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<h2 class="hd hd-2 unit-title">1. Second derivatives and higher</h2>
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<h2 class="hd hd-2 unit-title">2. Objectives</h2>
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<p><span style="color:#27408C"><b class="bf">Acceleration</b></span><br/><span style="color:#27408C"><b class="bf">Derivative of a derivative</b></span><br/><span style="color:#27408C"><b class="bf">A function</b></span><br/><span style="color:#27408C"><b class="bf">A rate of change of a rate of change</b></span></p><p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Compute <span style="color:#99182C"><b class="bf">second derivatives</b></span> and <span style="color:#99182C"><b class="bf">higher derivatives</b></span>. </p></li><li><p>
Relate the second derivative to the <span style="color:#99182C"><b class="bf">concavity</b></span> of a graph. </p></li><li><p>
Describe the second derivative of a position function as <span style="color:#99182C"><b class="bf">acceleration</b></span>. </p></li><li><p>
Interpret <span style="color:#99182C"><b class="bf">units</b></span> of higher derivatives. </p></li></ul><p><b class="bfseries">Contents: 16 pages</b></p><p>
6 videos (16 minutes 1x speed) 18 questions </p>
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<h2 class="hd hd-2 unit-title">3. Second derivative</h2>
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First derivative
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Let's say that [mathjaxinline]\ f(x) = x^3 - 6x[/mathjaxinline]. Calculate [mathjaxinline]\displaystyle \frac{df}{dx}[/mathjaxinline]. </p>
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(Type [mathjaxinline]*[/mathjaxinline] for multiplication; e.g. 2[mathjaxinline]*[/mathjaxinline]x for [mathjaxinline]2x[/mathjaxinline]. Type [mathjaxinline]\wedge[/mathjaxinline] for exponents; e.g. x[mathjaxinline]\wedge[/mathjaxinline]2 for [mathjaxinline]x^2[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{df}{dx}=[/mathjaxinline]</p>
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In the last question you computed the derivative of [mathjaxinline]\ f[/mathjaxinline]. It is itself a function, so let [mathjaxinline]g(x) = \displaystyle \frac{df}{dx}[/mathjaxinline]. What is the derivative of [mathjaxinline]g[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">4. Introducing higher derivatives</h2>
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<div class="video-player-post"></div>
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<h4 class="hd hd-5">Transcripts</h4>
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<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_6-tab4-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
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<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_6-tab4-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
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<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="75442c02ea6e11efbdc212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@der_6-tab4-text1">
<script type="json/xblock-args" class="xblock-json-init-args">
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<p>
The second derivative of a function [mathjaxinline]\ f(x)[/mathjaxinline] is the first derivative of [mathjaxinline]\ f'(x)[/mathjaxinline], and is denoted by [mathjaxinline]f^{\prime \prime }(x)[/mathjaxinline] or [mathjaxinline]\displaystyle \frac{d^2f}{dx^2}[/mathjaxinline]. </p>
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<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
The [mathjaxinline]n[/mathjaxinline]th derivative of a function [mathjaxinline]\ f(x)[/mathjaxinline] is the first derivative of [mathjaxinline]\ f^{(n-1)}(x)[/mathjaxinline], and is denoted by [mathjaxinline]f^{(n)}(x)[/mathjaxinline] or [mathjaxinline]\displaystyle \frac{d^ nf}{dx^ n}[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">5. A third derivative</h2>
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A third derivative
</h3>
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<div class="problem">
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<p>
Let [mathjaxinline]g(t) = t^4 - t^3 + 2t[/mathjaxinline]. Calculate [mathjaxinline]\displaystyle {\left.\frac{d^3g}{dt^3} \right|_{t=2}.}[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_der_6-tab5-problem1_2_1" class="inputtype formulaequationinput">
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\(\)
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<h2 class="hd hd-2 unit-title">6. Second derivative continued</h2>
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<h3 class="hd hd-3 problem-header" id="der_6-tab6-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_6-tab6-problem1-problem-progress" tabindex="-1">
Sign of a derivative
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@der_6-tab6-problem1-problem-progress"></div>
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<p>
Here's the graph of a function [mathjaxinline]\ f[/mathjaxinline] . </p>
<center>
<img src="/assets/courseware/v1/c0e373047bc74c025768d2b1b6d584fb/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_firstsectionx.svg" style="margin: 5px 5px 5px 5px; border:0px" width="400 px"/>
</center>
<p>
Consider left part of the graph drawn boldly in orange. (Ignore the portion of the curve faded out by the gray box.) </p>
<p>
Is [mathjaxinline]\ f'[/mathjaxinline] positive or negative on this section? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_6-tab6-problem1_2_1">
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<text> Positive</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_6-tab6-problem1_2_1" id="input_der_6-tab6-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_6-tab6-problem1_2_1-choice_2-label" for="input_der_6-tab6-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_6-tab6-problem1_2_1">
<text> Negative</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_6-tab6-problem1_2_1" id="input_der_6-tab6-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_6-tab6-problem1_2_1-choice_3-label" for="input_der_6-tab6-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab6-problem1_2_1">
