Lesson Plan #6
Lesson Plan #61
Class: AP Calculus Date: Monday February 27th, 2012
Topic: The derivative of [pic] Aim: How do we find the derivative of[pic]?
Objectives:
1) Students will be able to find the derivative of [pic]
HW# 61:
Page 170 #’s 1-10
Do Now:
1) Using your graphing calculator, sketch the graph of [pic]
2) Using your graphing calculator and/or picking values close to 0, evaluate [pic]
3) Recall
If[pic], let’s use the definition of the derivative to find[pic]. Ready? Let’s go!
Note: [pic]is considered to be a constant since the variable for the limit is [pic]. As such we can take out [pic]as a constant
Note: From earlier we know that the right hand side limit evaluates to 1
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
Example 1: Find the derivative of each of the following
A) [pic]
A
B) [pic]
B
C) [pic]
C
D) [pic]
D
E) [pic]
E
F) [pic]
F
G) [pic]
G
H) [pic]
H
Example:
For 1980 through 1983, the number [pic]of medical doctors in the United States could be modeled by [pic], where [pic]represents 1980. At what rate (this is rate of change or derivative) was the number of M.D.s changing in 1988?
Example:
The value V of an item [pic]years after is it purchased is [pic]. Find the rate of change of [pic]when [pic].
Example:
The spread of a flu in a certain school is modeled by the equation[pic], where [pic]is the total number of students infected [pic]days after the flu was first noticed.
A) Estimate the initial number of students infected with the flu
B) How fast is the flu spreading (this is rate of change or derivative) after 3 days?
Sample Test Questions:
1) If [pic], find [pic]
2) If [pic], find [pic]
3) If [pic], find [pic]
A) [pic] B) 1 C) [pic] D)[pic] [pic] E) [pic]
4) If [pic], find [pic]
A) [pic] B) [pic] C) [pic]
D) [pic] E) [pic]
5) If [pic], then [pic]equals
A) -2 B) -1 C) 0 D) 1 E) None of these
6) Find the indicated limits using L’Hopital’s Rule or recognizing the limit as a derivative. (There might be other ways to evaluate the limit)
[pic]
7) The tangent to the curve of [pic]is horizontal when [pic]is equal to?
A) 0 B) 1 C) -1 D) [pic] E) None of the other choices
8) If [pic], then [pic]equals
A) 2 B) -2 C) [pic] D) 0 E) -4
9) [pic]
A) 0 B) 1 C) [pic] D) [pic] E) [pic]
10) Evaluate [pic]
A) [pic] B) [pic] C) [pic] D) [pic] E) The limit does not exist
11) [pic]
A) 1 B) 0 C) e D) –e E) Nonexistent.
-----------------------
Theorem: The derivative of the Natural Exponential Function
Let [pic]be a differentiable function of[pic].
1. [pic] 2. [pic]
[pic]
[pic]
[pic]
[pic]
Definition of the Derivative of a Function: The derivative of a [pic]at [pic] is given by
[pic]
provided the limit exists
[pic]
[pic]
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