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Lesson Plan #28
Class: AP Calculus
Date: Wednesday November 17th, 2010
Topic: Derivatives of Sine and Cosine functions
functions?
Aim: How do we differentiate trigonometric
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:
and also suppose that:
Then
Illustration of the Squeeze Theorem
On your graphing calculator, set the mode to radian mode. In
y1 enter cosx, in y2 enter
sin x
, in y3 enter 1
x
sin x
1
x
sin x
1
Since there is an interval for which cos x 
x
and lim cos x  1
Verify that there is an interval for which cos x 
x0
and lim 1  1
x 0
then by the squeeze theorem lim
x0
sin 5 x
x 0
x
1) Find the limit lim
sin x
1
x
sin x
x0 5 x
2) Find the limit lim
1  cos x
0
x0
x
Comment: By the Squeeze Theorem it can also be shown that lim
2
3(1  cos x)
x0
x
3) Find the limit lim
sin( x   )
x0
x 
4) Find the limit lim
5) Find the limit lim
x0
sin x
x 2  3x
Note: Also do above question as x
4) If 3x  f ( x)  x3  2 for 0  x  2 , evaluate lim f ( x )
x1
Do Now:
In Precalculus we learned how to use the Sum and Difference formulas for trigonometric functions. For example, recall
sin( x  y )  sin x cos y  cos x sin y

o
o
o
Use this formula to find the exact value of sin 75 , assuming you know the exact values of sin 30 , cos 30 , sin 45 ,
cos 45o
2) On your graphing calculator, set you graphing window to an appropriate size and graph the function f ( x ) 
What y value is the graph approaching as
x 0?
sin x
x
What is another way to rewrite this question?
What is f (0) ?
3) On your graphing calculator, set you graphing window to an appropriate size and graph the function f ( x) 
What y value is the graph approaching as
x 0?
What is another way to rewrite this question?
What is f (0) ?
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
Definition of the Derivative of a Function: The derivative of a f at
f ' ( x)  lim
x 0
f ( x  x)  f ( x)
x
x is given by
provided the limit exists
sin x
1  cos x
x
d
sin( x  x)  sin x
[sin x]  lim

x

0
dx
x
Use the formula for the sin of the sum of angles to expand the first part of the limit
d
sinxcosx  cosxsinx  sin x
[sin x]  lim
x0
dx
x
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
d
cos x sin x  sin x  sin x cos x
[sin x]  lim

x

0
dx
x
Factor
sin x from the last two terms
d
cos x sin x  (sin x)(1  cos x)
[sin x]  lim

x

0
dx
x
Put the denominator under each term of the numerator

d
 sin x 
 1  cos x 
[sin x]  lim (cos x)
  (sin x)


x

0
dx
x 
 x 


Evaluate limit
d
[sin x]  (cos x)(1)  (sin x)(0)
dx
 cos x
So we have the derivative of sin
x.
d
[sin x]  cos x
dx
By other proofs we can get the derivatives of the other five trigonometric functions. Below are the derivatives of the six trig
functions.
d
[sin x]  cos x
dx
d
[cos x]   sin x
dx
d
[tan x]  sec 2 x
dx
d
[cot x]   csc 2 x
dx
d
[sec x]  sec x tan x
dx
d
[csc x]   csc x cot x
dx
Assignment:
I. Find the derivative of each of the following
1) y  2 sin x  tan x
3
4
2)
y  x 2  sec x
3) y  2 x cot x
4) y  x cos x
5)
y  4  x 2 sin x
6) y  x sin x  cos x
7) y  sec x csc x
III. Find the slope of the line tangent to y 
2  cot x
at x  1 radian
x
5
Sample Test Questions:
1) If f ( x ) 
2) Let
sin x
, find f ' ( x )
x2
g ( x)  cos x 1 . The maximum value attained
by g on the closed interval
A) -1
B) 0
C)

2
3) The period of f ( x)  sin
A)
1
3
B)
2
3
0,2  is for x equal to
D) 2
2
x is
3
3
C)
2
E)

D) 3
E) 6
4) Which of the following functions is not odd?
A) f ( x)  sin x
D) f ( x) 
B) f ( x)  sin 2 x
x
x 1
E) f ( x) 
2
3

2
B)

C)
f ( x)  x 3  1
2x
5) The smallest positive x for which the function
A)
C)
3
2
 x
f ( x)  sin    1 is maximum is
3
D) 3 
E) 6 
dy
5
if y  x tan x
dx
4
5
2
4
2
A) 5 x tan x
B) x sec x
C) 5 x sec x
4
2
4
5
2
D) 5 x  sec x
E) 5 x tan x  x sec x
6) Find
  
,  is
2 2
7) The equation of the tangent to the curve y  x sin x at the point 
A) y  x  
B) y 

2
C) y    x
D) y  x 

2
E) y  x
8) Evaluate lim sin x
x
A) -1
B) 
C) oscillates between -1 and 1
D) 0
E) Does not exist