<text> Sometimes positive and sometimes negative</text>
</label>
</div>
<span id="answer_der_6-tab6-problem1_2_1"/>
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<span class="status unanswered" id="status_der_6-tab6-problem1_2_1" data-tooltip="Not yet answered.">
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</span>
</div>
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<span id="solution_der_6-tab6-problem1_solution_1"/>
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Is the derivative increasing?
</h3>
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<p>
On the highlighted part, is [mathjaxinline]\ f'[/mathjaxinline] increasing or decreasing? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_6-tab6-problem2_2_1">
<fieldset aria-describedby="status_der_6-tab6-problem2_2_1">
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<input type="radio" name="input_der_6-tab6-problem2_2_1" id="input_der_6-tab6-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_6-tab6-problem2_2_1-choice_1-label" for="input_der_6-tab6-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_6-tab6-problem2_2_1">
<text> Increasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_6-tab6-problem2_2_1" id="input_der_6-tab6-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_6-tab6-problem2_2_1-choice_2-label" for="input_der_6-tab6-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_6-tab6-problem2_2_1">
<text> Decreasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_6-tab6-problem2_2_1" id="input_der_6-tab6-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_6-tab6-problem2_2_1-choice_3-label" for="input_der_6-tab6-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab6-problem2_2_1">
<text> Sometimes increasing, sometimes decreasing</text>
</label>
</div>
<span id="answer_der_6-tab6-problem2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_der_6-tab6-problem2_2_1" data-tooltip="Not yet answered.">
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</span>
</div>
</div></div>
</p>
<p>
What does that tell us about the sign of [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline] on this section? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_6-tab6-problem2_3_1">
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<text> It is positive</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_6-tab6-problem2_3_1" id="input_der_6-tab6-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_6-tab6-problem2_3_1-choice_2-label" for="input_der_6-tab6-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_6-tab6-problem2_3_1">
<text> It is negative</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_6-tab6-problem2_3_1" id="input_der_6-tab6-problem2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_6-tab6-problem2_3_1-choice_3-label" for="input_der_6-tab6-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab6-problem2_3_1">
<text> It is sometimes positive and sometimes negative</text>
</label>
</div>
<span id="answer_der_6-tab6-problem2_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_der_6-tab6-problem2_3_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_6-tab6-problem2_solution_1"/>
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<h2 class="hd hd-2 unit-title">7. Second derivative continued</h2>
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Graph of first derivative
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Consider the portion of the graph drawn boldly in orange. (Ignore the parts of the curve in blue covered by gray boxes.) </p>
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What is most likely to be the graph of [mathjaxinline]\ f'[/mathjaxinline] over this interval? </p>
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Sign of second derivative
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Given the correct answer above, should we expect [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline] to be positive, negative, or zero over this interval? </p>
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<h2 class="hd hd-2 unit-title">8. Concavity</h2>
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More signs of derivatives
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Consider the portion of the graph drawn boldly in orange. (Ignore the parts of the curve in blue covered by gray boxes.) </p>
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What are the signs of [mathjaxinline]\ f'[/mathjaxinline] and [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline] in this interval? </p>
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<text> Both are positive</text>
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<text> Both are negative</text>
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Even more signs of derivatives
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What are the signs of [mathjaxinline]\ f'[/mathjaxinline] and [mathjaxinline]\ f^{\prime \prime }[/mathjaxinline] in this interval? </p>
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<h2 class="hd hd-2 unit-title">9. Second derivative and concavity</h2>
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<h2 class="hd hd-2 unit-title">10. Determining concavity</h2>
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Determine concavity
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For each of these functions, determine whether its graph is always concave up, always concave down, sometimes concave up and sometimes concave down, or none of these. </p>
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<p style="display:inline">[mathjaxinline]\ f(x) = x^3 - 2x[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]g(x) = 30 + 7x - 2x^{2} - x^{10}[/mathjaxinline] </p>
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<option value="Always concave up"> Always concave up</option>
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<p style="display:inline">[mathjaxinline]h(x) = 3x - 2[/mathjaxinline] </p>
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Zero second derivative?
