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<h2 class="hd hd-2 unit-title">1. Derivative as a function: Graphical Intuition</h2>
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<h3 class="hd hd-2">The derivative as a function</h3>
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<h2 class="hd hd-2 unit-title">2. Objectives</h2>
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<ul class="itemize"><li><p>
Instantaneous rate of change </p></li><li><p>
Slope of the tangent line at a point </p></li><li><p><span style="color:#27408C"><b class="bf">A function</b></span></p></li></ul><p><b class="bfseries">Objectives</b></p><p>
At the end of this video, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Understand the difference between a <span style="color:#99182C"><b class="bf">derivative at a point</b></span> and the <span style="color:#99182C"><b class="bf">derivative as a function</b></span>. </p></li><li><p>
Correlate graph of [mathjaxinline]\ f[/mathjaxinline] to intervals of positive and negative values of the derivative. </p></li><li><p>
Be able to <span style="color:#99182C"><b class="bf">sketch the graph</b></span> of [mathjaxinline]\ f'[/mathjaxinline] given the graph of [mathjaxinline]\ f[/mathjaxinline]. </p></li></ul><p><b class="bfseries">Contents: 16 pages</b></p><p>
4 videos (19 minutes 1x speed) 15 questions </p>
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<h2 class="hd hd-2 unit-title">3. Tangents lines at every point</h2>
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<p>
In calculus, we often think of the function and its graph as being the same object. So to study functions, we often study their graphs. </p><p>
In the last segment, we learned that the derivative at a point is the slope of the tangent line to the graph through that point. But when our graph is nice and smooth, without any discontinuities, corners, or other weird behavior, we can find the slope of the tangent line at any point. </p><p>
Use the derivative function applet below to approximate the derivative using the tangent line approximation at all of the points marked using the sliders below. Do you notice any patterns? Press the “show derivative" button to display the actual values at the sliders, as well as for each point in between. </p><iframe src="https://mathlets1801.surge.sh/derivativeFunction.html" width="1100 px" height="630 px" style="border:0px; display: block;"/><p>
Use the dropdown menu to choose different functions. Find the derivatives, see what types of patterns emerge! </p>
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Slopes that lie on a line
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<h2 class="hd hd-2 unit-title">4. Begin example</h2>
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Derivative at 0
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Here's the graph of a typical function [mathjaxinline]\ f[/mathjaxinline]. Let's start thinking about what the graph of [mathjaxinline]\ f'[/mathjaxinline] might look like. </p>
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<img alt="Function increasing when x is less than 3, decreasing when x is between 3 and 6, and increasing when x is greater than 6. When x equals -0.1, the function is approximately 1.1. When x equals 0.1, the function is approximately 1.7." src="/assets/courseware/v1/87d158d1119b1161488242b990a7d18a/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_originalfunctionwithgridlines.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
What would be the best estimate for [mathjaxinline]\ f'(0)[/mathjaxinline]? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab4-problem1_2_1">
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<h2 class="hd hd-2 unit-title">5. Continuing example</h2>
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Interpreting the derivative at 1
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<img alt="Function increasing when x is less than 3, decreasing when x is between 3 and 6, and increasing when x is greater than 6. Tangent line plotted as line with slope 3 and just touching the graph of f at x equals 0. When x equals 0.9, the function is approximately 3.8. When x equals 1.1, the function is approxiamtely 4.2." src="/assets/courseware/v1/dbb3ff01db90ddc837c7280716fa1df3/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_tanlineatzero.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
If we estimate that [mathjaxinline]\ f'(0) = 3[/mathjaxinline], that means that we should plot the point [mathjaxinline](0,3)[/mathjaxinline] on the graph of [mathjaxinline]\ f'[/mathjaxinline]. </p>
<center>
<img alt="Set of axes with the vertical axis labeled f prime of x and the horizontal axis labeled x. The point x equals 0, f prime equals 3 is plotted." src="/assets/courseware/v1/84b977e7feb4ed37a10396532271016f/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_beginningofprime.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
This one point on the graph of [mathjaxinline]\ f'[/mathjaxinline] represents the slope of the corresponding tangent line to the graph of [mathjaxinline]\ f[/mathjaxinline]. </p>
<p>
Let's now look at [mathjaxinline]\ f'(1)[/mathjaxinline]. Which of the following is true of [mathjaxinline]\ f'(1)[/mathjaxinline]? </p>
<p>
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<text> Greater than [mathjaxinline]\ f'(0)[/mathjaxinline]</text>
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<text> Equal to [mathjaxinline]\ f'(0)[/mathjaxinline]</text>
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<text> Less than [mathjaxinline]\ f'(0)[/mathjaxinline] but still positive</text>
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<text> Zero</text>
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<input type="radio" name="input_der_3-tab5-problem1_2_1" id="input_der_3-tab5-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="der_3-tab5-problem1_2_1-choice_5-label" for="input_der_3-tab5-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_der_3-tab5-problem1_2_1">
<text> Negative</text>
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<h2 class="hd hd-2 unit-title">6. Continuing example, part 2</h2>
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When is the derivative zero?
