Lesson Plan #6



Lesson Plan #36

Class: AP Calculus Date: Tuesday December 7th, 2010

Topic: Tangent Line Approximation Aim: How do we use tangent line approximations?

Objectives:

1) Students will be able to use tangent line approximations or linearizations.

HW# 36:

Page 229 #1, 2, 3

Do Now:

1) What is the point slope form equation of a line?

Consider a function[pic]that is differentiable at [pic]. What is the equation for the line tangent to [pic]at the point [pic]? (Use the point slope form of a line)

At right we have the graph of [pic]and the graph of the line tangent to[pic]at the point[pic]. Use what you have learned this semester to find the equation of the line tangent to [pic] at the point[pic].

So hopefully you did the work above first and discovered that the line tangent to [pic] at the point[pic] is [pic].

What can you tell about the tangent line and the function around the area of the point[pic]?

Procedure:

Write the Aim and Do Now

Get students working!

Take attendance

Give back work

Go over the HW

Collect HW

Go over the Do Now

Complete the table below

|[pic] |-0.5 |-0.1 |-0.01 |0 |0.01 |01 |.5 |

|[pic] | | | | | | | |

|[pic] | | | | | | | |

We see that the [pic]-value on the tangent line is an approximation for the actual or true value of[pic]. This is called tangent-line approximation or local linearization.

Example #1:

Find an equation of the line tangent to [pic]at the point [pic]. Use this linear approximation to complete the table below.

|[pic] |1.9 |1.99 |2 |2.01 |2.1 |

| | | | | | |

| | | | | | |

Example #2:

1) On your graphing calculator, set your viewing rectangle to Xmin = 0, Xmax=3, Ymin=-5, Ymax=3. Then sketch the graph of [pic]

2) Find the equation of the line tangent to[pic], at[pic].

3) On your graphing calculator, sketch the graph of the line tangent to [pic]at [pic]

4) How close are the curve and the tangent line to each other at?

At[pic], the curve and its tangent line appear to coincide. We say that the graph is locally straight at[pic].

To simplify work, sometimes a function [pic]that is locally straight near a point, [pic], is replaced with the equation of the tangent line [pic].

The above equation is in the point slope form of a line, where the point is [pic]and the slope at that point is[pic].

Solving for [pic]gives us the equation [pic]

Example #3:

Find the linearization of [pic]at [pic]

Example #4:

The tangent line approximation for [pic]near [pic]is

A) [pic] B) [pic] C) [pic]

D) [pic] E) [pic]

Example #5:

The best linear approximation for [pic] near [pic] is?

A) [pic] B) [pic] C) [pic]

D) [pic] E) [pic]

Example #6:

Find the linearization of [pic]at [pic]

Example #6:

Verify the tangent line approximation of the function [pic]at the point [pic]is [pic]

Example #7:

The curve [pic]passes through[pic]. Use local linearization to estimate the value of [pic]at [pic]. The value is

A) [pic] B) [pic] C) [pic] D) [pic] E) [pic]

Example #8:

When the local linearization of [pic]near 0 is used, an estimate of [pic]is

A) [pic] B) [pic] C) [pic] D) [pic] E) [pic]

Sample Test Questions:

1) Evaluate [pic]

A) -5 B) [pic] C) 0 D) 5 E) 1

2) The only function that does not satisfy the Mean Value Theorem on the interval specified is

A) [pic]on [pic] B) [pic]on [pic] C) [pic] on [pic]

D) [pic]on [pic] E) [pic]on [pic]

-----------------------

[pic]

[pic]

Definition: Linearization

If [pic]is differentiable[pic], then [pic]is the linearization of [pic]at[pic]. The approximation [pic]is the standard linear approximation of [pic]at[pic], and we say the graph of [pic]is locally straight at [pic].

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