Overview
We now start Chapter 3: “Using the Derivative”. The theme of this chapter centers on what we can learn from key information regarding the derivative of a function. In the first section, Section 3.1, we focus on how the derivative detects extreme values of functions. That is, we investigate how information from the derivative function can tell us whether the original function has a relative maximum or relative minimum at a given point. While many of the ideas in this section will be natural and intuitive (and ones we’ve discussed briefly to some extent earlier in the course), there is considerable new language and reasoning to learn and understand.
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 the concepts of local maximum, local minimum, global maximum, and global minimum. Explain the difference between local extrema and global extrema.
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State the definition of a critical number.
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Find all critical numbers of a function, given its derivative.
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State the First Derivative Test and explain both its purpose and how it is used.
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State and understand the following result: If the function \(f\) has a local extremum at \(x=c\), then \(c\) is a critical number of \(f\). Understand why the opposite is false: If \(c\) is a critical number of \(f\), then there is not necessarily a local extremum at \(c\).
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State the Second Derivative Test and explain both its purpose and how it is used.
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Define the concept of an inflection point.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Make a sign chart for the first derivative and then use it to determine the intervals of the increasing/decreasing behaviour, as well as the location of extreme values of a function.
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Make a sign chart for the second derivative and then use it to determine the intervals of the concave up/concave down behaviour, as well as the location of inflection points of a function.
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Classify all critical points of a function as local max, min, or neither, using the first or second derivative test.
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Use a sign chart to determine the intervals on which a function is concave up or concave down and to find inflection points.
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Use sign charts for the first and second derivative to sketch the graph of a function. (Construct sign charts for functions to find where they are increasing and decreasing, and concave up or concave down, and consequently to find extreme values and inflection points.)
To prepare for class
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Read the introduction of Section 3.1 in Active Calculus up to the Preview Activity.
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Watch the following video (make yourself a list of at least 3 things you learned from this and the following videos and include it in your feedback):
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Do questions 1-4 of the Preview Activity for 3.1 (on WeBWork if required by your teacher). If you get stuck on questions 3-4, continue with the instructions until after the next video, then go back to finish them.
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Read the subsection Critical numbers and the first derivative test, stopping before Activity 3.1.2.
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Watch the following video which shows how to identify critical points on a graph:
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Do questions 3 and 4 of the Preview Activity for 3.1, if you didn’t finish them before.
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Watch the following video which shows how to compute critical numbers given the expression of a function:
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Do question 5 of the Preview Activity for 3.1 (this exists only on WeBWorK).
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Read the First Derivative Test and the Second Derivative Test in the text and watch the following video which summarizes the main ideas of the section:
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Do some experimentation with the following interactive applets:
- First derivative test: Reconstruct \(f\) from its First Derivative
- Second derivative test: Reconstruct \(f\) from its Second Derivative
- Derivatives and the Shape of a Graph
After class
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Watch the following video which shows how to apply the First Derivative Test in practice, by using the sign table of the derivative to determine the intervals of increasing/decreasing behaviour of a function:
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Watch the following video which shows how to apply the Second Derivative Test in practice (video created at CCSL):
