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start on section 3.3 #12

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1c33404
start on section 3.3
oscarlevin Oct 30, 2025
c0ac789
add shells for the rest of ch 3
oscarlevin Nov 6, 2025
3984439
finish ch 3
oscarlevin Nov 7, 2025
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<title>Using Derivatives</title>
<xi:include href="./sec-3-1-rel-rates.xml" />
<xi:include href="./sec-3-2-LHR.xml" />
<!-- <xi:include href="./sec-3-3-tests.xml" /> -->
<!-- <xi:include href="./sec-3-4-families.xml" /> -->
<!-- <xi:include href="./sec-3-5-optimization.xml" /> -->
<!-- <xi:include href="./sec-3-6-applied-opt.xml" /> -->
<xi:include href="./sec-3-3-tests.xml" />
<xi:include href="./sec-3-4-families.xml" />
<xi:include href="./sec-3-5-optimization.xml" />
<xi:include href="./sec-3-6-applied-opt.xml" />
</chapter>

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<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec-3-4-families">
<title>Using derivatives to describe families of functions</title>
<objectives>
<!-- <objectives>
<ul>
<li>
<p>
Expand All @@ -18,263 +18,20 @@
</p>
</li>
</ul>
</objectives>

<subsection><title>Introduction</title>
<p>
Mathematicians are often interested in making general observations to describe patterns that hold in a large number of related situations.
Think about the Pythagorean Theorem:
it doesn't tell us something about a single right triangle,
but rather a fact about <em>every</em> right triangle.
In the next part of our studies,
we use calculus to make general observations about families of functions that depend on one or more parameters.
People who use applied mathematics,
such as engineers and economists,
often encounter the same types of functions where only small changes to certain constants occur.
These constants are called <em>parameters</em>.
</p>

<figure xml:id="F-3-2-SineFam">
<caption>The graph of <m>f(t) = a \sin(b(t-c)) + d</m> based on parameters <m>a</m>, <m>b</m>, <m>c</m>, and <m>d</m>.</caption>
<image width="65%" source="images/3_2_SineFam"/>
</figure>

<p>
You are already familiar with certain families of functions.
For example,
<me>
f(t) = a \sin(b(t-c)) + d
</me>
is a stretched and shifted version of the sine function with amplitude <m>a</m>,
period <m>\frac{2\pi}{b}</m>,
phase shift <m>c</m>, and vertical shift <m>d</m>.
We know that <m>a</m> affects the size of the oscillation,
<m>b</m> the rapidity of oscillation,
and <m>c</m> where the oscillation starts,
as shown in <xref ref="F-3-2-SineFam">Figure</xref>,
while <m>d</m> affects the vertical positioning of the graph.
</p>

<p>
As another example, every function of the form
<m>y = mx + b</m> is a line with slope <m>m</m> and <m>y</m>-intercept <m>(0,b)</m>.
The value of <m>m</m> affects the line's steepness,
and the value of <m>b</m> situates the line vertically on the coordinate axes.
These two parameters describe all possible non-vertical lines.
</p>

</objectives> -->
<subsection xml:id="subsec-families-foundations">
<title>Foundations</title>
<p>
For other less familiar families of functions,
we can use calculus to discover where key behavior occurs:
where members of the family are increasing or decreasing,
concave up or concave down,
where relative extremes occur, and more,
all in terms of the parameters involved.
To get started,
we revisit a common collection of functions to see how calculus confirms things we already know.
</p>

<xi:include href="./previews/PA-3-4-WW.xml" />
<xi:include href="./previews/PA-3-4.xml" />
</subsection>

<xi:include href="./proteus/proteus-3-2.xml" />

<subsection>
<title>Describing families of functions in terms of parameters</title>
<p>
Our goal is to describe the key characteristics of the overall behavior of each member of a family of functions in terms of its parameters.
By finding the first and second derivatives and constructing sign charts
(each of which may depend on one or more of the parameters),
we can often make broad conclusions about how each member of the family will appear.
</p>

<example xml:id="Ex-3-2-1">
<statement>
<p>
Consider the two-parameter family of functions given by <m>g(x) = axe^{-bx}</m>,
where <m>a</m> and <m>b</m> are positive real numbers.
Fully describe the behavior of a typical member of the family in terms of <m>a</m> and <m>b</m>,
including the location of all critical numbers,
where <m>g</m> is increasing, decreasing, concave up,
and concave down, and the long term behavior of <m>g</m>.
</p>
</statement>
<solution>
<p>
We begin by computing <m>g'(x)</m>.
By the product rule,
<me>
g'(x) = ax \frac{d}{dx}\left[e^{-bx}\right] + e^{-bx} \frac{d}{dx}[ax]
</me>.
By applying the chain rule and constant multiple rule,
we find that
<me>
g'(x) = axe^{-bx}(-b) + e^{-bx}(a)
</me>.
</p>

<p>
To find the critical numbers of <m>g</m>,
we solve the equation <m>g'(x) = 0</m>.
By factoring <m>g'(x)</m>, we find
<me>
0 = ae^{-bx}(-bx + 1)
</me>.
</p>

<p>
Since we are given that <m>a \ne 0</m> and we know that
<m>e^{-bx} \ne 0</m> for all values of <m>x</m>,
the only way this equation can hold is when <m>-bx + 1 = 0</m>.
Solving for <m>x</m>, we find <m>x = \frac{1}{b}</m>,
and this is therefore the only critical number of <m>g</m>.
</p>

<p>
We construct the first derivative sign chart for <m>g</m> that is shown in <xref ref="F-3-2-signchartg">Figure</xref>.
</p>

