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<h2 class="hd hd-2 unit-title">1. Modeling a Zipline</h2>
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<p>
Note that this section is OPTIONAL. There is no due date, and it does not affect your grade. </p><p><h3>Objectives</h3></p><ul class="itemize"><li><p>
Use the geometric definition of <span style="color:#27408C"><b class="bf">hyperbolic sine</b></span> and <span style="color:#27408C"><b class="bf">hyperbolic cosine</b></span> to derive <span style="color:#27408C"><b class="bf">differential equations</b></span>. </p></li><li><p>
Identify the solution to the differential equation modeling the hanging cable as a hyperbolic cosine. </p></li></ul><p><h3>Higher Goals</h3></p><ul class="itemize"><li><p>
Understand how <span style="color:#99182C"><b class="bf">physical systems</b></span> give rise to differential equations. </p></li><li><p>
Figure out how to set up a differential equation from a physical system. </p></li></ul><p><h3>Contents: 4 pages</h3></p><p>
4 videos (27 minutes 1x speed) 5 questions 1 mathlet </p>
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<h2 class="hd hd-2 unit-title">2. Modeling</h2>
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We keep using the word modeling. You may be wondering, what exactly is mathematical modeling? </p><p><span style="color:#99182C"><b class="bf">Modeling</b></span> is the process of translating a real world problem into mathematics. The <span style="color:#99182C"><b class="bf">model</b></span> is the resulting mathematics. </p><p>
In order to make the mathematical model, we typically make several simplifying assumptions. Once a model is created, it is tested and changed based on experimental data. The model is used to make predictions, which can be used to validate the model through experiment. The advantage of mathematical models is that it is cheaper and faster to run a simulation than to build an experiment. </p><p>
This section will show you the process of creating a model for the shape of a hanging cable. This section is at a higher level than the rest of the course content and is <b class="bf">optional</b>. </p>
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<h2 class="hd hd-2 unit-title">3. Hyperbolic sine and cosine</h2>
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<p><b class="bfseries">Hyperbolic sine and cosine</b></p><p>
Before we explore how differential equations show up in real world modeling scenarios, let's recall some facts about the hyperbolic trig functions. </p><p>
Recall that the hyperbolic cosine and sine are defined by the relationships </p><table id="a0000000596" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000597"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \cosh (t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac12\left(e^{t} + e^{-t}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.194)</td></tr><tr id="a0000000598"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sinh (t)[/mathjaxinline]
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[mathjaxinline]\displaystyle = \frac12\left(e^{t} - e^{-t}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.195)</td></tr></table><p>
The basic trigonometric functions are related to the geometry of a circle. These hyperbolic trig functions are related to the geometry of a hyperbola. </p><p><h3>Geometric description of trig functions</h3></p><p>
The point labeled in the image, [mathjaxinline](\cos (\theta ), \sin (\theta ) ),[/mathjaxinline] is defined to be the point on the circle [mathjaxinline]x^2 + y^2 =1[/mathjaxinline] such that the shaded area is [mathjaxinline]\theta[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/85f60221c6170ad3475f59b1824178c6/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3DiffEqCircle.svg" width="400px" alt="A circle of radius 1 centered at the origin is plotted in the x y plane. A point is indicated on the circle in the first quadrant and makes an angle of theta with the positive x axis. A portion of the circle is shaded that indicates the area swept over the range of angles from minus theta to theta." style="margin: 10px 25px 25px 25px"/></center><p>
Recall that the total area of the circle with radius 1 is [mathjaxinline]\pi[/mathjaxinline], and the circumference is [mathjaxinline]2\pi[/mathjaxinline]. So while the shaded region is bounded by an arc of arc length [mathjaxinline]2\theta[/mathjaxinline], the area of the shaded region is [mathjaxinline]\theta[/mathjaxinline]. </p><p><h3>Geometric description of hyperbolic trig functions</h3></p><p>
The point labeled in the image, [mathjaxinline](\cosh (t), \sinh (t) ),[/mathjaxinline] is defined to be the point on the hyperbola [mathjaxinline]x^2 - y^2 =1[/mathjaxinline] such that the area of the shaded region is [mathjaxinline]t[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/b998431cb1da603826409f0f60c3f308/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3DiffEqHyperbola.svg" width="400px" alt="Half of a hyperbola is plotted in the first and fourth quadrants of the x y plane. A point is indicated on the hyperbola in the first quadrant representing the point hyperbolic cosine of t comma hyperbolic sine. A region is shaded that connects the indicated point to the origin on the left of the hyperbola and extends to the mirror image of the indicated point over the positive x axis." style="margin: 10px 25px 25px 25px"/></center>
