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Differentiablilty & Equation of Tangent Line

# Differentiablilty & Equation of Tangent Line - -3 x x-4...

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1.4B & 1.5B Where is a function not differentiable Equation of a Tangent line

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1) A function f ( x ) is not differentiable at a point x = a , if there is a “corner” (or sharp point) at a . Where a Function is Not Differentiable :
2) A function f ( x ) is not differentiable at a point x = a , if there is a vertical tangent at a . Where a Function is Not Differentiable:

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3) A function f ( x ) is not differentiable at a point x = a , if it is not continuous at a . Example: g ( x ) is not continuous at –2, so g ( x ) is not differentiable at x = –2. Where a Function is Not Differentiable:
Find the points on the graph of f at which the tangent line is horizontal.

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Recall that: the derivative is the slope of the tangent line the slope of a horizontal line is 0. Therefore, we wish to find all the points on the graph of f where the derivative of f equals 0.
First find the derivative of f: f ( x ) = - 3 x 3 - 1 + 6 2 x 2 - 1 f ( x ) = - 3 x 2 + 12 x Find points at which the tangent line to is horizontal. f ( x ) = - x 3 + 6 x 2

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Then find where m = 0 by setting To obtain ( 29 0 f x

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Unformatted text preview: -3 x ( x-4) =-3 x = x-4 = x = x = 4 Now to find the points, you must find the corresponding y-values for these x-values by substituting back into x=0 x=4 Thus, the tangent line to the graph of f is horizontal at the points (0, 0) and (4, 32). f ( x ) = -x 3 + 6 x 2 . f (0) = -3 + 6 ⋅ 2 f (0) = f (4) = -4 3 + 6 ⋅ 4 2 f (4) = 32 Graph for this example Write the equation of the tangent line at a given point on the graph Write the equation of the tangent line to the graph of at the point (-3,9) First find the slope using m = . For this function . At the point (-3,9), f x ( 29 = x 2 , ′ f x ( 29 = 2 x ( 29 ( 29 3 2 3 6 m f ′ =-=-= -( 29 f x ′ Now use m = -6 and the point (-3,9) to write the equation for the tangent line ( 29 ( 29 y x y x y x y x- = -- -- = -+- = --= --9 6 3 9 6 3 9 6 18 6 9 Thus the tangent line to the graph of at the point (-3,9) is the line f x ( 29 = x 2 , y x = --6 9...
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