4.4 – Notes – Day 2
Average Value of a Function and 2nd Fundamental Theorem of Calculus
Homework: Assign # 7: Worksheet
Learning Target: I can evaluate a definite integral using the Fundamental Theorem of Calculus (FTC).
 I understand the connection between the Mean Value Theorem for integrals and the average value of a
function.
 I can find the average value of a function over a given interval.
Learning Target: I understand and can interpret functions defined by integrals such as f ( x) 



b
a
f '( x)dx .
I can evaluate an integral-defined function for a given value of x.
I can find the derivative of an integral-defined function using the Second Fundamental Theorem of Calculus.
On the graph below, f  x  , the area under the curve between 0 and 2 can be written as
the fundamental theorem of calculus, we can calculate the exact area to be
. Using
14
. Can you draw a rectangle whose base is
3
2 which is exactly equal to this integral? What are the dimensions?
Recall the MVT (for derivatives): If f is continuous on  a, b and differentiable on  a, b  , then there exists at least one
c   a, b  such that f '(c) 
f (b)  f (a)
.
ba
f
(Instantaneous rate of change = average rate of change)
a
b
Today, we look at the MVT for integrals: If f is continuous on  a, b , then there exists at least one c   a, b  such that
Accumulation (area) =
Average value:

b
a
f ( x) dx 

f (c ) 
b
a
f ( x) dx
ba
a
b
Ex 1. Given f ( x)  x 2  1 on the interval 0, 2 , find
a.
The average value of the function.
b. The value of c guaranteed by the MVT for integrals.
Ex 2. Given f ( x)  sin x on the interval  0,   , find
a. The average value of the function.
Ex 3. Use the FTC to find
 t
x
2
1
b. The value of c guaranteed by the MVT for integrals.
 4t  1 dt
Now, take the derivative of the answer you just found. What do you notice?
THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
If f is continuous on an open interval I containing a, then for every x in the interval,
d  x
  f ( x) and d  u f (t ) dt   f (u ) du
f
(
t
)
dt


dx  a
dx  a
dx
Use the Second Fundamental Theorem of Calculus to find the derivatives of the following functions.
Ex 4. Find F '( x) if f ( x) 
Ex 6.
d  x
t  2 dt 

dx  1

x
1
4
t dt
Ex 5. Find F '( x) if f ( x) 
Ex 6.
d  2x  t  

 dt
dx  1  2t  3  

x
1
t2
dt
t 2 1