxinline] and [mathjaxinline]\cos ^2[/mathjaxinline] over any number of half wavelengths is [mathjaxinline]1/2[/mathjaxinline], so the integral from [mathjaxinline]0[/mathjaxinline] to [mathjaxinline]L[/mathjaxinline] will always be [mathjaxinline]L/2[/mathjaxinline]. Furthermore, recall that the integral of a product of two trig functions with different values of [mathjaxinline]k_ m[/mathjaxinline] over any number of half wavelengths (for example [mathjaxinline]\int \sin \left(k_ i x\right)\sin \left(k_ j x\right) dx[/mathjaxinline]) is always zero. </p>
<p>
For each of the following, the function [mathjaxinline]\psi (x,0)[/mathjaxinline] is given over the range [mathjaxinline]x=0[/mathjaxinline] to [mathjaxinline]x=L[/mathjaxinline] (with [mathjaxinline]\psi (x,0)=0[/mathjaxinline] otherwise). The function has fixed endpoints at [mathjaxinline]x=0[/mathjaxinline] and [mathjaxinline]x=L[/mathjaxinline]. For each function,<br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;determine the number of modes that comprise [mathjaxinline]\psi (x,0)[/mathjaxinline] <br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;identify [mathjaxinline]A_{m}[/mathjaxinline] for the highest mode [mathjaxinline]m_{\mathrm{max}}[/mathjaxinline] <br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;identify [mathjaxinline]k_{m}[/mathjaxinline] for the highest mode [mathjaxinline]m_{\mathrm{max}}[/mathjaxinline] <br/>where [mathjaxinline]m_{\mathrm{max}}[/mathjaxinline] corresponds to the mode with the highest value of [mathjaxinline]k_ m[/mathjaxinline]. </p>
<p>
Express your answer in terms of <code>A</code>, <code>L</code>, and <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline] as needed. </p>
<p><b class="bfseries">(Part a)</b> [mathjaxinline]\psi (x,0)=A \sin \left(\frac{2\pi x}{L}\right)[/mathjaxinline] </p>
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<p style="display:inline">Number of modes:</p>
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<p style="display:inline">[mathjaxinline]A_{m_\mathrm {max}} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]k_{m_\mathrm {max}} =[/mathjaxinline] </p>
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Fourier Decomposition Examples - part b
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<p><b class="bfseries">(Part b)</b> [mathjaxinline]\psi (x,0)=A \sin \left(\frac{2\pi x}{L}\right) + \frac{1}{2}A\sin \left(\frac{4\pi x}{L}\right) + \frac{1}{3}A\sin \left(\frac{6\pi x}{L}\right)[/mathjaxinline] </p>
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<p style="display:inline">Number of modes:</p>
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<p style="display:inline">[mathjaxinline]A_{m_\mathrm {max}} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]k_{m_\mathrm {max}} =[/mathjaxinline] </p>
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<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L15Q3: Fourier Decomposition III</h2>
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Fourier Decomposition Example
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Calculate [mathjaxinline]A_{m}[/mathjaxinline] given [mathjaxinline]\psi (x,0)[/mathjaxinline] for the piecewise function defined over the range [mathjaxinline]x=0[/mathjaxinline] to [mathjaxinline]x=L[/mathjaxinline] ([mathjaxinline]\psi (x,0)=0[/mathjaxinline] otherwise), with fixed endpoints at [mathjaxinline]x=0[/mathjaxinline] and [mathjaxinline]x=L[/mathjaxinline]: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\psi (x,0)= \begin{cases} \frac{2Ax}{L} &amp; 0 \leq x \leq \frac{L}{2} \\ 0 &amp; \frac{L}{2} \lt x \leq L \\ \end{cases}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Furthermore, the velocity initial condition is [mathjaxinline]\frac{\partial }{\partial t}{\psi }(x,t=0)=0[/mathjaxinline]. </p>
<p>
You may find the following integration "trick" helpful: </p>
<table id="a0000000003" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000004">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int x \sin (kx)dx[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\int \frac{d}{dk} \cos (kx)dx[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
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&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\frac{d}{dk} \int \cos (kx)dx[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
<tr id="a0000000006">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
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[mathjaxinline]\displaystyle = -\frac{d}{dk} \frac{\sin (kx)}{k}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<td style="width:40%; border:none">&#160;</td>
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&#160;
</td>
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[mathjaxinline]\displaystyle = \frac{\sin (kx)}{k^{2}} - \frac{x\cos (kx)}{k}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
Express your answer in terms of <code>A</code>, <code>m</code> and <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]A_{m} =[/mathjaxinline] </p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>