Overview
In practice, we often have to deal with functions which depend on one or several unknown constants, or “parameters”, and we want to study their behaviour (for example local extrema, increasing/decreasing intervals, inflection points and concavity intervals) in relation to these parameters. Since we actually get different functions for different values for the parameter(s), we say that we are dealing with a “family of functions” (we also sometimes specify the number of parameters, such as a one-parameter family, or a two-parameter family of functions).
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Define what is meant by a parameter in the context of a function, or give an example of a function which has a parameter. Describe how a parameter is different from a variable.
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Compute derivatives of functions involving one or more parameters.
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Construct the sign chart(s) for a function involving one or more parameters.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Fully describe the behavior of a typical member of a family of functions in terms of its parameter(s), including the location of all critical numbers, where the function is increasing, decreasing, concave up, and concave down, its long term behavior, and its asymptotes (horizontal, vertical, oblique). Sketch the graph.
To prepare for class
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Read the beginning of section 3.2 in Active Calculus, up to the Preview Activity.
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Do the Preview Activity for 3.2 (on WeBWorK if required by your teacher).
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Watch the following videos, making yourself a list of at least 3 things you learned from the videos or your reading - and include this as part of your feedback:
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Watch the following video, but ignore (and skip) the segment from 3:40 to 6:45, which is unnecessarily complicated (see my note below on how to do this simplification much faster and easier):
Note: The simplification starting at 3:40 in the video can (and should) be done much simpler by factoring out \((b-x^2)^{-3/2}\):
$$\begin{aligned}[t] &-a\left[(b-x^2)^{-1/2} +x^2(b-x^2)^{-3/2} \right] \\&= -a(b-x^2)^{-3/2}\left[(b-x^2)^{-1/2+3/2}+x^2 \right] \\&= -a(b-x^2)^{-3/2} \left[ (b-x^2) + x^2 \right] \\&= \frac{-ab}{(b-x^2)^{3/2}} \end{aligned}$$
