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3 | 3 |
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4 | 4 | <section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec-3-4-families"> |
5 | 5 | <title>Using derivatives to describe families of functions</title> |
6 | | - <objectives> |
| 6 | + <!-- <objectives> |
7 | 7 | <ul> |
8 | 8 | <li> |
9 | 9 | <p> |
|
18 | 18 | </p> |
19 | 19 | </li> |
20 | 20 | </ul> |
21 | | - </objectives> |
22 | | - |
23 | | - <subsection><title>Introduction</title> |
24 | | - <p> |
25 | | - Mathematicians are often interested in making general observations to describe patterns that hold in a large number of related situations. |
26 | | - Think about the Pythagorean Theorem: |
27 | | - it doesn't tell us something about a single right triangle, |
28 | | - but rather a fact about <em>every</em> right triangle. |
29 | | - In the next part of our studies, |
30 | | - we use calculus to make general observations about families of functions that depend on one or more parameters. |
31 | | - People who use applied mathematics, |
32 | | - such as engineers and economists, |
33 | | - often encounter the same types of functions where only small changes to certain constants occur. |
34 | | - These constants are called <em>parameters</em>. |
35 | | - </p> |
36 | | - |
37 | | - <figure xml:id="F-3-2-SineFam"> |
38 | | - <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> |
39 | | - <image width="65%" source="images/3_2_SineFam"/> |
40 | | - </figure> |
41 | | - |
42 | | - <p> |
43 | | - You are already familiar with certain families of functions. |
44 | | - For example, |
45 | | - <me> |
46 | | - f(t) = a \sin(b(t-c)) + d |
47 | | - </me> |
48 | | - is a stretched and shifted version of the sine function with amplitude <m>a</m>, |
49 | | - period <m>\frac{2\pi}{b}</m>, |
50 | | - phase shift <m>c</m>, and vertical shift <m>d</m>. |
51 | | - We know that <m>a</m> affects the size of the oscillation, |
52 | | - <m>b</m> the rapidity of oscillation, |
53 | | - and <m>c</m> where the oscillation starts, |
54 | | - as shown in <xref ref="F-3-2-SineFam">Figure</xref>, |
55 | | - while <m>d</m> affects the vertical positioning of the graph. |
56 | | - </p> |
57 | | - |
58 | | - <p> |
59 | | - As another example, every function of the form |
60 | | - <m>y = mx + b</m> is a line with slope <m>m</m> and <m>y</m>-intercept <m>(0,b)</m>. |
61 | | - The value of <m>m</m> affects the line's steepness, |
62 | | - and the value of <m>b</m> situates the line vertically on the coordinate axes. |
63 | | - These two parameters describe all possible non-vertical lines. |
64 | | - </p> |
65 | | - |
| 21 | + </objectives> --> |
| 22 | + <subsection xml:id="subsec-families-foundations"> |
| 23 | + <title>Foundations</title> |
66 | 24 | <p> |
67 | | - For other less familiar families of functions, |
68 | | - we can use calculus to discover where key behavior occurs: |
69 | | - where members of the family are increasing or decreasing, |
70 | | - concave up or concave down, |
71 | | - where relative extremes occur, and more, |
72 | | - all in terms of the parameters involved. |
73 | | - To get started, |
74 | | - we revisit a common collection of functions to see how calculus confirms things we already know. |
75 | | - </p> |
76 | | - |
77 | | - <xi:include href="./previews/PA-3-4-WW.xml" /> |
78 | | - <xi:include href="./previews/PA-3-4.xml" /> |
79 | | - </subsection> |
80 | | - |
81 | | - <xi:include href="./proteus/proteus-3-2.xml" /> |
82 | | - |
83 | | - <subsection> |
84 | | - <title>Describing families of functions in terms of parameters</title> |
85 | | - <p> |
86 | | - 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. |
87 | | - By finding the first and second derivatives and constructing sign charts |
88 | | - (each of which may depend on one or more of the parameters), |
89 | | - we can often make broad conclusions about how each member of the family will appear. |
90 | | - </p> |
91 | | - |
92 | | - <example xml:id="Ex-3-2-1"> |
93 | | - <statement> |
94 | | - <p> |
95 | | - Consider the two-parameter family of functions given by <m>g(x) = axe^{-bx}</m>, |
96 | | - where <m>a</m> and <m>b</m> are positive real numbers. |
