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Linearity and Local Linearity

Linearity and Local Linearity . Linear Functions. Linear Functions. Linear Functions. Our slope of tells us that a change of  x in our independent variable . . . .  x. . . . elicits a change of in our dependent variable. . Linear Functions.

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Linearity and Local Linearity

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  1. Linearity and Local Linearity

  2. Linear Functions

  3. Linear Functions

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

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

  6. 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 

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

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

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

  10. 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 ___.

  11. 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 .

  12. A nice curvy graph

  13. A nice curvy graph Consider a small portion of the graph . . . . . . . shown here in blue.

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

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

  16. “Zooming In” Keep it up. . .

  17. “Zooming In”

  18. “Zooming In”

  19. “Zooming In”

  20. Typical Behavior

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

  22. Local Linearity

  23. Local Linearity 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 faround 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.

  24. Differentiability

  25. The Derivative of f at a Informal Definition: When fis 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 = aand is denoted, symbolically by f ’(a).

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