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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Related Rates</h3>
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<h2 class="hd hd-2 unit-title">2. Related rates</h2>
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<p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Set up and solve related rates problems. </p></li></ul><p><b class="bfseries">Contents: 11 pages</b></p><p>
7 videos (45 minutes 1x speed) 12 questions </p>
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<h2 class="hd hd-2 unit-title">3. Oil spill problem</h2>
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Oil spill
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An oil tanker has ruptured and is spilling oil into the ocean at 2 million liters per hour. We can model the spill as a very thin cylinder on the surface of the ocean, the volume of which is given by [mathjaxinline]V = \pi h r^2[/mathjaxinline], where [mathjaxinline]r[/mathjaxinline] is the radius of the spill and [mathjaxinline]h[/mathjaxinline] is the height, or thickness, of the oil. Suppose that the spilled oil has a constant thickness [mathjaxinline]h[/mathjaxinline] of [mathjaxinline]2[/mathjaxinline] cm. At this moment, the spill's radius is 250 meters. We would like to know at what speed is the outer perimeter of the spill is moving. </p>
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Which of the following strategies will result in the answer that we are seeking? </p>
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<h2 class="hd hd-2 unit-title">4. Oil spill continued</h2>
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Implicitly differentiating
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We want to implicitly differentiate the equation [mathjaxinline]V= \pi h r^2[/mathjaxinline] with respect to [mathjaxinline]t[/mathjaxinline], where [mathjaxinline]h[/mathjaxinline] is constant. On the left we will get [mathjaxinline]\displaystyle \frac{dV}{dt}[/mathjaxinline]. What will we get on the right? </p>
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<text> [mathjaxinline]\displaystyle 2 \pi r^2[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \pi r^2 + 2 h \pi r[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle 2 \pi r \frac{dr}{dt}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle 2 \pi r \frac{dh}{dt} \frac{dr}{dt}[/mathjaxinline]</text>
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<h3 class="hd hd-2">Oil Spill Solution</h3>
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<h2 class="hd hd-2 unit-title">5. Related rates strategy</h2>
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<p>
To solve related rates problems, it is useful to follow this strategy: </p><ol class="enumerate"><li value="1"><p>
Start with a good picture! </p></li><li value="2"><p>
Identify the relevant variables and rates. </p></li><li value="3"><p>
Find an equation relating the relevant variables that always holds. </p></li><li value="4"><p>
Differentiate implicitly. </p></li><li value="5"><p>
Plug in and solve! </p></li></ol>
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<h2 class="hd hd-2 unit-title">6. Prison break</h2>
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Planning a prison break!
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<p>
Your prison's yard is a square with 400 meter sides. At the center is a large spotlight that casts a beam which makes a bright spot on the walls of the yard, and rotates at 2 revolutions per minute. For simplicity we will treat the beam as a straight line. </p>
<p>
Your escape plan requires you to outrun the bright spot along the wall, so you need to figure out the fastest speed at which it travels. By symmetry, we only need to think about how the spot travels along just one of the walls. </p>
<p>
Here's a picture of the yard, with the beam in orange marked by [mathjaxinline]z[/mathjaxinline]. </p>
<center>
<img alt="Diagram depicts the yard as a square with side length of 400 meters. The light is indicated by a point in the center. The center point is also the corner of a right triangle whose height lies along the right side of the yard. The height of this triangle is labeled y, the base is labeled x, and the hypotenuse is labeled z. The angle formed by the sides x and z is labeled theta." src="/assets/courseware/v1/8fdaf74efd18504cb9cdb7a5ec611aa4/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app4_prisonyard.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
</center>
<p>
The rate of change that is given in the problem is which of these rates of change? </p>
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<text> [mathjaxinline]\displaystyle \frac{dx}{dt}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{dz}{dt}[/mathjaxinline]</text>
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What rate of change do we want to determine? </p>
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Using the variables from above, is it true that </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\frac{dy}{dt} = \frac{dy}{d\theta } \cdot \frac{d\theta }{dt} ?[/mathjax]</td>
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A 6-foot tall man is walking down the street. Directly behind him is a 10-foot tall lamp post that casts a shadow onto the street. You are stationary, 40 feet from the post. When the tip of the shadow passes you, the tip is moving at 3 feet per second. We wish to determine how fast the man is moving. </p>
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Using the labeled variables in the diagram, what rate of change is given to us? </p>
