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<h2 class="hd hd-2 unit-title">1. What is the derivative?</h2>
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<p>
In the United States, you cannot actually get a speeding ticket this way. A law was passed that prevents law enforcement from using toll booth data to give speeding tickets retroactively. Otherwise, no one would purchase automated toll booth passes! </p>
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<h2 class="hd hd-2 unit-title">2. Objectives</h2>
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<ul class="itemize"><li><p><span style="color:#27408C"><b class="bf">Instantaneous rate of change</b></span></p></li></ul><p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Describe what is meant by an <b class="bf"><span style="color:#99182C">average rate of change</span></b>, and compute them with appropriate <span style="color:#99182C"><b class="bf">units</b></span>. </p></li><li><p>
Describe the difference and relationship between the average rate of change and an <b class="bf"><span style="color:#99182C">instantaneous rate of change</span></b>. </p></li><li><p>
Use a <b class="bf"><span style="color:#99182C">limit</span></b> to find the instantaneous rate of change, also known as the <b class="bf"><span style="color:#99182C">derivative</span></b> at a point. </p></li><li><p>
Interpret the <span style="color:#99182C"><b class="bf">sign of a derivative</b></span> - positive, negative, or zero - as having real-world meaning. </p></li></ul><p><b class="bfseries">Contents: 13 pages</b></p><p>
6 videos (23 minutes 1x speed) 17 questions </p>
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<h2 class="hd hd-2 unit-title">3. Rates of change</h2>
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<p>
Derivatives are all about rates of change. What do we mean by a rate of change? Let's take an example. </p><p>
Using the fact that you went through the toll booth at the 50 mile marker on the Turnpike at 8am, and you went through a second toll booth at the 220 mile marker at 10am. </p>
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On average, how fast were you traveling between 8 AM and 10 AM? </p>
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OK, you entered a number there, and hopefully you got the right answer. But this is supposed to be measuring something real - a velocity, not just a number. In what units did we just measure our speed? </p>
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<h2 class="hd hd-2 unit-title">4. Average vs. Instantaneous</h2>
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We calculated a velocity of 85 miles per hour, so we were definitely speeding. Does that mean that at the exact instant of 8 AM, the speed of the car was exactly 85 mph? </p>
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Given that at 8:00 our position was 50 miles, and at 8:01 our position was 51 miles, what was our average velocity over the one minute from 8:00 to 8:01? Express your answer in miles per hour. </p>
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Pressure changes
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The pressure of a volume of gas changes as we change the temperature. If we measure the pressure in units of pascals, and temperature in degrees Celsius, in what units would we be measuring a rate of change of pressure with respect to temperature? </p>
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<text> degrees Celsius per pascal</text>
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<text> pascals per degree Celsius</text>
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Pressure rate of change
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If pressure is 210000 pascals at 30 degrees Celsius, and 220000 pascals at 32 degrees Celsius, what is the average rate of change of pressure over this range of temperature? </p>
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(Express answer as a number in the units you found in the previous problem.) </p>
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<h2 class="hd hd-2 unit-title">6. Getting closer to instantaneous</h2>
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Getting closer to instantaneous
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We've calculated the average velocity of our car between 8:00 and 8:01. Great! But again we have the problem - this is merely the average velocity over a period of one minute. We still haven't found the velocity at the precise instant of 8am. </p>
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Which of the following expressions would give us the precise velocity at <b class="bf">exactly</b> 8am? </p>
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<text> [mathjaxinline]\displaystyle \frac{f(8+1/3600) - f(8)}{8+1/3600 - 8}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">7. Finding a formula</h2>
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Finding a formula
