Overview
We have spent a lot of time so far focusing on the derivative of a function, which we will in the future also call the “first derivative”: its definition, its meaning, and various interpretations in applied contexts. The first derivative, in essence, tells us whether a given function is increasing or decreasing at a certain point. Next, we are interested in learning how a function is increasing or decreasing at a given point, and the key tool to doing so is the second derivative of the function: Here, we consider what happens when we consider the derivative of the derivative of a function.
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.
- Understand the formal definition of the terms increasing and decreasing and their connection to the first derivative of a function.
- Tell whether a function is increasing or decreasing given information about the sign of \(f'\) (the first derivative).
- Begin to think about what we can learn by taking the derivative of the derivative of a function.
- Define what is meant by the second derivative of a function.
- Define what it means for a function to be concave up or concave down on an interval, and graphically identify intervals on which a function is concave up or concave down, given a graph of the function.
- Tell whether a function is concave up, concave down, or linear given information about the sign of \(f''\) (the second derivative).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Compute the second derivative of a function, using the limit definition.
- Sketch the graph of the second derivative \(f''\), given the graph of \(f\).
- Given a function \(y = f(x)\) and the units of measure of both \(x\) and \(y\), state the units of \(f''(x)\).
- Explain what the six different combinations of increasing/decreasing and concave up/concave down/linear mean in real-life terms (such as “increasing at a decreasing rate”).
To prepare for class
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Do the Preview Activity for section 1.6 (on WeBWorK if required by your teacher).
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Watch the following video on the second derivative of a function:
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Read subsection 1.6.3 in Active Calculus, up until and including the definition of “concave up” and “concave down”. Make sure to think about the relationship between concavity (up or down) and the rate of change of the first derivative.
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Watch the following video on determining the concavity of a function from its graph:
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Do some experimentation with the following interactive applet: Identify a Function and its First and Second Derivatives
After class
- (Re-)Read all of Section 1.6 in Active Calculus.
- Finish any in-class activities you might not have finished during class.
- Do the problems on the WeBWorK assignment for this section.
