Module 8  Derivative of a Function  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 8.2: Local Linearity  
In Lesson 8.1 you found the equation of the line tangent to f(x) = 2x – x^{2} at (0.5, 0.75). In this lesson you will explore the appearance of the graphs of y = f(x) and its tangent line as you zoom in around the point (0.5, 0.75). First reset the factory defaults. Click here to see the keystroke sequence to do this. Zooming In
The TI89 prompts you for the zoom center.
The graphs are redrawn in a smaller window whose center is approximately (0.5, 0.75). Zooming in Again The local linearity of the function can be better illustrated by zooming in repeatedly.
In the last window the graphs of the function and the tangent line appear to coincide. That is, the curve appears to be linear on a small scale. Local Linearity and Point Differentiability
If a curve looks linear around a point after zooming in repeatedly, it is said to be locally linear near that point. If a curve is locally linear near a point, the curve is
Determining Differentiability Graphically One of the following functions is differentiable at x = 0 and the other is not: and Displaying the graphs of the functions illustrates which one is differentiable.
The graphs appear to coincide in this viewing window. You may want to use Trace to convince yourself that both graphs are present.
Setting the Zoom Factors Zoom in around the point (0,1) to see which graph is locally linear there. You may want a higher level of magnification to make the process faster. This can be accomplished by changing the Zoom factors.
Zooming In Using the New Factors
At first the graphs will appear to be identical, but as you continue to zoom in, you should see a difference.
8.2.1 Based on the graphs above, which function is differentiable at x = 0 and which is not? Click here for the answer. 

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