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<input type="radio" name="input_app_4-tab7-problem1_2_1" id="input_app_4-tab7-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="app_4-tab7-problem1_2_1-choice_1-label" for="input_app_4-tab7-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{da}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_2_1" id="input_app_4-tab7-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_4-tab7-problem1_2_1-choice_2-label" for="input_app_4-tab7-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{db}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_2_1" id="input_app_4-tab7-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_4-tab7-problem1_2_1-choice_3-label" for="input_app_4-tab7-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{dc}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_2_1" id="input_app_4-tab7-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="app_4-tab7-problem1_2_1-choice_4-label" for="input_app_4-tab7-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{d\theta }{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_2_1" id="input_app_4-tab7-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="app_4-tab7-problem1_2_1-choice_5-label" for="input_app_4-tab7-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_2_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_app_4-tab7-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_app_4-tab7-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
What rate of change do we want to determine? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_4-tab7-problem1_3_1">
<fieldset aria-describedby="status_app_4-tab7-problem1_3_1">
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_3_1" id="input_app_4-tab7-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="app_4-tab7-problem1_3_1-choice_1-label" for="input_app_4-tab7-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{da}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_3_1" id="input_app_4-tab7-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_4-tab7-problem1_3_1-choice_2-label" for="input_app_4-tab7-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{db}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_3_1" id="input_app_4-tab7-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_4-tab7-problem1_3_1-choice_3-label" for="input_app_4-tab7-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{dc}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_3_1" id="input_app_4-tab7-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="app_4-tab7-problem1_3_1-choice_4-label" for="input_app_4-tab7-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{d\theta }{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab7-problem1_3_1" id="input_app_4-tab7-problem1_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="app_4-tab7-problem1_3_1-choice_5-label" for="input_app_4-tab7-problem1_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_app_4-tab7-problem1_3_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_app_4-tab7-problem1_3_1"/>
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<div class="solution-span">
<span id="solution_app_4-tab7-problem1_solution_1"/>
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Finish up
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<p>
Find a relationship between the two variables, then differentiate and solve. </p>
<p>
How fast is the man moving, in feet per second? </p>
<p>
(Enter exact solution as a fraction or decimal.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_app_4-tab7-problem2_2_1" class=" capa_inputtype textline">
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<h2 class="hd hd-2 unit-title">8. Fire truck problem</h2>
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<h3 class="hd hd-3 problem-header" id="app_4-tab8-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_4-tab8-problem1-problem-progress" tabindex="-1">
Fire truck
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<div class="problem">
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<p>
A fire truck is backing up to a building at 0.5 feet per second. It has its ladder out and leaning against the side of the building, and the ladder is being lengthened at a rate of [mathjaxinline]0.2[/mathjaxinline] feet per second. The ladder starts from the back of the truck at a height of 6 feet. </p>
<p>
The top of the ladder is going to hit a gargoyle 30 feet up on the side of the building, when the truck is 7 feet away from the building. We want to know how fast the top of the ladder will be moving when it does that. </p>
<center>
<img alt="Diagram depicts a fire truck facing away from a building with a ladder extended from the back of the truck. The back of the fire truck is a distance x from the building. The ladder is leaning against the building and has a length of z. The height along the building from the base of the ladder to the top of the ladder is labeled y. This forms a right triangle." src="/assets/courseware/v1/79947f31d3d9e253cd33fdf190bb40f7/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app4_fire.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
What rate of change do we want to determine? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_4-tab8-problem1_2_1">
<fieldset aria-describedby="status_app_4-tab8-problem1_2_1">
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_2_1" id="input_app_4-tab8-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="app_4-tab8-problem1_2_1-choice_1-label" for="input_app_4-tab8-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{dx}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_2_1" id="input_app_4-tab8-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_4-tab8-problem1_2_1-choice_2-label" for="input_app_4-tab8-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{dy}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_2_1" id="input_app_4-tab8-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_4-tab8-problem1_2_1-choice_3-label" for="input_app_4-tab8-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{dz}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_2_1" id="input_app_4-tab8-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="app_4-tab8-problem1_2_1-choice_4-label" for="input_app_4-tab8-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{d\theta }{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_2_1" id="input_app_4-tab8-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="app_4-tab8-problem1_2_1-choice_5-label" for="input_app_4-tab8-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_2_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_app_4-tab8-problem1_2_1"/>
</fieldset>
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<span class="status unanswered" id="status_app_4-tab8-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
Given the diagram, [mathjaxinline]0.2[/mathjaxinline] feet per second is equal to which of these rates? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_4-tab8-problem1_3_1">
<fieldset aria-describedby="status_app_4-tab8-problem1_3_1">