97 | | - Fully describe the behavior of a typical member of the family in terms of <m>a</m> and <m>b</m>, |
98 | | - including the location of all critical numbers, |
99 | | - where <m>g</m> is increasing, decreasing, concave up, |
100 | | - and concave down, and the long term behavior of <m>g</m>. |
101 | | - </p> |
102 | | - </statement> |
103 | | - <solution> |
104 | | - <p> |
105 | | - We begin by computing <m>g'(x)</m>. |
106 | | - By the product rule, |
107 | | - <me> |
108 | | - g'(x) = ax \frac{d}{dx}\left[e^{-bx}\right] + e^{-bx} \frac{d}{dx}[ax] |
109 | | - </me>. |
110 | | - By applying the chain rule and constant multiple rule, |
111 | | - we find that |
112 | | - <me> |
113 | | - g'(x) = axe^{-bx}(-b) + e^{-bx}(a) |
114 | | - </me>. |
115 | | - </p> |
116 | | - |
117 | | - <p> |
118 | | - To find the critical numbers of <m>g</m>, |
119 | | - we solve the equation <m>g'(x) = 0</m>. |
120 | | - By factoring <m>g'(x)</m>, we find |
121 | | - <me> |
122 | | - 0 = ae^{-bx}(-bx + 1) |
123 | | - </me>. |
124 | | - </p> |
125 | | - |
126 | | - <p> |
127 | | - Since we are given that <m>a \ne 0</m> and we know that |
128 | | - <m>e^{-bx} \ne 0</m> for all values of <m>x</m>, |
129 | | - the only way this equation can hold is when <m>-bx + 1 = 0</m>. |
130 | | - Solving for <m>x</m>, we find <m>x = \frac{1}{b}</m>, |
131 | | - and this is therefore the only critical number of <m>g</m>. |
132 | | - </p> |
133 | | - |
134 | | - <p> |
135 | | - We construct the first derivative sign chart for <m>g</m> that is shown in <xref ref="F-3-2-signchartg">Figure</xref>. |
136 | | - </p> |
137 | | - |
138 | | - <figure xml:id="F-3-2-signchartg"> |
139 | | - <caption>The first derivative sign chart for <m>g(x) = axe^{-bx}</m>.</caption> |
140 | | - <image width="45%" source="images/3_2_signchartg"/> |
141 | | - </figure> |
142 | | - |
143 | | - <p> |
144 | | - Because the factor <m>ae^{-bx}</m> is always positive, |
145 | | - the sign of <m>g'</m> depends on the linear factor <m>(1-bx)</m>, |
146 | | - which is positive for |
147 | | - <m>x \lt \frac{1}{b}</m> and negative for <m>x \gt \frac{1}{b}</m>. |
148 | | - Hence we can conclude that <m>g</m> is always increasing for |
149 | | - <m>x \lt \frac{1}{b}</m> and decreasing for <m>x \gt \frac{1}{b}</m>, |
150 | | - and also that <m>g</m> has a global maximum at |
151 | | - <m>(\frac{1}{b}, g(\frac{1}{b}))</m> and no local minimum. |
152 | | - </p> |
153 | | - |
154 | | - <p> |
155 | | - We turn next to analyzing the concavity of <m>g</m>. |
156 | | - With <m>g'(x) = -abxe^{-bx} + ae^{-bx}</m>, |
157 | | - we differentiate to find that |
158 | | - <me> |
159 | | - g''(x) = -abxe^{-bx}(-b) + e^{-bx}(-ab) + ae^{-bx}(-b) |
160 | | - </me>. |
161 | | - </p> |
162 | | - |
163 | | - <p> |
164 | | - Combining like terms and factoring, we now have |
165 | | - <me> |
166 | | - g''(x) = ab^2xe^{-bx} - 2abe^{-bx} = abe^{-bx}(bx - 2) |
167 | | - </me>. |
168 | | - </p> |
169 | | - |
170 | | - <figure xml:id="F-3-2-signchartg2"> |
171 | | - <caption>The second derivative sign chart for <m>g(x) = axe^{-bx}</m>.</caption> |
172 | | - <image width="45%" source="images/3_2_signchartg2"/> |
173 | | - </figure> |
174 | | - |
175 | | - <p> |
176 | | - We observe that <m>abe^{-bx}</m> is always positive, |
177 | | - and thus the sign of <m>g''</m> depends on the sign of <m>(bx-2)</m>, |
178 | | - which is zero when <m>x = \frac{2}{b}</m>. |
179 | | - Since <m>b</m> is positive, |
180 | | - the value of <m>(bx-2)</m> is negative for |
181 | | - <m>x \lt \frac{2}{b}</m> and positive for <m>x \gt \frac{2}{b}</m>. |
182 | | - The sign chart for <m>g''</m> is shown in <xref ref="F-3-2-signchartg2">Figure</xref>. |
183 | | - Thus, <m>g</m> is concave down for all |
184 | | - <m>x \lt \frac{2}{b}</m> and concave up for all <m>x \gt \frac{2}{b}</m>. |
185 | | - </p> |
186 | | - |
187 | | - <p> |
188 | | - Finally, we analyze the long term behavior of <m>g</m> by considering two limits. |
189 | | - First, we note that |
190 | | - <me> |
191 | | - \lim_{x \to \infty} g(x) = \lim_{x \to \infty} axe^{-bx} = \lim_{x \to \infty} \frac{ax}{e^{bx}} |
192 | | - </me>. |
193 | | - </p> |
194 | | - |
195 | | - <p> |
