11.3 The Tangent Line Problem

11.3 The Tangent Line Problem Spring 2011

At the end of this lesson, you should be able to • Use a tangent line to approximate the slope of a graph at a point. • Use the limit definition of slope to find exact slopes of graphs. • Find derivatives of functions and use derivatives to find slopes of graphs.

Tangent Line to a Graph (page 763) y x

Tangent Line to a Graph (page 763) The tangent line to a graph of a function f at a point is the line that … To determine the rate at which a graph rises or falls at a single point, ….

Slope of Secant Line We can calculate the slope of a line given two points y Calculate the slope of the line between the given point P (5, 4) and another point on the curve, say Q(2, 1). The line is called a secant line. P(5,4) Q(2,1) x .

Slope and the Limit Process (page 765) y y=f(x) ( , ) ( , ) x 0 ________________

Definition of a Slope of a Graph (page 765) The slope mof the graph of f(x) at the point (x, f(x)) is equal to the slope of its tangent line at (x, f(x)), and is given by provided this limit exists.

Example 1Find the slope of the graph of at the point (-2,4)

Example 2 (You try!)Find the slope of the graph of at the point (3, 12)

Example 2 (Solution)Find the slope of the graph of at the point (3, 12)

Example 3 Find the slope of the graph of

Example 3 (Discussion) Why do you think the slope of is a constant number? Explain.

Example 4 (You try!) Find a formula for the slope of the graph of What are the slopes at points (-1,3) and (0,4)?

The Derivative of a Function (page 768-769) The derivative is the formula which gives the slope of the tangent line at any point x for f(x)

Example 5 Find the derivative of

Example 6 (You try!)Find the derivative of

Example 7Find the derivative of

Example 8Find the derivative of

Practice p.770-771 #s 1-45, every four

11.3 The Tangent Line Problem