Lesson Plan #6
Lesson Plan #28
Class: AP Calculus Date: Wednesday November 17th, 2010
Topic: Derivatives of Sine and Cosine functions Aim: How do we differentiate trigonometric functions?
Objectives:
1) Students will be able to differentiate trigonometric functions
HW# 28:
Page 140 #’s 1-9 (odd)
Squeeze Theorem:
The squeeze theorem is formally stated as follows.
Let I be an interval containing the point a. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:
[pic]
and also suppose that:
[pic]
Then [pic]
Illustration of the Squeeze Theorem
On your graphing calculator, set the mode to radian mode. In [pic]enter cosx, in [pic]enter [pic], in [pic]enter 1
Verify that there is an interval for which [pic]
Since there is an interval for which [pic]
and [pic]
and [pic]
then by the squeeze theorem [pic]
1) Find the limit [pic] 2) Find the limit [pic]
Comment: By the Squeeze Theorem it can also be shown that [pic]
3) Find the limit [pic] 4) Find the limit [pic] 5) Find the limit [pic]
Note: Also do above question as x((
4) If [pic] for [pic], evaluate [pic]
Do Now:
In Precalculus we learned how to use the Sum and Difference formulas for trigonometric functions. For example, recall [pic]
Use this formula to find the exact value of [pic], assuming you know the exact values of [pic], [pic], [pic], [pic]
2) On your graphing calculator, set you graphing window to an appropriate size and graph the function [pic]
What [pic]value is the graph approaching as [pic]?
What is another way to rewrite this question?
What is [pic]
3) On your graphing calculator, set you graphing window to an appropriate size and graph the function [pic]
What [pic]value is the graph approaching as [pic]?
What is another way to rewrite this question?
What is [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
Recall the long way to find the derivative. Let’s use this long way to find the derivative of [pic]
[pic]
Use the formula for the sin of the sum of angles to expand the first part of the limit
[pic]
Rewrite the right side of the equation so that the middle term is first, the last term is in the middle and the middle term is last
[pic]
Factor [pic]from the last two terms
[pic]
Put the denominator under each term of the numerator
[pic]
Evaluate limit
[pic]
[pic]
So we have the derivative of[pic].
[pic]
By other proofs we can get the derivatives of the other five trigonometric functions. Below are the derivatives of the six trig functions.
Assignment:
I. Find the derivative of each of the following
1) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
7) [pic]
III. Find the slope of the line tangent to [pic]at [pic]radian
Sample Test Questions:
1) If [pic], find [pic]
2) Let [pic]. The maximum value attained by [pic]on the closed interval [pic] is for [pic]equal to
A) -1 B) 0 C) [pic] D) 2 E) [pic]
3) The period of [pic]is
A) [pic] B) [pic] C) [pic] D) 3 E) 6
4) Which of the following functions is not odd?
A) [pic] B) [pic] C) [pic]
D) [pic] E) [pic]
5) The smallest positive x for which the function [pic] is maximum is
A) [pic] B) [pic] C) [pic] D) 3[pic] E) 6[pic]
6) Find [pic]if [pic]
A) [pic] B) [pic] C) [pic]
D) [pic] E) [pic]
7) The equation of the tangent to the curve [pic]at the point [pic]is
A) [pic] B) [pic] C) [pic] D) [pic] E) [pic]
8) Evaluate [pic]
A) -1 B) [pic] C) oscillates between -1 and 1 D) 0 E) Does not exist
-----------------------
[pic] [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
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