196 | | - This limit has indeterminate form <m>\frac{\infty}{\infty}</m>, so |
197 | | - we apply L'Hôpital's Rule and find that <m>\lim_{x \to \infty} g(x) = 0</m>. |
198 | | - In the other direction, |
199 | | - <me> |
200 | | - \lim_{x \to -\infty} g(x) = \lim_{x \to -\infty} axe^{-bx} = -\infty |
201 | | - </me>, |
202 | | - because <m>ax \to -\infty</m> and |
203 | | - <m>e^{-bx} \to \infty</m> as <m>x \to -\infty</m>. |
204 | | - Hence, as we move left on its graph, |
205 | | - <m>g</m> decreases without bound, |
206 | | - while as we move to the right, <m>g(x) \to 0</m>. |
207 | | - </p> |
208 | | - |
209 | | - <p> |
210 | | - All of this information now helps us produce the graph of a typical member of this family of functions without using a graphing utility |
211 | | - (and without choosing particular values for <m>a</m> and <m>b</m>), |
212 | | - as shown in <xref ref="F-3-2-SurgeFam">Figure</xref>. |
213 | | - </p> |
214 | | - |
215 | | - <figure xml:id="F-3-2-SurgeFam"> |
216 | | - <caption>The graph of <m>g(x) = axe^{-bx}</m>.</caption> |
217 | | - <image width="80%" source="images/3_2_SurgeFam"/> |
218 | | - </figure> |
219 | | - |
220 | | - <p> |
221 | | - Note that the value of <m>b</m> controls the horizontal location of the global maximum and the inflection point, |
222 | | - as neither depends on <m>a</m>. |
223 | | - The value of <m>a</m> affects the vertical stretch of the graph. |
224 | | - For example, |
225 | | - the global maximum occurs at the point <m>(\frac{1}{b}, g(\frac{1}{b})) = (\frac{1}{b}, \frac{a}{b}e^{-1})</m>, |
226 | | - so the larger the value of <m>a</m>, |
227 | | - the greater the value of the global maximum. |
228 | | - </p> |
229 | | - </solution> |
230 | | - </example> |
231 | | - |
232 | | - <p> |
233 | | - The work we've completed in <xref ref="Ex-3-2-1">Example</xref> |
234 | | - can often be replicated for other families of functions that depend on parameters. |
235 | | - Normally we are most interested in determining all critical numbers, |
236 | | - a first derivative sign chart, |
237 | | - a second derivative sign chart, |
238 | | - and the limit of the function as <m>x \to \infty</m>. |
239 | | - Throughout, |
240 | | - we prefer to work with the parameters as arbitrary constants. |
241 | | - In addition, |
242 | | - we can experiment with some particular values of the parameters present to reduce the algebraic complexity of our work. |
243 | | - 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. |
244 | | - </p> |
245 | | - |
246 | | - <xi:include href="./activities/act-3-4-1.xml"/> |
247 | | - |
248 | | - <xi:include href="./activities/act-3-4-2.xml"/> |
249 | | - |
250 | | - <xi:include href="./activities/act-3-4-3.xml"/> |
251 | | - </subsection> |
252 | | - |
253 | | - <subsection> |
254 | | - <title>Summary</title> |
255 | | - <p> |
256 | | - <ul> |
257 | | - <li> |
258 | | - <p> |
259 | | - Given a family of functions that depends on one or more parameters, |
260 | | - by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, |
261 | | - we can often accurately describe the shape of the function in terms of the parameters. |
262 | | - </p> |
263 | | - </li> |
264 | | - |
265 | | - <li> |
266 | | - <p> |
267 | | - In particular, |
268 | | - just as we can create first and second derivative sign charts for a single function, |
269 | | - we can often do so for entire families of functions where critical numbers and possible inflection points depend on arbitrary constants. |
270 | | - These sign charts then reveal where members of the family are increasing or decreasing, |
271 | | - concave up or concave down, |
272 | | - and help us to identify relative extremes and inflection points. |
273 | | - </p> |
274 | | - </li> |
275 | | - </ul> |
| 25 | + Coming soon. |
276 | 26 | </p> |
277 | 27 | </subsection> |
278 | 28 |
|
279 | | - <xi:include href="./exercises/ez-3-4.xml"/> |
| 29 | + <exercises xml:id="cp-families"> |
| 30 | + <title>Calculus Practice</title> |
| 31 | + <introduction> |
| 32 | + <p> |
| 33 | + Coming soon. |
| 34 | + </p> |
| 35 | + </introduction> |
| 36 | + </exercises> |
280 | 37 | </section> |
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