</h3>
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<img alt="Function increasing when x is less than 3, decreasing when x is between 3 and 6, and increasing when x is greater than 6. Tangent line at x equals 0 is plotted as line with slope 3 and just touching the graph of f at x equals 0. Tangent line at x equals 1 is plotted as line with slope approximately 2 and just touching the graph of f at x equals 1." src="/assets/courseware/v1/989fd5974776182f2a41fa345976ecbf/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_tanlineatzeroandonev2.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
The tangent line to the graph of [mathjaxinline]\ f[/mathjaxinline] at [mathjaxinline]x=1[/mathjaxinline] is not as steep as the tangent line at [mathjaxinline]x=0[/mathjaxinline]. Therefore [mathjaxinline]\ f'(1)[/mathjaxinline] is smaller than [mathjaxinline]\ f'(0)[/mathjaxinline]. Our graph of [mathjaxinline]\ f'[/mathjaxinline] goes downward like this: </p>
<center>
<img alt="Set of axes with the vertical axis labeled f prime of x and the horizontal axis labeled x. Line segment decreasing from x equals 0 to x equals 1 is plotted." src="/assets/courseware/v1/a83d1750b0f2de3a1c0993c52e38e4ed/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_derivativeplot2.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
Where, if ever, does [mathjaxinline]\ f'[/mathjaxinline] first become zero? Enter the number. If there is no point at which the derivative is zero, type [mathjaxinline]N[/mathjaxinline] into the answer box. (Make sure it is a capital [mathjaxinline]N[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]\ f'[/mathjaxinline] first becomes zero at [mathjaxinline]x=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">7. Continuing example, part 3</h2>
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Beyond the derivative at a point
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<img alt="Function increasing when x is less than 3, decreasing when x is between 3 and 6, and increasing when x is greater than 6. Tangent line at x equals 3 is plotted as a horizontal line and just touching the graph of f at x equals 3." src="/assets/courseware/v1/c2463fc4ff47dc26100c468629d7b0d8/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_tanlineatthree.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
We know that [mathjaxinline]\ f'(3) = 0[/mathjaxinline] due to the horizontal tangent line at [mathjaxinline]x=3[/mathjaxinline]. So [mathjaxinline]\ f'[/mathjaxinline] gets smaller and smaller until it becomes zero at [mathjaxinline]x=3[/mathjaxinline]. </p>
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<img alt="Set of axes with the vertical axis labeled f prime of x and the horizontal axis labeled x. Line segment plotted which begins at x equals 0, f prime equals 3 and decreases to the point x equals 3, f prime equals 0." src="/assets/courseware/v1/0f97a86ca4f19b057138dcf7e9dc3d64/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_derivativeplot3.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
Which of these pictures best depicts the graph of [mathjaxinline]\ f'[/mathjaxinline] when [mathjaxinline]x[/mathjaxinline] passes 3? </p>
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<text> It goes back up: <img alt=" " src="/assets/courseware/v1/e32bb182a62ff16e43bf828a58c4a582/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_decoy1.svg" style="margin: 10px 25px 25px 25px" width="200px"/></text>
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<text> It stays zero: <img alt=" " src="/assets/courseware/v1/9e5c1d4f5ff25b2767c31630fe9a3962/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_decoy2.svg" style="margin: 10px 25px 25px 25px" width="200px"/></text>
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<text> It keeps going down: <img alt=" " src="/assets/courseware/v1/415deef49f70d45f40013edaf5ddcd74/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_decoy3.svg" style="margin: 10px 25px 25px 25px" width="200px"/></text>
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<h2 class="hd hd-2 unit-title">8. Continuing example, part 4</h2>
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More and more negative?
