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

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Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park

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What function is this? Does the window setting matter?

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ZoomTrig view of the function!

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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!)…

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

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

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

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Example: Find at x = 2. nDeriv( x ^ 3, x, 2 ) ENTER returns 12 From your home screen, the MATH 8 command calculates the derivative of y 1 at a point; the syntax is: nDeriv(function, independent variable, coordinate) y 1 = x^3 y 2 = nDeriv(y 1, x, x) From your y= screen, the MATH 8 command calculates and plots the derivative of function y 1 at all x values in the window: the SLOPES along y 1 are graphed as the HEIGHTS on y 2

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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

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Graphing Derivatives Graph: This graph looks like: Y 1 =nDeriv(lnx, x, x) You may recognize the patterns of some derivative graphs!

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There are two theorems on page 110: If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative: limit = f(a) = limit x a- xa+ then each function that has a derivative is 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!!!)

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

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