Assignment 4 Section 3.1 The Derivative and Tangent Line Problem.

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Presentation transcript:

Assignment 4 Section 3.1 The Derivative and Tangent Line Problem

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 pointand a slope

How do you find the slope when the line is a tangent line?

First, we approximate with the secant line.

How do we make the approximation better? Choose h smaller… And smaller… How close to zero can it get? Infinitely

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:

Example: #6—Find the slope of the tangent line to the graph of the function at the given point. (-2, -2)

The limit that is the slope of the tangent line is actually much more.. 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.

Notations for derivative

Find the derivative by the limit process. #20 #24

Find an equation of the tangent line to th graph of f at the given point. #26 » ( - 3, 4)

#34 Find an equation of the line that is tangent to the graph of f and parallel to the given line.

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