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Which of the following types of functions have a second derivative equal to zero at all points? </p>
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(Select all that apply.) </p>
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<text>All quadratic functions (parabolas) [mathjaxinline]\ f(x) = ax^2 + bx + c[/mathjaxinline]</text>
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<text>All constant functions [mathjaxinline]h(x) = c[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">11. Second derivative and concavity summary</h2>
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On intervals where [mathjaxinline]\ f^{\prime \prime } > 0[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is concave up. (Depicted in bold orange below.) </p><center><img src="/assets/courseware/v1/51e4935d434a86fb05fcb5a7ee76c350/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_concaveupgraph.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
On intervals where [mathjaxinline]\ f^{\prime \prime } < 0[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is concave down. (Depicted in bold orange below.) </p><center><img src="/assets/courseware/v1/ce792b66460f76e31c504bb3b8169028/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_concavedowngraph.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
Points where the graph of a function changes from concave up to concave down, or vice versa, are called <span style="color:#27408C"><b class="bf">inflection points</b></span>. </p>
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<h2 class="hd hd-2 unit-title">12. Picture problems</h2>
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Signs of a function&#39;s derivatives
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Given this graph of a function [mathjaxinline]\ f[/mathjaxinline], determine the sign of the first and second derivatives at the marked points. (The graph of [mathjaxinline]\ f[/mathjaxinline] is meant to be straight when [mathjaxinline]x&lt;0[/mathjaxinline].) <p style="display:inline">[mathjaxinline]\ f'(-1)[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input inline">
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<option value="Positive"> Positive</option>
<option value="Negative"> Negative</option>
<option value="Zero"> Zero</option>
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</div></div> <br/><p style="display:inline">[mathjaxinline]\ f^{\prime \prime }(-1)[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input inline">
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<option value="Positive"> Positive</option>
<option value="Negative"> Negative</option>
<option value="Zero"> Zero</option>
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<option value="Positive"> Positive</option>
<option value="Negative"> Negative</option>
<option value="Zero"> Zero</option>
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Changing concavity
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<p>
Locate all of the values of [mathjaxinline]x[/mathjaxinline] where the graph changes from concave up to concave down, or vice versa. </p>
<p>
(If there is more than one, separate them with a comma; use integers only. If there aren't any, type <b class="bf">none</b>.) </p>
<p>
Points where the graph of a function changes from concave up to concave down, or vice versa, are called <span style="color:#27408C"><b class="bf">inflection points</b></span>. </p>
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<img src="/assets/courseware/v1/dc86053389226fd761f0c73292902ee1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_inflectionid.svg" style="margin: 5px 5px 5px 5px; border:0px" width="400 px"/>
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<h2 class="hd hd-2 unit-title">13. Real world meaning</h2>
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Headline news
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<p>
Let's go back to our headline, &#8220;Rate of job growth slows". </p>
<p>
If this headline occurred at the end of the second quarter, which of these might describe the performance of the economy over quarters 1 and 2? </p>
<p>
(Select all that are possible. Assume the headline writer is being accurate! ) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_6-tab13-problem1_2_1">