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<img alt="Function increasing when x is less than 3, decreasing when x is between 3 and 6, and increasing when x is greater than 6. Tangent line at x equals 3.5 is plotted as line with negative slope and just touching the graph of f at x equals 3.5." src="/assets/courseware/v1/c4a10b5e1bc0087e2f121f987df7dc2f/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_tanlineatthreefive.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
We see that [mathjaxinline]\ f'[/mathjaxinline] becomes negative when [mathjaxinline]x[/mathjaxinline] passes [mathjaxinline]x=3[/mathjaxinline] - the tangent lines to the graph of [mathjaxinline]\ f[/mathjaxinline] have a negative slope. Thus the graph of [mathjaxinline]\ f'[/mathjaxinline] should look like this: </p>
<center>
<img alt="Set of axes with the vertical axis labeled f prime of x and the horizontal axis labeled x. Decreasing line segment plotted which begins at x equals 0, f prime equals 3. It crosses the x axis at x equals 3, f prime equals 0 and continues downwards until approximately x equals 3.5, f prime equals negative one half." src="/assets/courseware/v1/0739d2a696de4d61e12583995ac5c484/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_derivativeplot4.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
At what point does [mathjaxinline]\ f'[/mathjaxinline] become most negative? </p>
<p>
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<text> Near [mathjaxinline]x=3[/mathjaxinline]</text>
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<text> Near [mathjaxinline]x=4[/mathjaxinline]</text>
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<text> Near [mathjaxinline]x=5[/mathjaxinline]</text>
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<text> Near [mathjaxinline]x=6[/mathjaxinline]</text>
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<text> Near [mathjaxinline]x=7[/mathjaxinline]</text>
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<text> There is no point at which it is most negative</text>
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<h2 class="hd hd-2 unit-title">9. Finishing example</h2>
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Finishing up the example
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<img alt="Function increasing when x is less than 3, decreasing when x is between 3 and 6, and increasing when x is greater than 6. Tangent line at x equals 5 is plotted as a negatively sloped line and touching the graph of f at x equals 5." src="/assets/courseware/v1/12788fd9b301c93d883eb12f58ab4745/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_tanlineatfive.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
The steepest negative slope of a tangent line occurs when [mathjaxinline]x = 5[/mathjaxinline]. The graph of [mathjaxinline]\ f'[/mathjaxinline] hits its most negative value at that point. </p>
<center>
<img alt="Set of axes with the vertical axis labeled f prime of x and the horizontal axis labeled x. Decreasing function is plotted which begins at the point x equals 0, f prime equals 3. It crosses the x axis at x equals 3, f prime equals 0 and continues downwards until approximately x equals 5, f prime equals negative 2. Then, the function starts to increase." src="/assets/courseware/v1/90053ce8a2340d093f23999026bea913/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_derivativeplot5.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
What happens to [mathjaxinline]\ f'[/mathjaxinline] near [mathjaxinline]x=6[/mathjaxinline]? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab9-problem1_2_1">
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<text> [mathjaxinline]\ f'(6)[/mathjaxinline] is still negative</text>
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<text> [mathjaxinline]\ f'(6) = 0[/mathjaxinline], then [mathjaxinline]\ f'(x)[/mathjaxinline] becomes positive when [mathjaxinline]x &gt; 6[/mathjaxinline]</text>
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<text> [mathjaxinline]\ f'(6) = 0[/mathjaxinline], then [mathjaxinline]\ f'(x)[/mathjaxinline] becomes negative when [mathjaxinline]x &gt; 6[/mathjaxinline]</text>
</label>
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<text> [mathjaxinline]\ f'(6)[/mathjaxinline] is undefined</text>
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<text> None of the above</text>
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<h2 class="hd hd-2 unit-title">10. The graph of the derivative</h2>
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<center><img src="/assets/courseware/v1/87d158d1119b1161488242b990a7d18a/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_originalfunctionwithgridlines.svg" width="400px" alt="Plot of f of x from the video and previous examples. See transcript above." style="margin: 10px 25px 25px 25px"/></center><center><img src="/assets/courseware/v1/e33d1e3049a203edc4bbcaab9606fc5d/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_finalderivativegraph.svg" width="400px" alt="Plot of f prime of x from the video. See transcript above." style="margin: 10px 25px 25px 25px"/></center>
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<h2 class="hd hd-2 unit-title">11. A second example</h2>
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Initial value
</h3>
