Calculus Section 4.4 Mean Value and 2nd Fund. Thm of Calculus
-Understand and use the Mean Value Theorem
-Find the average value of a function over a closed interval
-Understand and use the 2nd Fundamental Theorem of Calculus
Homework: page 291 #’s 47-50, 81-92
Mean Value Theorem
If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that
b
 f ( x)dx  f (c)(b  a)
a
In other words, there exists a c between [a, b] such that a rectangle of height h = f(c) would have the same area as
the area found under the curve. This theorem only tells you there is a c, it doesn’t tell you how to find c.
Average Value of a Function
If f is integrable on the closed interval [a, b], then the average value of f on the interval is
b
1
f ( x)dx
b  a a
This is the height h = f(c) value that was found to exist in the Mean Value Theorem above. It is the average value
of f(x) on the interval [a, b], and it’s the value that makes the area of the rectangle equal the area under the curve.
Example)
Find the average value of f ( x)  3x 2  2 x on the interval [1, 4].
Evaluate the integrals:
0
 /6
 /4
 /3
 /2
0
0
0
0
0
 cos tdt  cos tdt
 cos tdt
 cos tdt  cos tdt
The 2nd Fundamental Theorem of Calculus
If f is continuous on an open interval I containing a, then, for every x in the interval:
x

d 
  f (t )dt   f ( x)
dx  a

The 2nd Fundamental Theorem of Calculus tells that that if a function is continuous, we can be sure that there
exists an ________________________ for the function.
Example)
Evaluate
x

d 
2
  t  1dt 
dx  0

x3
Find the derivative of F ( x) 
 cos tdt
/2