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<text>[mathjaxinline]+200,000[/mathjaxinline] jobs/month in first quarter, [mathjaxinline]+300,000[/mathjaxinline] jobs/month in second quarter</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_der_6-tab13-problem1_2_1[]" id="input_der_6-tab13-problem1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="der_6-tab13-problem1_2_1-choice_1-label" for="input_der_6-tab13-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_6-tab13-problem1_2_1">
<text>[mathjaxinline]+200,000[/mathjaxinline] jobs/month in first quarter, [mathjaxinline]+200,000[/mathjaxinline] jobs/month in second quarter</text>
</label>
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<input type="checkbox" name="input_der_6-tab13-problem1_2_1[]" id="input_der_6-tab13-problem1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="der_6-tab13-problem1_2_1-choice_2-label" for="input_der_6-tab13-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_6-tab13-problem1_2_1">
<text>[mathjaxinline]+200,000[/mathjaxinline] jobs/month in first quarter, [mathjaxinline]+100,000[/mathjaxinline] jobs/month in second quarter</text>
</label>
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<div class="field">
<input type="checkbox" name="input_der_6-tab13-problem1_2_1[]" id="input_der_6-tab13-problem1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="der_6-tab13-problem1_2_1-choice_3-label" for="input_der_6-tab13-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab13-problem1_2_1">
<text>[mathjaxinline]+200,000[/mathjaxinline] jobs/month in first quarter, [mathjaxinline]-100,000[/mathjaxinline] jobs/month in second quarter</text>
</label>
</div>
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<input type="checkbox" name="input_der_6-tab13-problem1_2_1[]" id="input_der_6-tab13-problem1_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="der_6-tab13-problem1_2_1-choice_4-label" for="input_der_6-tab13-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_der_6-tab13-problem1_2_1">
<text>[mathjaxinline]+200,000[/mathjaxinline] jobs/month in first quarter, [mathjaxinline]-300,000[/mathjaxinline] jobs/month in second quarter</text>
</label>
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<div class="field">
<input type="checkbox" name="input_der_6-tab13-problem1_2_1[]" id="input_der_6-tab13-problem1_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="der_6-tab13-problem1_2_1-choice_5-label" for="input_der_6-tab13-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_der_6-tab13-problem1_2_1">
<text>[mathjaxinline]-100,000[/mathjaxinline] jobs/month in first quarter, [mathjaxinline]-200,000[/mathjaxinline] jobs/month in second quarter</text>
</label>
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<input type="checkbox" name="input_der_6-tab13-problem1_2_1[]" id="input_der_6-tab13-problem1_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="der_6-tab13-problem1_2_1-choice_6-label" for="input_der_6-tab13-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_der_6-tab13-problem1_2_1">
<text>[mathjaxinline]-100,000[/mathjaxinline] jobs/month in first quarter, [mathjaxinline]-50,000[/mathjaxinline] jobs/month in second quarter</text>
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<span id="answer_der_6-tab13-problem1_2_1"/>
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Signs of first and second derivatives
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A newspaper article says that &#8220;Rate of population loss in Capitol City eases." If [mathjaxinline]p(t)[/mathjaxinline] is population as a function of time, what is the article implying about the signs of [mathjaxinline]p^{\prime }(t)[/mathjaxinline] and [mathjaxinline]p^{\prime \prime }(t)[/mathjaxinline]? </p>
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<text> Both [mathjaxinline]p^{\prime }(t)[/mathjaxinline] and [mathjaxinline]p^{\prime \prime }(t)[/mathjaxinline] are positive</text>
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<text> Both [mathjaxinline]p^{\prime }(t)[/mathjaxinline] and [mathjaxinline]p^{\prime \prime }(t)[/mathjaxinline] are negative</text>
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<input type="radio" name="input_der_6-tab13-problem2_2_1" id="input_der_6-tab13-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_6-tab13-problem2_2_1-choice_3-label" for="input_der_6-tab13-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab13-problem2_2_1">
<text> [mathjaxinline]p^{\prime }(t)[/mathjaxinline] is positive but [mathjaxinline]p^{\prime \prime }(t)[/mathjaxinline] is negative</text>
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</div>
<div class="field">
