Pre-Calculus Honors Section 11.3 The Tangent Line Problem

Standard: T 7.0

Objectives:

1.  To use a tangent line to approximate the slope of a graph.

2.  To use the limit definition of slope to find exact slopes of graphs.

Lesson:

·  Calculus is a branch of mathematics that studies rates of change of functions.

·  The slope of a line indicates the rate at which a line rises or falls.

- For a line this rate (or slope) is the same for every point on the line.

- For graphs other that lines, the rate at which the graph rises or falls changes from point to point.

For example: A parabola

·  The graph is rising more quickly at point than at point .

·  To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at that point.

The tangent line to a graph of a function f at point Pis the line that best approximates the slope of the graph at the point.

Other tangent lines:

If you want to find the slope of a graph at a point, simply find the slope of the tangent line at the point.

Ex. 1) Approximate the slope of the curve at the point .

at point (1, 1)

Slope and the Limit Process

In example 1 we “eyeballed” the tangent line at the point of tangency. However, we can use a more precise method of approximating tangent lines by using a secant line through the point of tangency and a second point on the graph.

Difference Quotient

The slope is better approximated by choosing a second point that is closer and closer to the point of tangency.

·  As h approaches 0, the secant line approaches the tangent line.

·  Using the limit process we can find the exact slope of the tangent line at

Ex. 2) Use the limit process to find the slope of the graph of the function at the specified point.

at point (-2, 4)

Ex. 3) Find the slope of the function.

Ex. 4) Find a formula for the slope of the graph. Then use it to find the slope at the two points.

a) (-1, 2) b) (2, 5)

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Pre-Calculus Honors Section 11.3 Continued Day 2

Yesterday we were given a function and used the limit process to derive another function , that represents the slope of the graph of f at the point . This derived function is called the derivative of f at x. It is denoted by , “f prime of x.”

Ex. 1) Find the derivative of the function.

a.

b.

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