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_3_1" id="input_app_4-tab8-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="app_4-tab8-problem1_3_1-choice_1-label" for="input_app_4-tab8-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{dx}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_3_1" id="input_app_4-tab8-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_4-tab8-problem1_3_1-choice_2-label" for="input_app_4-tab8-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{dy}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_3_1" id="input_app_4-tab8-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_4-tab8-problem1_3_1-choice_3-label" for="input_app_4-tab8-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{dz}{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_3_1" id="input_app_4-tab8-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="app_4-tab8-problem1_3_1-choice_4-label" for="input_app_4-tab8-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{d\theta }{dt}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_4-tab8-problem1_3_1" id="input_app_4-tab8-problem1_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="app_4-tab8-problem1_3_1-choice_5-label" for="input_app_4-tab8-problem1_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_app_4-tab8-problem1_3_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_app_4-tab8-problem1_3_1"/>
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Given the diagram, [mathjaxinline]0.5[/mathjaxinline] feet per second is equal to which of these rates? </p>
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The chain rule? 2
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<img alt="Diagram depicts a fire truck facing away from a building with a ladder extended from the back of the truck. The back of the fire truck is a distance x from the building. The ladder is leaning against the building and has a length of z. The height along the building from the base of the ladder to the top of the ladder is labeled y. This forms a right triangle." src="/assets/courseware/v1/79947f31d3d9e253cd33fdf190bb40f7/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app4_fire.svg" style="margin: 10px 25px 25px 25px" width="250px"/>
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Using the variables from above, is it true that </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} ?[/mathjax]</td>
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Using the variables from above, is it true that </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\frac{dy}{dt} = \frac{dy}{dz} \cdot \frac{dz}{dt} ?[/mathjax]</td>
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<h2 class="hd hd-2 unit-title">9. Volume of an iceberg</h2>
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Iceberg cone
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We want to think about the volume of an iceberg, and we've decided to model it as a cone. Note that the volume of a cone is given by [mathjaxinline]V = \frac{1}{3} \pi r^2 h[/mathjaxinline], where [mathjaxinline]r[/mathjaxinline] is the radius of the base and [mathjaxinline]h[/mathjaxinline] is the height. </p>
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Right now, the radius is 300 meters and the height is 60 meters. We've determined that the height is decreasing by [mathjaxinline]0.10[/mathjaxinline] meters per day, and the radius decreasing by [mathjaxinline]0.50[/mathjaxinline] meters per day. We want to determine the rate of change of the volume. </p>
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When implicitly differentiating the right hand side of [mathjaxinline]V = \frac{1}{3} \pi r^2 h[/mathjaxinline] with respect to [mathjaxinline]t[/mathjaxinline], what rules will we need to use? Check as many as apply. </p>
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Final answer
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<p>
Given the data presented, what are the natural units to use for our final answer? </p>
<p>
(Use [mathjaxinline]m[/mathjaxinline] for meters and [mathjaxinline]d[/mathjaxinline] for days. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponentiation.) </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_app_4-tab9-problem2_2_1" class="inputtype formulaequationinput">
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\(\)
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<p>
In those units, what is the rate of change of the volume of the iceberg? </p>
<p>
(Enter answer with 2 significant figures, i.e. 2 nonzero digits.) </p>
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\(\)
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<table class="formulainput">
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> 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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
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<h2 class="hd hd-2 unit-title">10. More open ended practice: recitation videos</h2>
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The following problems leave it to you to draw the diagram, label variables, and solve the related rates problems. This is much more open ended than the problems you've done in this lecture. Watch the videos to get the problem, then pause to work through the problem yourself. You can start watching the video again if you get stuck, or to compare your answer with the instructor answer! </p>
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<h3 class="hd hd-2">Filling a ballon</h3>
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class="video closed"
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<h3 class="hd hd-2">Another ladder problem</h3>
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<h2 class="hd hd-2 unit-title">11. Summary</h2>
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<p><b class="bfseries">Related Rates Strategy</b></p><p>
To solve related rates problems, it is useful to follow this strategy: </p><ol class="enumerate"><li value="1"><p>
Start with a good picture! </p></li><li value="2"><p>
Identify the relevant variables and rates. </p></li><li value="3"><p>
Find an equation relating the relevant variables that always holds. </p></li><li value="4"><p>
Differentiate implicitly. </p></li><li value="5"><p>
Plug in and solve! </p></li></ol>
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