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<img alt="The function g of x which is linear and increasing for x less than negative 1 and decreasing for x between negative 1 and 1. At x equals negative 1, g has a sharp corner. For x greater than or equal to 1, g is constant. When x equals negative 3.1, g is approximately 0.8. When x equals negative 2.9, g is approximately 1.2." src="/assets/courseware/v1/67e04d675f59cbedb02d6ad787366d99/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_secondexample.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
What is [mathjaxinline]g'(-3)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab11-problem1_2_1">
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<text> 2</text>
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<text> 1</text>
</label>
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<input type="radio" name="input_der_3-tab11-problem1_2_1" id="input_der_3-tab11-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab11-problem1_2_1-choice_3-label" for="input_der_3-tab11-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem1_2_1">
<text> 0.5</text>
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<text> -0.5</text>
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<text> -1</text>
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First part
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What does [mathjaxinline]g'[/mathjaxinline] do when [mathjaxinline]-3&lt;x&lt;-1[/mathjaxinline]? </p>
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<text> Increase as [mathjaxinline]x[/mathjaxinline] goes from left to right</text>
</label>
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<input type="radio" name="input_der_3-tab11-problem2_2_1" id="input_der_3-tab11-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_3-tab11-problem2_2_1-choice_2-label" for="input_der_3-tab11-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem2_2_1">
<text> Decrease as [mathjaxinline]x[/mathjaxinline] goes from left to right</text>
</label>
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<input type="radio" name="input_der_3-tab11-problem2_2_1" id="input_der_3-tab11-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab11-problem2_2_1-choice_3-label" for="input_der_3-tab11-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem2_2_1">
<text> Stay constant</text>
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Evaluate at negative 1
</h3>
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<p>
Which of these options describes [mathjaxinline]g'(-1)[/mathjaxinline]? </p>
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<input type="radio" name="input_der_3-tab11-problem3_2_1" id="input_der_3-tab11-problem3_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_3-tab11-problem3_2_1-choice_1-label" for="input_der_3-tab11-problem3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem3_2_1">
<text> It is positive</text>
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</div>
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<input type="radio" name="input_der_3-tab11-problem3_2_1" id="input_der_3-tab11-problem3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_3-tab11-problem3_2_1-choice_2-label" for="input_der_3-tab11-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem3_2_1">
<text> It is negative</text>
</label>
</div>
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<input type="radio" name="input_der_3-tab11-problem3_2_1" id="input_der_3-tab11-problem3_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab11-problem3_2_1-choice_3-label" for="input_der_3-tab11-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem3_2_1">
<text> It is zero</text>
</label>
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<input type="radio" name="input_der_3-tab11-problem3_2_1" id="input_der_3-tab11-problem3_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="der_3-tab11-problem3_2_1-choice_4-label" for="input_der_3-tab11-problem3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem3_2_1">
<text> It is both positive and negative</text>
</label>
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<input type="radio" name="input_der_3-tab11-problem3_2_1" id="input_der_3-tab11-problem3_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="der_3-tab11-problem3_2_1-choice_5-label" for="input_der_3-tab11-problem3_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_der_3-tab11-problem3_2_1">
<text> It does not exist</text>
</label>
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<h2 class="hd hd-2 unit-title">12. Second part</h2>
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<h3 class="hd hd-3 problem-header" id="der_3-tab12-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab12-problem1-problem-progress" tabindex="-1">
Second part
</h3>
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<div class="problem">
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<center>
<img alt="The function g of x which is linear and increasing for x less than negative 1 and decreasing for x between negative 1 and 1. At x equals negative 1, g has a sharp corner. For x greater than or equal to 1, g is constant. When x equals negative 3.1, g is approximately 0.8. When x equals negative 2.9, g is approximately 1.2." src="/assets/courseware/v1/67e04d675f59cbedb02d6ad787366d99/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_secondexample.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