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<text> [mathjaxinline]p^{\prime }(t)[/mathjaxinline] is negative but [mathjaxinline]p^{\prime \prime }(t)[/mathjaxinline] is positive</text>
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Graphing
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Which of the following could be the graph of [mathjaxinline]p[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">14. Acceleration</h2>
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<h3 class="hd hd-2">Acceleration</h3>
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<h2 class="hd hd-2 unit-title">15. Acceleration of pumpkin at the top</h2>
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Pumpkin velocity at the top
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One thing we did not talk about in the video is acceleration at the instant when the pumpkin reaches the top of its trajectory. We will get to that in the next few questions. </p>
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First, what is the pumpkin's instantaneous <b class="bf">velocity</b> at the moment it reaches its highest point? </p>
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Zero velocity and acceleration
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True or False: Suppose that [mathjaxinline]\ f^{\prime }(5) = 0[/mathjaxinline]. Then [mathjaxinline]\ f^{\prime \prime } (5)[/mathjaxinline] must equal 0. </p>
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What is acceleration at the top?
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True or False: When the pumpkin is at the top of its trajectory, its acceleration is 0. </p>
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<h2 class="hd hd-2 unit-title">16. Acceleration problems</h2>
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Acceleration units
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Suppose that position is given in meters [mathjaxinline]m[/mathjaxinline], and time in seconds [mathjaxinline]s[/mathjaxinline]. </p>
<p>
What are the units of velocity? </p>
<p>
(Type m for meters. Type s for seconds. Type [mathjaxinline]*[/mathjaxinline] for multiplication; e.g. a[mathjaxinline]*[/mathjaxinline]b for [mathjaxinline]ab[/mathjaxinline]. Type [mathjaxinline]\wedge[/mathjaxinline] for exponents; e.g. a[mathjaxinline]\wedge[/mathjaxinline]2 for [mathjaxinline]a^2[/mathjaxinline]. Type / for division; e.g. a/b for [mathjaxinline]\displaystyle \frac{a}{b}.[/mathjaxinline]) </p>
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What are the units of acceleration? </p>
<p>
(Type m for meters. Type s for seconds. Type [mathjaxinline]*[/mathjaxinline] for multiplication; e.g. a[mathjaxinline]*[/mathjaxinline]b for [mathjaxinline]ab[/mathjaxinline]. Type [mathjaxinline]\wedge[/mathjaxinline] for exponents; e.g. a[mathjaxinline]\wedge[/mathjaxinline]2 for [mathjaxinline]a^2[/mathjaxinline]. Type / for division; e.g. a/b for [mathjaxinline]\displaystyle \frac{a}{b}.[/mathjaxinline]) </p>
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Parachute
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Suppose that a skydiver jumps out of an airplane at time [mathjaxinline]t=0[/mathjaxinline] seconds. Her height as a function of time is given by [mathjaxinline]h(t)[/mathjaxinline]. Assume that the initial vertical velocity [mathjaxinline]h^{\prime }(0)[/mathjaxinline] is zero. The parachute is closed to begin with, and after 15 seconds the skydiver reaches her maximum downward (terminal) velocity. At [mathjaxinline]t=30[/mathjaxinline] seconds, the parachute is opened, and by [mathjaxinline]t=45[/mathjaxinline] seconds, the velocity has stabilized once again. </p>
<p>
Sketch a graph of [mathjaxinline]h'(t)[/mathjaxinline] between [mathjaxinline]t=0[/mathjaxinline] and [mathjaxinline]t=60[/mathjaxinline]. (Hint: It may be easier to draw the function in several segments. You do not need to draw it with one continuous line.) </p>
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Parachute acceleration
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<p>
For the function [mathjaxinline]h[/mathjaxinline] described above, what is the sign of [mathjaxinline]h^{\prime \prime }(10)[/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_6-tab16-problem3_2_1">
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<text> Positive</text>
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<input type="radio" name="input_der_6-tab16-problem3_2_1" id="input_der_6-tab16-problem3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_6-tab16-problem3_2_1-choice_2-label" for="input_der_6-tab16-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_6-tab16-problem3_2_1">
