Presentation on theme: "THE DERIVATIVE AND THE TANGENT LINE PROBLEM Section 2.1."— Presentation transcript:
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 A.Y = -5x + 19 B.Y = 5x - 11 C.There is not enough information to solve this problem.
A secant line represents the A.Instantaneous rate of change of a function. B.The average rate of change of a function. C.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 A.4 B.9 C.1 D.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 A.-1 B.0 C.1 D.Does not exist
THEOREM: Differentiability Implies Continuity If f is differentiable at then f is continuous at