When [mathjaxinline]x&lt;-1[/mathjaxinline], [mathjaxinline]g'(x) = 2[/mathjaxinline], so the graph of [mathjaxinline]g'[/mathjaxinline] is a horizontal line. We use an open circle to denote the fact that [mathjaxinline]g'(-1)[/mathjaxinline] does not exist due to the &#8220;corner" in the graph of [mathjaxinline]g[/mathjaxinline]. The point [mathjaxinline](-1,2)[/mathjaxinline] is not on the graph of [mathjaxinline]g'[/mathjaxinline]. </p>
<center>
<img alt="Graph of g prime of x for x less than negative 1. The function g prime is 2 for x less than negative 1 and has an open circle at x equals negative 1." src="/assets/courseware/v1/4ebd3fb1ab931165a08bcc3626457f2a/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2ndexamplederivativetildot.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
Which of the following pictures best depicts the graph of [mathjaxinline]g'(x)[/mathjaxinline] in the range [mathjaxinline]-3&lt;x&lt;1[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab12-problem1_2_1">
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<input type="radio" name="input_der_3-tab12-problem1_2_1" id="input_der_3-tab12-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_3-tab12-problem1_2_1-choice_1-label" for="input_der_3-tab12-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_3-tab12-problem1_2_1">
<text>
<img alt="For x less than negative 1, g prime is constant and equal to 2. At x equals negative 1, g prime has a removable discontinuity. For x greater than negative 1, g prime decreases starting from 2 until it reaches 0 at x equals 1." src="/assets/courseware/v1/61fe99dc4f65c9f6008bd6a61ee45bed/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2decoy1small.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab12-problem1_2_1" id="input_der_3-tab12-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_3-tab12-problem1_2_1-choice_2-label" for="input_der_3-tab12-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab12-problem1_2_1">
<text>
<img alt="For x less than negative 1, g prime is constant and equal to 2. At x equals negative 1, g prime has a jump discontinuity. For x greater than negative 1, g prime decreases starting from 0 down until x equals 1." src="/assets/courseware/v1/d51f24a8d622085662f2e5164502a435/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2decoy2small.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab12-problem1_2_1" id="input_der_3-tab12-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab12-problem1_2_1-choice_3-label" for="input_der_3-tab12-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab12-problem1_2_1">
<text>
<img alt="For x less than negative 1, g prime is constant and equal to 2. At x equals negative 1, g prime has a jump discontinuity. For x greater than negative 1, g prime decreases starting from a negative value down until x equals 1." src="/assets/courseware/v1/17370baa7055a74233df91b517d7f903/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2decoy3small.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab12-problem1_2_1" id="input_der_3-tab12-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="der_3-tab12-problem1_2_1-choice_4-label" for="input_der_3-tab12-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_3-tab12-problem1_2_1">
<text>
<img alt="For x less than negative 1, g prime is constant and equal to 2. At x equals negative 1, g prime has a jump discontinuity. For x greater than negative 1, g prime increases starting from a negative value until x equals 1 and g prime remains negative." src="/assets/courseware/v1/7b4e6d50864f1a21648aa4adb6724a48/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2decoy4small.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab12-problem1_2_1" id="input_der_3-tab12-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="der_3-tab12-problem1_2_1-choice_5-label" for="input_der_3-tab12-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_der_3-tab12-problem1_2_1">
<text>
<img alt="For x less than negative 1, g prime is constant and equal to 2. At x equals negative 1, g prime has a jump discontinuity. For x greater than negative 1, g prime increases starting from a negative value until x equals 1 and g prime reaches 0." src="/assets/courseware/v1/77218ece726c8eb519d2da41c2d81bdc/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2decoy5small.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
</text>
</label>
</div>
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<input type="radio" name="input_der_3-tab12-problem1_2_1" id="input_der_3-tab12-problem1_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="der_3-tab12-problem1_2_1-choice_6-label" for="input_der_3-tab12-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_der_3-tab12-problem1_2_1">
<text>
<img alt="For x less than negative 1, g prime is constant and equal to 2. At x equals negative 1, g prime has a removable discontinuity. For x greater than negative 1, g prime increases starting from 2." src="/assets/courseware/v1/579a1b8de51a5653f2ffd64f5188ccef/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2decoy6small.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab12-problem1_2_1" id="input_der_3-tab12-problem1_2_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="der_3-tab12-problem1_2_1-choice_7-label" for="input_der_3-tab12-problem1_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_der_3-tab12-problem1_2_1">
