Linearity and Local Linearity. Linear Functions.

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Presentation transcript:

Linearity and Local Linearity

Linear Functions

Our slope of tells us that a change of  x in our independent variable... xx... elicits a change of in our dependent variable.

Linear Functions If we increase our x-value from 1 to 4, our y-value will_______.  increase  decrease

Linear Functions If we increase our x-value from 1 to 4, our y-value will increase. If we decrease our x-value from 1 to -5, our y-value will______.  increase  decrease

Linear Functions If we increase our x-value from 1 to 4, our y-value will increase from 2 to ____. If we decrease our x-value from 1 to -5, our y-value will decrease from 2 to ___.

Linear Functions  x=3 If we increase our x-value from 1 to 4, our y-value will _______ from 2 to ______.

Linear Functions If we increase our x-value from 1 to 4, our y-value will increase from 2 to 4.

Linear Functions If we increase our x-value from 1 to 4, our y-value will increase from 2 to 4.  x= -6 If we decrease our x-value from 1 to -5, our y-value will _______from 2 to ___.

Linear Functions If we increase our x-value from 1 to 4, our y-value will increase from 2 to 4. If we decrease our x-value from 1 to -5, our y-value will decrease from 2 to -2.

A nice curvy graph

Consider a small portion of the graph shown here in blue.

Zooming Now “zoom in” on the blue part of the graph...

“Zooming In” And repeat the process by zooming in on the part colored in pink...

“Zooming In” Keep it up...

“Zooming In”

Typical Behavior

In general... When we zoom in on a “sufficiently nice” function, we see a straight line.

Informal Definition: A function f is said to be locally linear at x = a, provided that if we "zoom in sufficiently far" on the graph of f around the point (a, f (a)), the graph of f "looks like a straight line." It is locally linear, provided that it is locally linear at every point. Local Linearity

Informal Definition: When f is locally linear at x = a, we have a name for the slope of the line that we see when we zoom in on the graph of f around the point (a, f (a)). This number is called the derivative of f at x = a and is denoted, symbolically by f ’(a). The Derivative of f at a