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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