Presentation on theme: "THE DERIVATIVE AND THE TANGENT LINE PROBLEM"— Presentation transcript:
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 pointUse the limit definition to find the derivative of a functionUnderstand 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 + 19Y = 5x - 11There 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 byprovided 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 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
8 Find the slope of the graph of at 491Does not exist
9 Alternative 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.