<text><b class="bf">G.</b> None of the above</text>
</label>
</div>
<span id="answer_der_3-tab12-problem1_2_1"/>
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<span class="status unanswered" id="status_der_3-tab12-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>
<div class="solution-span">
<span id="solution_der_3-tab12-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
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<h2 class="hd hd-2 unit-title">13. Even more derivative</h2>
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Evaluate at 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab13-problem1-problem-progress"></div>
<div class="problem">
<div>
<center>
<img alt="The function g of x which is linear and increasing for x less than negative 1 and decreasing for x between negative 1 and 1. At x equals negative 1, g has a sharp corner. For x greater than or equal to 1, g is constant." src="/assets/courseware/v1/67e04d675f59cbedb02d6ad787366d99/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_secondexample.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
What is [mathjaxinline]g'(1)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab13-problem1_2_1">
<fieldset aria-describedby="status_der_3-tab13-problem1_2_1">
<div class="field">
<input type="radio" name="input_der_3-tab13-problem1_2_1" id="input_der_3-tab13-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_3-tab13-problem1_2_1-choice_1-label" for="input_der_3-tab13-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem1_2_1">
<text> Positive</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab13-problem1_2_1" id="input_der_3-tab13-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_3-tab13-problem1_2_1-choice_2-label" for="input_der_3-tab13-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem1_2_1">
<text> Negative</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab13-problem1_2_1" id="input_der_3-tab13-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab13-problem1_2_1-choice_3-label" for="input_der_3-tab13-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem1_2_1">
<text> Zero</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab13-problem1_2_1" id="input_der_3-tab13-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="der_3-tab13-problem1_2_1-choice_4-label" for="input_der_3-tab13-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem1_2_1">
<text> It does not exist</text>
</label>
</div>
<span id="answer_der_3-tab13-problem1_2_1"/>
</fieldset>
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<span class="status unanswered" id="status_der_3-tab13-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>
<div class="solution-span">
<span id="solution_der_3-tab13-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
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data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab13-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab13-problem2/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="der_3-tab13-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab13-problem2-problem-progress" tabindex="-1">
Third part
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab13-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
What best describes [mathjaxinline]g'[/mathjaxinline] for [mathjaxinline]1&lt;x &lt; 3[/mathjaxinline] ? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab13-problem2_2_1">
<fieldset aria-describedby="status_der_3-tab13-problem2_2_1">
<div class="field">
<input type="radio" name="input_der_3-tab13-problem2_2_1" id="input_der_3-tab13-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_3-tab13-problem2_2_1-choice_1-label" for="input_der_3-tab13-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem2_2_1">
<text> It is positive and increasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab13-problem2_2_1" id="input_der_3-tab13-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_3-tab13-problem2_2_1-choice_2-label" for="input_der_3-tab13-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem2_2_1">
<text> It is positive and decreasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab13-problem2_2_1" id="input_der_3-tab13-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab13-problem2_2_1-choice_3-label" for="input_der_3-tab13-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem2_2_1">
<text> It is positive and constant</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab13-problem2_2_1" id="input_der_3-tab13-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="der_3-tab13-problem2_2_1-choice_4-label" for="input_der_3-tab13-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem2_2_1">
<text> It is constantly zero</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab13-problem2_2_1" id="input_der_3-tab13-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="der_3-tab13-problem2_2_1-choice_5-label" for="input_der_3-tab13-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_der_3-tab13-problem2_2_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_der_3-tab13-problem2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_der_3-tab13-problem2_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>
<div class="solution-span">
