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 1 Does not exist

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