1. THE DERIVATIVE AND THE TANGENT LINE PROBLEM
Section 2.1

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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

8. Find the slope of the graph of at4 9 1
Does not exist

9. 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.

10. Evaluate the derivative of-1 1
Does not exist

11. THEOREM: Differentiability Implies Continuity
If f is differentiable at
then f is continuous at