<span id="solution_der_3-tab13-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Third part" />
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<span class="sr">Loading…</span>
</p>
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<h2 class="hd hd-2 unit-title">14. Wrap-up of example 2</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="52c5d340ea7511efbdc212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Graphing the Derivative of g</h3>
<div
id="video_der_3-tab14-video1"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:fwepXQJQio4", "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
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<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_der_3-tab14-video1">Downloads and transcripts</h3>
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<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@der_3-tab14-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
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<center><img src="/assets/courseware/v1/67e04d675f59cbedb02d6ad787366d99/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_secondexample.svg" width="400px" alt="Plot of g of x from the video and previous examples. See transcript above." style="margin: 10px 25px 25px 25px"/></center><center><img src="/assets/courseware/v1/509c6f6b68e50f2d4d546b8b17d2c1e3/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2ndexamplederivativecomplete.svg" width="400px" alt="Plot of g prime of x from the video. See transcript above." style="margin: 10px 25px 25px 25px"/></center>
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<h2 class="hd hd-2 unit-title">15. Review questions</h2>
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<h3 class="hd hd-3 problem-header" id="der_3-tab15-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem1-problem-progress" tabindex="-1">
Review question 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Suppose [mathjaxinline]\ f[/mathjaxinline] is differentiable at [mathjaxinline]x=3[/mathjaxinline]. Then [mathjaxinline]\ f'(3)[/mathjaxinline] is a [mathjaxinline]\dotsc[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab15-problem1_2_1">
<fieldset aria-describedby="status_der_3-tab15-problem1_2_1">
<div class="field">
<input type="radio" name="input_der_3-tab15-problem1_2_1" id="input_der_3-tab15-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_3-tab15-problem1_2_1-choice_1-label" for="input_der_3-tab15-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem1_2_1">
<text> Number</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab15-problem1_2_1" id="input_der_3-tab15-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_3-tab15-problem1_2_1-choice_2-label" for="input_der_3-tab15-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem1_2_1">
<text> Line</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab15-problem1_2_1" id="input_der_3-tab15-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab15-problem1_2_1-choice_3-label" for="input_der_3-tab15-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem1_2_1">
<text> Point</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab15-problem1_2_1" id="input_der_3-tab15-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="der_3-tab15-problem1_2_1-choice_4-label" for="input_der_3-tab15-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem1_2_1">
<text> Function</text>
</label>
</div>
<span id="answer_der_3-tab15-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_der_3-tab15-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>
<div class="solution-span">
<span id="solution_der_3-tab15-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
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<div class="action">
<input type="hidden" name="problem_id" value="Review question 1" />
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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>
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<span class="sr">Loading…</span>
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<h3 class="hd hd-3 problem-header" id="der_3-tab15-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem2-problem-progress" tabindex="-1">
Review question 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
If [mathjaxinline]\ f[/mathjaxinline] is a function, then [mathjaxinline]\ f'[/mathjaxinline] is a [mathjaxinline]\dotsc[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab15-problem2_2_1">
<fieldset aria-describedby="status_der_3-tab15-problem2_2_1">
<div class="field">
<input type="radio" name="input_der_3-tab15-problem2_2_1" id="input_der_3-tab15-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_3-tab15-problem2_2_1-choice_1-label" for="input_der_3-tab15-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem2_2_1">
<text> Number</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab15-problem2_2_1" id="input_der_3-tab15-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_3-tab15-problem2_2_1-choice_2-label" for="input_der_3-tab15-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem2_2_1">
<text> Line</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab15-problem2_2_1" id="input_der_3-tab15-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_3-tab15-problem2_2_1-choice_3-label" for="input_der_3-tab15-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem2_2_1">