<figure xml:id="F-3-2-signchartg">
<caption>The first derivative sign chart for <m>g(x) = axe^{-bx}</m>.</caption>
<image width="45%" source="images/3_2_signchartg"/>
</figure>

<p>
Because the factor <m>ae^{-bx}</m> is always positive,
the sign of <m>g'</m> depends on the linear factor <m>(1-bx)</m>,
which is positive for
<m>x \lt \frac{1}{b}</m> and negative for <m>x \gt \frac{1}{b}</m>.
Hence we can conclude that <m>g</m> is always increasing for
<m>x \lt \frac{1}{b}</m> and decreasing for <m>x \gt \frac{1}{b}</m>,
and also that <m>g</m> has a global maximum at
<m>(\frac{1}{b}, g(\frac{1}{b}))</m> and no local minimum.
</p>

<p>
We turn next to analyzing the concavity of <m>g</m>.
With <m>g'(x) = -abxe^{-bx} + ae^{-bx}</m>,
we differentiate to find that
<me>
g''(x) = -abxe^{-bx}(-b) + e^{-bx}(-ab) + ae^{-bx}(-b)
</me>.
</p>

<p>
Combining like terms and factoring, we now have
<me>
g''(x) = ab^2xe^{-bx} - 2abe^{-bx} = abe^{-bx}(bx - 2)
</me>.
</p>

<figure xml:id="F-3-2-signchartg2">
<caption>The second derivative sign chart for <m>g(x) = axe^{-bx}</m>.</caption>
<image width="45%" source="images/3_2_signchartg2"/>
</figure>

<p>
We observe that <m>abe^{-bx}</m> is always positive,
and thus the sign of <m>g''</m> depends on the sign of <m>(bx-2)</m>,
which is zero when <m>x = \frac{2}{b}</m>.
Since <m>b</m> is positive,
the value of <m>(bx-2)</m> is negative for
<m>x \lt \frac{2}{b}</m> and positive for <m>x \gt \frac{2}{b}</m>.
The sign chart for <m>g''</m> is shown in <xref ref="F-3-2-signchartg2">Figure</xref>.
Thus, <m>g</m> is concave down for all
<m>x \lt \frac{2}{b}</m> and concave up for all <m>x \gt \frac{2}{b}</m>.
</p>

<p>
Finally, we analyze the long term behavior of <m>g</m> by considering two limits.
First, we note that
<me>
\lim_{x \to \infty} g(x) = \lim_{x \to \infty} axe^{-bx} = \lim_{x \to \infty} \frac{ax}{e^{bx}}
</me>.
</p>

<p>
This limit has indeterminate form <m>\frac{\infty}{\infty}</m>, so
we apply L'Hôpital's Rule and find that <m>\lim_{x \to \infty} g(x) = 0</m>.
In the other direction,
<me>
\lim_{x \to -\infty} g(x) = \lim_{x \to -\infty} axe^{-bx} = -\infty
</me>,
because <m>ax \to -\infty</m> and
<m>e^{-bx} \to \infty</m> as <m>x \to -\infty</m>.
Hence, as we move left on its graph,
<m>g</m> decreases without bound,
while as we move to the right, <m>g(x) \to 0</m>.
</p>

<p>
All of this information now helps us produce the graph of a typical member of this family of functions without using a graphing utility
(and without choosing particular values for <m>a</m> and <m>b</m>),
as shown in <xref ref="F-3-2-SurgeFam">Figure</xref>.
</p>

<figure xml:id="F-3-2-SurgeFam">
<caption>The graph of <m>g(x) = axe^{-bx}</m>.</caption>
<image width="80%" source="images/3_2_SurgeFam"/>
</figure>

<p>
Note that the value of <m>b</m> controls the horizontal location of the global maximum and the inflection point,
as neither depends on <m>a</m>.
The value of <m>a</m> affects the vertical stretch of the graph.
For example,
the global maximum occurs at the point <m>(\frac{1}{b}, g(\frac{1}{b})) = (\frac{1}{b}, \frac{a}{b}e^{-1})</m>,
so the larger the value of <m>a</m>,
the greater the value of the global maximum.
</p>
</solution>
</example>

<p>
The work we've completed in <xref ref="Ex-3-2-1">Example</xref>
can often be replicated for other families of functions that depend on parameters.
Normally we are most interested in determining all critical numbers,
a first derivative sign chart,
a second derivative sign chart,
and the limit of the function as <m>x \to \infty</m>.
Throughout,
we prefer to work with the parameters as arbitrary constants.
In addition,
we can experiment with some particular values of the parameters present to reduce the algebraic complexity of our work.
The following activities offer several key examples where we see that the values of the parameters substantially affect the behavior of individual functions within a given family.
</p>

<xi:include href="./activities/act-3-4-1.xml"/>

<xi:include href="./activities/act-3-4-2.xml"/>

<xi:include href="./activities/act-3-4-3.xml"/>
</subsection>

<subsection>
<title>Summary</title>
<p>
<ul>
<li>
<p>
Given a family of functions that depends on one or more parameters,
by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters,
we can often accurately describe the shape of the function in terms of the parameters.
</p>
</li>

<li>
<p>
In particular,
just as we can create first and second derivative sign charts for a single function,
we can often do so for entire families of functions where critical numbers and possible inflection points depend on arbitrary constants.
These sign charts then reveal where members of the family are increasing or decreasing,
concave up or concave down,
and help us to identify relative extremes and inflection points.
</p>
</li>
</ul>
Coming soon.
</p>
</subsection>

<xi:include href="./exercises/ez-3-4.xml"/>
<exercises xml:id="cp-families">
<title>Calculus Practice</title>
<introduction>
<p>
Coming soon.
</p>
</introduction>
</exercises>
</section>
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