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Identify the differential equation describing the circle by implicitly differentiating the equation [mathjaxinline]x^2 + y^2 = 1[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{dy}{dx} =[/mathjaxinline]</p>
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Verify that [mathjaxinline]y=\sin (\theta )[/mathjaxinline] and [mathjaxinline]x=\cos (\theta )[/mathjaxinline] solves the differential equation you found. </p>
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Identify all the differential equations that hold for [mathjaxinline]\displaystyle y=\cosh \left( x \right)[/mathjaxinline] by implicitly differentiating the equation for the hyperbola. </p>
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Perform substitution to identify the differential equation that holds for [mathjaxinline]\displaystyle y=A \cosh \left(\frac{x}{A}\right)[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">4. Modeling with differential equations</h2>
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Identifying tangent theta
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Given the image of the segment of the hanging cable, identify [mathjaxinline]\tan (\theta )[/mathjaxinline] in terms of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
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<img alt="A concave up, increasing function is plotted in the first quadrant and denoted by y equals f of x. The origin is indicated, and the point x comma y is indicated on the curve where x and y are positive values. A line segment extends to the right of the point x comma y. The angle between this horizontal segment and the curve f of x is indicated by theta." src="/assets/courseware/v1/332d97b2a38594bc558232a93110300e/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_hangingcable.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<text> [mathjaxinline]\displaystyle \frac{y}{x}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{dy}{dx}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \left. \frac{dy}{dx}\right|_{(x,y)}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \left. \frac{dy}{dx}\right|_{(s,\theta )}[/mathjaxinline]</text>
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<p><b class="bfseries">Hyperbolic Sine and Cosine</b></p><p>
Recall that the hyperbolic cosine and sine are defined by the relationships </p><table id="a0000000628" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000629"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \cosh (t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac12\left(e^{t} + e^{-t}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.217)</td></tr><tr id="a0000000630"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sinh (t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac12\left(e^{t} - e^{-t}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.218)</td></tr></table><p>
The basic trigonometric functions are related to the geometry of a circle. These hyperbolic trig functions are related to the geometry of a hyperbola. </p><p><b class="bfseries">Geometric description of trig functions</b></p><p>
The point labeled in the image, [mathjaxinline](\cos (\theta ), \sin (\theta ) ),[/mathjaxinline] is defined to be the point on the circle [mathjaxinline]x^2 + y^2 =1[/mathjaxinline] such that the shaded area is [mathjaxinline]\theta[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/85f60221c6170ad3475f59b1824178c6/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3DiffEqCircle.svg" width="200 px" style="margin: 5px 5px 5px 5px; border:0px"/>A circle of radius 1 centered at the origin is plotted in the x y plane. A point is indicated on the circle in the first quadrant and makes an angle of theta with the positive x axis. A portion of the circle is shaded that indicates the area swept over the range of angles from minus theta to theta. </center><p><b class="bfseries">Geometric description of hyperbolic trig functions</b></p><p>
The point labeled in the image, [mathjaxinline](\cosh (t), \sinh (t) ),[/mathjaxinline] is defined to be the point on the hyperbola [mathjaxinline]x^2 - y^2 =1[/mathjaxinline] such that the area of the shaded region is [mathjaxinline]t[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/b998431cb1da603826409f0f60c3f308/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3DiffEqHyperbola.svg" width="200px" alt="Half of a hyperbola is plotted in the first and fourth quadrants of the x y plane. A point is indicated on the hyperbola in the first quadrant representing the point hyperbolic cosine of t comma hyperbolic sine. A region is shaded that connects the indicated point to the origin on the left of the hyperbola and extends to the mirror image of the indicated point over the positive x axis." style="margin: 10px 25px 25px 25px"/></center>
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