1
Lesson Plan #58
Class: Intuitive Calculus
Date: Tuesday March 16th, 2010
Topic: Second Fundamental Theorem of Calculus
Aim: How do we use the Second Fundamental Theorem of Calculus?
Objectives:
1) Students will be able to use the Second Fundamental Theorem of Calculus to find the derivative of a function.
HW# 58:
Page 392 #’s 29, 32 (For these two questions), the expression given is equal to F ( x ) . Find F '( x)
Page 393 # 56
Do Now:
1) Find the area of the shaded region.
y  x 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
So far we have defined a definite integral with constants for the lower and upper limits of integration. If we wanted, we could
leave the upper limit of integration as a variable, let’s say x and leave the lower limit as a constant. For example
x
F ( x)   cos tdt
0
|0x sin t
= sin x  sin 0
= sin x
We could now evaluate
sin x for different values of x .
So to reiterate, we have F ( x)  sin x .
2
Now find F '( x )
The derivative of
sin x is _______________
So F '( x ) = _________
So we have F '( x)  cos x
Rewriting what we had above, we get
x
F ( x)   cos tdt
0
F ( x)  sin x
F '( x)  cos x
x

d 
  cos tdt   cos x
dx  0

Theorem: The Second Fundamental Theorem of Calculus
x

d 
If f is continuous on an open interval I containing a , then, for every x in the interval,
  f (t )dt   f ( x)
dx  a

u
In general, if we have F ( x) 
 f (t )dt , then
F '( x) 
a
dF du

du dx
u
 du
d 
F '( x) 
  f (t )dt 
du  a
 dx
Example #1:
Integrate to find F as a function of x and demonstrate the Second Fundamental Theorem of Calculus by differentiating the result.
x
1)
0
F ' ( x)  x  2
x
2)
t2
 2t 
0 2
F ( x)   (t  2)dt Solution: F ( x)  |
F ( x)   3 tdt
8
x
F ( x) 
x2
 2x
2
3
Example #2: Use the Second Fundamental Theorem of Calculus right away to find F '( x )
x
1) F ( x) 
 (t
2
 2t )dt
2
x
2) F ( x ) 

t 4  1dt
1
Example #3:
x3
Find the derivative of
 cos tdt

2
Example #4:
2x
If F ( x ) 
1
 1 t
1
3
dt then F '( x) equals
4