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THE DERIVATIVE AND THE TANGENT LINE PROBLEM

- Section 2.1

When you are done with your homework, you should

be able to

- Find the slope of the tangent line to a curve at

a point - Use the limit definition to find the derivative

of a function - Understand the relationship between

differentiability and continuity

The Tangent Line Problem

- How do we find an equation of the tangent line to

a graph at point P? - We can approximate this slope using a secant line

through the point of tangency and a second point

on the curve.

Find the equation of the secant line to the

function at and

- Y -5x 19
- Y 5x - 11
- There is not enough information to solve this

problem.

A secant line represents the

- Instantaneous rate of change of a function.
- The average rate of change of a function.
- Line tangent to a function.

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

this limit exists, f is a function of x.

Definition of Tangent Line with Slope m

- If f is defined on an open interval containing c,

and if the limit - exists, then the line passing through f with

slope m is the tangent line to the graph of at

the point - The slope of the tangent line to the graph of f

at the point c is also called the slope of the

graph of f at

Find the slope of the graph of at

- 4
- 9
- 1
- Does not exist

Alternative limit form of the derivative

- The existence of the limit in this alternative

form requires that the following one-sided limits

- and

- exist and are equal.
- These one-sided limits are called the derivatives

from the left and from the right, respectively.

It follows that f is differentiable on the closed

interval if it is differentiable on and

if the derivatives from the right at a and the

derivative from the left at b

- both exist.

Evaluate the derivative of

- -1
- 0
- 1
- Does not exist

THEOREM Differentiability Implies Continuity

- If f is differentiable at
- then f is continuous at

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