AP Calculus AB
Day 10
Section 4.4
7/29/2017
Perkins
The Mean Value Theorem for Integrals
Gives the average value of f(x) on [a,b].
4
sum of all y-values
avg y-value 
# of y-values
2
b
a
b
5

 f  x  dx
a
ba
b
1
Average y-value 
f  x  dx

ba a
1.
Find the average value of y  3 x 2  2x on [1,4].
Then find where f(x) obtains this average value.
4
1
2
3
x
 2 x  dx
Avg val 


4 1 1
1 3
2 4
  x  x 
1
3
1
1


  64  16    0     48   16
3
3
3 x 2  2 x  16
3 x 2  2 x  16  0
3x  8 x  2  0
x  83 and x  2
...but only x  83 is in 1,4.
This value is obtained when…
2.
x

d  1 3 3 2 x
d  2
 3 t  2 t 
  t  3t dt  

2

dx
dx  2



d  1 3 3
2
3
2
3
1




x

x

2

2








2
2
 3
 
dx   3
d 1 3 3 2 1 3 3 2

x  2 x  3  2  2  2
3


dx
 x 2  3x  0
 x 2  3x
Using the 2nd Fundamental Theorem of Calculus:
x

d  2
  t  3t dt  
dx  2



 x 2  3  x  1


 x 2  3x
Second Fundamental Theorem of Calculus

d 
du
  f  t  dt   f  u 
dx  a
dx

u
In order to use this shortcut we must have…
a derivative…
of an integral…
whose lower bound is constant…
and whose upper bound is a function of x.

d 
3
3.
  t  t dt 
dx  

x3
We don’t know how to
integrate this function!
Using the 2nd Fundamental Theorem of Calculus:

x 
 3x
3
2
3
2

  x  3 x 
3
x x
9
3
AP Calculus AB
Day 10
Section 4.4
Perkins
The Mean Value Theorem for Integrals
4
2
a
b
5
1.
Find the average value of y  3 x 2  2x on [1,4].
Then find where f(x) obtains this average value.
2.
x

d  2
  t  3t dt  
dx  2



Second Fundamental Theorem of Calculus

d 
  f  t  dt  
dx  a

u

d 
3
3.
  t  t dt 
dx  

x3