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All of the choices were of the form [mathjaxinline]\displaystyle \frac{f(b) - f(8)}{b-8}[/mathjaxinline]. When [mathjaxinline]b[/mathjaxinline] was 10, we know that gives us the average velocity over the span of time from 8:00 to 10:00, but that won't necessarily match the instantaneous velocity at 8:00. When we choose [mathjaxinline]b[/mathjaxinline] to be one minute past 8, [mathjaxinline]8+1/60[/mathjaxinline], we probably get a more accurate measurement, but it's still an average velocity over a minute, not the instantaneous velocity. When [mathjaxinline]b[/mathjaxinline] is only one second away, [mathjaxinline]8+1/3600[/mathjaxinline], even more accurate, but again, not exact. </p>
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<h2 class="hd hd-2 unit-title">8. Derivative at a point</h2>
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<h2 class="hd hd-2 unit-title">9. Definition of the derivative</h2>
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The <span style="color:#99182C"><b class="bf">derivative</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] at a point [mathjaxinline]x = a[/mathjaxinline] is defined to be </p><table id="a0000000047" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle f'(a) = \lim _{b\rightarrow a} \frac{f(b) - f(a)}{b-a}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Alternative definition<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
This alternative definition is also common. </p><p>
The <span style="color:#99182C"><b class="bf">derivative</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] at a point [mathjaxinline]x = a[/mathjaxinline] is defined to be </p><table id="a0000000048" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle f'(a) = \lim _{h\rightarrow 0} \frac{f(a+h) - f(a)}{h}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></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+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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Units of the derivative
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If [mathjaxinline]x[/mathjaxinline] is temperature, measured in degrees, and [mathjaxinline]\ f(x)[/mathjaxinline] is pressure, measured in pascals, what units would [mathjaxinline]\ f'(30)[/mathjaxinline] be measured in? </p>
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The height of the pumpkin at time [mathjaxinline]t[/mathjaxinline] is [mathjaxinline]\ f(t) = 100 + 20t - 5t^2[/mathjaxinline]. We got a negative number for [mathjaxinline]\ f'(3)[/mathjaxinline]. Does this make sense? </p>
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Now you calculate [mathjaxinline]\ f'(2)[/mathjaxinline]. Use the same limit definition as we did above! </p>
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<p style="display:inline">[mathjaxinline]\ f'(2)=[/mathjaxinline]</p>
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<p><span style="color:#99182C"><b class="bf">Food for thought:</b></span> What does an instantaneous velocity of 0 mean here? </p>
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<h2 class="hd hd-2 unit-title">12. Zero derivative</h2>
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<h2 class="hd hd-2 unit-title">13. Units of derivatives</h2>
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Units of derivatives question
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A truck is going to make a trip of 500 km. If it is loaded with [mathjaxinline]x[/mathjaxinline] pounds of cargo, its fuel efficiency on the trip will be [mathjaxinline]\ f(x)[/mathjaxinline] miles per gallon. What units is [mathjaxinline]\ f'(5000)[/mathjaxinline] measured in? </p>
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(Type mi for miles, km for kilometers, g for gallons, and p for pounds. Use [mathjaxinline]*[/mathjaxinline] for multiplication and / for division.) </p>
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Do you expect [mathjaxinline]\ f'(5000)[/mathjaxinline] to be positive, negative, or zero? </p>
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Make sure that you can explain your thought process in solving this problem. This sort of reasoning is tested on the AP exam! </p>
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<h2 class="hd hd-2 unit-title">14. Summary</h2>
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<p><b class="bfseries">The Definition of the average rate of change</b></p><p>
The <span style="color:#27408C"><b class="bf">average rate of change</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] over an interval [mathjaxinline]a \leq x \leq b[/mathjaxinline] is defined to be </p><table id="a0000000056" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{f(b) - f(a)}{b-a}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Definition of the derivative</b></p><p>
The <span style="color:#27408C"><b class="bf">derivative</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] at a point [mathjaxinline]x = a[/mathjaxinline] is defined to be </p><table id="a0000000057" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle f'(a) = \lim _{b\rightarrow a} \frac{f(b) - f(a)}{b-a}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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