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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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3.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

3.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

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