Section 3.1 slides - Definition of the Derivative of a...

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Assignment 4 Section 3.1 The Derivative and Tangent Line Problem
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The Basic Question is… How do you find the equation of a line that is tangent to a function y=f(x) at an arbitrary point P? To find the equation of a line you need: a point and a slope
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How do you find the slope when the line is a tangent line?
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First, we approximate with the secant line. h x f h x f m ) ( ) ( sec - + =
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How do we make the approximation better? Choose h smaller… And smaller… And smaller… And smaller… How close to zero can it get? Infinitely
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Definition of slope of the tangent line If f(x) is defined on an open interval (a,b) then the slope of the tangent line to the graph of y=f(x) at an arbitrary point (x,f(x)) is given by: h x f h x f m h ) ( ) ( lim 0 - + =
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Example: #6—Find the slope of the tangent line to the graph of the function at the given point. (-2, -2) 2 5 ) ( x x g - =
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The limit that is the slope of the tangent line is actually much more. .
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Unformatted text preview: Definition of the Derivative of a Function The derivative of f at x is given by Provided the limit exists. For all x for which the limit exists, is a function of x. h x f h x f x f h ) ( ) ( lim ) ( '-+ = ' f Notations for derivative ] [ )] ( [ ' ) ( ' y D x f dx d y dx dy x f x Find the derivative by the limit process. #20 #24 2 3 ) ( x x x f + = x x f 4 ) ( = Find an equation of the tangent line to th graph of f at the given point. #26 ( - 3, 4) 1 2 ) ( 2 + + = x x x f #34 Find an equation of the line that is tangent to the graph of f and parallel to the given line. 2 ) ( 3 + = x x f 4 3 =--y x Sketch the graph of f #46 What destroys the derivative at a point? a) Cusps b) Corners c) Vertical tangents And Points of Discontinuity Fact: If a function is differentiable at x=c, then f is continuous at x=c...
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Section 3.1 slides - Definition of the Derivative of a...

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