Module 10  Derivative of a Function  
Introduction  Lesson 1  Lesson 2  Lesson 3  Self Test  
Lesson 10.2: Local Linearity  
In Lesson 10.1 you found an equation of the line tangent to a function at a given point. In this lesson you will explore the appearance of the graphs of the function and its tangent line as you zoom in around the point. A function is locally linear around a point if it looks like a line in a small region around the point. You can often determine if a function has local linearity at a point by viewing the graph of the function in a small window centered at that point. You can not be sure how small your window should be to make the determination. Also the limitations of technology can cause graphs to be misrepresented in very small windows. Zooming In Determine if the function f(x) = 2x  x^{2} is locally linear at (0.5, 0.75) by zooming in around that point.
The TI83 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 would look linear around a point in a small enough window, 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:
Display the graphs of the functions to illustrate which one is differentiable at x = 0.
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.
10.2.1 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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