<text> Negative</text>
</label>
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<input type="radio" name="input_der_6-tab16-problem3_2_1" id="input_der_6-tab16-problem3_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_6-tab16-problem3_2_1-choice_3-label" for="input_der_6-tab16-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab16-problem3_2_1">
<text> Zero</text>
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What is the sign of [mathjaxinline]h^{\prime \prime }(20)[/mathjaxinline]? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_der_6-tab16-problem3_3_1">
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<input type="radio" name="input_der_6-tab16-problem3_3_1" id="input_der_6-tab16-problem3_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="der_6-tab16-problem3_3_1-choice_1-label" for="input_der_6-tab16-problem3_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_der_6-tab16-problem3_3_1">
<text> Positive</text>
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<text> Negative</text>
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<input type="radio" name="input_der_6-tab16-problem3_3_1" id="input_der_6-tab16-problem3_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_6-tab16-problem3_3_1-choice_3-label" for="input_der_6-tab16-problem3_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab16-problem3_3_1">
<text> Zero</text>
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What is the sign of [mathjaxinline]h^{\prime \prime }(40)[/mathjaxinline]? </p>
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<text> Positive</text>
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<input type="radio" name="input_der_6-tab16-problem3_4_1" id="input_der_6-tab16-problem3_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="der_6-tab16-problem3_4_1-choice_2-label" for="input_der_6-tab16-problem3_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_der_6-tab16-problem3_4_1">
<text> Negative</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_der_6-tab16-problem3_4_1" id="input_der_6-tab16-problem3_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="der_6-tab16-problem3_4_1-choice_3-label" for="input_der_6-tab16-problem3_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_der_6-tab16-problem3_4_1">
<text> Zero</text>
</label>
</div>
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<h2 class="hd hd-2 unit-title">17. Summary</h2>
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<p><b class="bfseries">Second derivative</b></p><p>
The second derivative of a function [mathjaxinline]\ f(x)[/mathjaxinline] is the first derivative of [mathjaxinline]\ f'(x)[/mathjaxinline], and is denoted by [mathjaxinline]f^{\prime \prime }(x)[/mathjaxinline] or [mathjaxinline]\displaystyle \frac{d^2f}{dx^2}[/mathjaxinline]. </p><p><b class="bfseries">Higher derivatives</b></p><p>
The [mathjaxinline]n[/mathjaxinline]th derivative of a function [mathjaxinline]\ f(x)[/mathjaxinline] is the first derivative of [mathjaxinline]\ f^{(n-1)}(x)[/mathjaxinline], and is denoted by [mathjaxinline]f^{(n)}(x)[/mathjaxinline] or [mathjaxinline]\displaystyle \frac{d^ nf}{dx^ n}[/mathjaxinline]. </p><p><b class="bfseries">Second derivative and concavity summary</b></p><p>
On intervals where [mathjaxinline]\ f^{\prime \prime } > 0[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is concave up. </p><center><img src="/assets/courseware/v1/51e4935d434a86fb05fcb5a7ee76c350/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_concaveupgraph.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
On intervals where [mathjaxinline]\ f^{\prime \prime } < 0[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is concave down. </p><center><img src="/assets/courseware/v1/ce792b66460f76e31c504bb3b8169028/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_concavedowngraph.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
Points where the graph of a function changes from concave up to concave down, or vice versa, are called <span style="color:#27408C"><b class="bf">inflection points</b></span>. </p><p><b class="bfseries">Position, velocity, acceleration</b></p><p>
If [mathjaxinline]x(t)[/mathjaxinline] is a function that describes position as a function of time, then: </p><ul class="itemize"><li><p>
[mathjaxinline]x'(t)[/mathjaxinline] is the velocity, and </p></li><li><p>
[mathjaxinline]x^{\prime \prime }(t)[/mathjaxinline] is the acceleration. </p></li></ul>
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