<text> Point</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_3-tab15-problem2_2_1" id="input_der_3-tab15-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="der_3-tab15-problem2_2_1-choice_4-label" for="input_der_3-tab15-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem2_2_1">
<text> Function</text>
</label>
</div>
<span id="answer_der_3-tab15-problem2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_der_3-tab15-problem2_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>
<div class="solution-span">
<span id="solution_der_3-tab15-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Review question 2" />
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data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem3" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem3/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="der_3-tab15-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem3-problem-progress" tabindex="-1">
Review question 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_3-tab15-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
The graph of [mathjaxinline]\ f(x)[/mathjaxinline] is smooth (without corners or discontinuities) and going down as we travel from left to right. </p>
<p>
Select as many of the following that <b class="bf">must</b> be true about the graph of [mathjaxinline]\ f'(x)[/mathjaxinline]. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_3-tab15-problem3_2_1">
<fieldset aria-describedby="status_der_3-tab15-problem3_2_1">
<div class="field">
<input type="checkbox" name="input_der_3-tab15-problem3_2_1[]" id="input_der_3-tab15-problem3_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="der_3-tab15-problem3_2_1-choice_0-label" for="input_der_3-tab15-problem3_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem3_2_1">
<text>Constant</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_der_3-tab15-problem3_2_1[]" id="input_der_3-tab15-problem3_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="der_3-tab15-problem3_2_1-choice_1-label" for="input_der_3-tab15-problem3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem3_2_1">
<text>Straight</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_der_3-tab15-problem3_2_1[]" id="input_der_3-tab15-problem3_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="der_3-tab15-problem3_2_1-choice_2-label" for="input_der_3-tab15-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem3_2_1">
<text>Going down</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_der_3-tab15-problem3_2_1[]" id="input_der_3-tab15-problem3_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="der_3-tab15-problem3_2_1-choice_3-label" for="input_der_3-tab15-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem3_2_1">
<text>Going up</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_der_3-tab15-problem3_2_1[]" id="input_der_3-tab15-problem3_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="der_3-tab15-problem3_2_1-choice_4-label" for="input_der_3-tab15-problem3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem3_2_1">
<text>Above the [mathjaxinline]x[/mathjaxinline]-axis</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_der_3-tab15-problem3_2_1[]" id="input_der_3-tab15-problem3_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="der_3-tab15-problem3_2_1-choice_5-label" for="input_der_3-tab15-problem3_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_der_3-tab15-problem3_2_1">
<text>Below the [mathjaxinline]x[/mathjaxinline]-axis</text>
</label>
</div>
<span id="answer_der_3-tab15-problem3_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_der_3-tab15-problem3_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>
<div class="solution-span">
<span id="solution_der_3-tab15-problem3_solution_1"/>
</div></p>
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<h2 class="hd hd-2 unit-title">16. More practice: recitation video</h2>
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<h3 class="hd hd-2">Graphing the derivative of a function</h3>
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<h2 class="hd hd-2 unit-title">17. Summary</h2>
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<p><b class="bfseries">Recall secant and tangent lines</b></p><p>
In calculus, we often think of the function and its graph as being the same object. So to study functions, we often study their graphs. </p><p>
In the last segment, we learned that the derivative at a point is the slope of the tangent line to the graph through that point. But when our graph is nice and smooth, without any discontinuities, corners, or other weird behavior, we can find the slope of the tangent line at any point. Thus we can think of the derivative of a function as a function. </p><p><b class="bfseries">Example: The graph of a function and its derivative</b></p><center><img src="/assets/courseware/v1/67e04d675f59cbedb02d6ad787366d99/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_secondexample.svg" width="400px" alt="Plot of g of x from previous examples." style="margin: 10px 25px 25px 25px"/></center><center><img src="/assets/courseware/v1/509c6f6b68e50f2d4d546b8b17d2c1e3/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der3_2ndexamplederivativecomplete.svg" width="400px" alt="Plot of g prime of x from previous examples." style="margin: 10px 25px 25px 25px"/></center>
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