Overview
We are about to wrap up Chapter 1 of the text, titled “Understanding the Derivative.” One key aspect of understanding the derivative is how a differentiable function is locally linear. That is, how a differentiable function, up close, looks like a line. This enables us to use linear functions – the simplest functions in all of mathematics – as an effective tool to estimate the values of a differentiable function for \(x\)-values near a certain point where we know key information. Here, we are basically using some sophisticated ideas from calculus to do something natural: if we can see or identify a trend in how a function is changing at a given point, what might we predict for the future? Following a tangent line is a good approach to doing so.
This section covers the following concepts: Differentiability, Relationship between Differentiability and Continuity, Linearization 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.
- State informally what it means for a function to be differentiable at a point (use the expression locally linear).
- Determine whether a function is differentiable at a point by examining the graph of that function.
- Provide examples of functions that are continuous and yet not differentiable at a specific point.
- (Algebra review) Given the slope m of a line and a point \((x_0,y_0)\) (not necessarily the y-intercept) on that line, state an equation for that line in point-slope form and in slope-intercept form.
- Given the value of the derivative of \(f\) at a point \(x = a\) (that is, given \(f'(a)\)), write the equation of the tangent line to the graph of \(f\) at \(x = a\).
- Explain what is meant by the local linearization of a function \(f\) at the point \(x = a\).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Explain the relationship between continuity and differentiability.
- Compute the linear approximation of a function at a specific point.
- Use the linear approximation/local linearization of a function at \(x=a\) to approximate values of \(f\) near \(x=a\).
- Use the second-order derivative to determine whether an approximation is an overestimate or an underestimate. (If \(L(x)\) is the local linearization of a function \(f(x)\) at \(x = a\), and if \(b\) is some point near \(a\), determine whether \(L(b)\) is greater than, less than, or equal to \(f(b)\) and explain.)
- Explain how the concept of local linearity helps us explain that \(\displaystyle \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\).
To prepare for class
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Read subsection 1.7.3 in Active Calculus: “Being differentiable at a point”.
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Watch the following video:
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Read the first theorem and its proof on this page, which shows that if a function is differentiable at a point then it must be continuous at that point as well. Read the statement of this important result and try to understand the proof.
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Do the Preview Activity for Section 1.8 (on WeBWorK if required by your teacher).
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Watch the following videos to further increase your understanding of the material:
After class
- Finish any in-class activities you might not have finished during class.
- Do the problems on the WeBWorK assignment for this section.
