Presentation on theme: "THE DERIVATIVE AND THE TANGENT LINE PROBLEM"— Presentation transcript:
1THE DERIVATIVE AND THE TANGENT LINE PROBLEM Section 2.1
2When you are done with your homework, you should be able to… Find the slope of the tangent line to a curve at a pointUse the limit definition to find the derivative of a functionUnderstand the relationship between differentiability and continuity
3The 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.
4Find the equation of the secant line to the function at and Y = -5x + 19Y = 5x - 11There is not enough information to solve this problem.
5A secant line represents the Instantaneous rate of change of a function.The average rate of change of a function.Line tangent to a function.
6Definition of the Derivative of a Function The derivative of f at x is given byprovided the limit exists. For all x for which this limit exists, f’ is a function of x.
7Definition of Tangent Line with Slope m If f is defined on an open interval containing c, and if the limitexists, then the line passing through f with slope m is the tangent line to the graph of at the pointThe 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
8Find the slope of the graph of at 491Does not exist
9Alternative limit form of the derivative The existence of the limit in this alternative form requires that the following one-sided limitsandexist 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 bboth exist.