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3.2 Differentiability

Photo by Vickie Kelly, 2003. 3.2 Differentiability. Arches National Park. Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts. Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington.

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3.2 Differentiability

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  1. Photo by Vickie Kelly, 2003 3.2 Differentiability Arches National Park Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

  2. Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington Arches National Park

  3. What function is this? Does the window setting matter?

  4. ZoomTrig view of the function!

  5. For a function like y = x^5 – 6x… Zooming in makes the curve straighten out… “Differentiable functions are continuous and locally linear” (but not vertical!)…

  6. A derivative will fail to existat any x-value where the slope of f(x) changes drastically or is undefined, or at an x-value where f(x) is discontinuous: corner at x = 0 cusp at x = 0 discontinuity at x = 0 vertical tangent at x = 0

  7. Most of the functions we study in calculus will be differentiable!

  8. Derivatives on the TI-83/84: You must be able to calculate derivatives with the calculator and without (using limits) .

  9. nDeriv( x ^ 3, x, 2) ENTER Find at x = 2 Example: From your home screen, the MATH 8 command calculates the derivative of y1 “at” a point; the syntax is: nDeriv(function, independent variable, coordinate) 12 From your y= screen, the MATH 8 command calculates and plots the derivative of function y1at all xvalues in the window: the SLOPES along y1 are graphed as the HEIGHTS on y2 y1 = x^3 y2 = nDeriv(y1, x, x)

  10. W a r n I n g: The calculator can return an incorrect value if you try to evaluate a derivative at a point where the function is not differentiable (at a discontinuity, a cusp, a corner, or a vertical tangent location!). This is known as “grapher failure.” Examples: nDeriv(1/x,x,0) returns 1,000,000 (or some other large number!) nDeriv(abs(x),x,0) returns 0

  11. Graphing Derivatives You may recognize the patterns of some derivative graphs! Graph: Y1=nDeriv(lnx, x, x) This graph looks like:

  12. Iff has a derivative at x = a, thenf is continuous at x = a. There are two theorems on page 110: (Since a function must be continuous to have a derivative: limit = f(a) = limit x →a- x→a+ then each function that has a derivative is already continuous on its domain.) A typical logic error by beginning calculus students is to try to switch the “if” and the “then”, thereby creating the converse (which may or MAY NOT be true!!!)

  13. If a and b are any two x-values in an interval on which f is differentiable, then takes on every value between and . Intermediate Value Theorem for Derivatives The slope takes on every value between the slope at aand the slope at b …for this function, every slope between ½ and 3. p

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