Math 400 (Section 5.3)
Exploration: Fundamental Theorem of Calculus
Name ____________________________
Recall the Mean Value Theorem:
If f is differentiable on an interval ( a, b ) and continuous on [ a, b] , then there exists some
f (b ) − f ( a )
.
b−a
Now suppose that f is a function that is integrable on the closed interval [ a, b] , and suppose that
c ∈ ( a, b ) such that
f ′ (c ) =
F ( x ) = ∫ f ( x ) dx .
1. Does F satisfy the conditions of the Mean Value Theorem on [a, b]? Why or why not?
2. Split the interval into n subintervals of equal width. This will make the norm of the partition
∆ = ∆xk = [ xk − xk −1 ] =
__________
3. Does F satisfy the conditions of the Mean Value Theorem on each subinterval [ xk −1 , xk ] of
[a, b]? Why or why not?
4. For each subinterval [ xk −1 , xk ] , we can choose the sample point x k such that it satisfies the
conclusion of the Mean Value Theorem for F on that subinterval. (Why can we do this?)
5. Then the Riemann sum (i.e., the estimate for
b
∫ f ( x ) dx ) will be
a
n
Rn = ∑ f ( xk ) ∆x =
k =1
(The canceling effect here is called a telescoping argument).
Note that this result from the previous page will be independent of n as long as the xk are chosen
so that they satisfy the conclusion of the MVT for F(x). Thus
lim R n = ______________
∆x → 0
But this limit is the definite integral
b
∫ f ( x ) dx .
a
Thus we have the Fundamental Theorem of Calculus:
If f is integrable on the interval [ a, b ] , and F ( x ) = ∫ f ( x ) dx , then
b
∫ f ( x ) dx =
a
_________________
There is an alternative form of this theorem which is also very useful. Recall that the area
function that gives the area bounded by the function f(t) on the interval [a, x] is
x
A ( x ) = ∫ f ( t ) dt .
a
6. The graph at right shows the small change in A(x)
as t changes from x to x+h. Find an expression for
this change.
∆A =
.
7. In this case, ∆t = h , so we see that
∆A
=
∆t
8. What is the name for the limit of the expression from Problem 7 as h approaches 0?
9. Now let m be the minimum value of f ( t ) on the interval [ x, x + h ] , and let M be the
maximum value of f ( t ) on the interval [ x, x + h ] . (Let's assume that f is continuous, so the
Extreme Value Theorem guarantees that such m and M exist.) Then we see that
mh ≤
≤ Mh
10. Dividing both sides by h gives the compound inequality
≤
A( x + h) − A( x)
h
≤
11. But as h → 0 , it is clear that both m and M approach the same value, _________. Thus, using
the Squeeze Theorem, we can see that
A( x + h) − A( x)
lim
= _______ = _______
h →0
h
12. Thus A ( x ) is an ________________ of f. Thus if F is any _________________, then
F ( x ) = A ( x ) + C ( C constant ) , so F ′ ( x ) = A′ ( x ) = f ( x ) . This last result gives the
alternative form of the Fundamental Theorem of Calculus:
d x
f ( t ) dt = f ( x ) ( a constant )
dx ∫a
Let us now consider what the Fundamental Theorem of Calculus is saying.
Fundamental Theorem of Calculus (Standard Form)
If f is integrable on [a, b], and if F is any antiderivative of f, then
b
∫ f ( x ) dx = F ( b ) − F ( a )
a
Let f ( x ) be any differentiable function. Then it is certainly an antiderivative of f ′ ( x ) . Thus (by
the Fundamental Theorem of Calculus,
∫
b
a
f ′ ( x ) dx = f ( b ) − f ( a ) .
This is really saying that the accumulated change in f over the interval [a, b] is the definite
integral of f ′ ( x ) from a to b. This has huge implications and many applications. Let's consider
some examples to illustrate this.
13. Suppose an object is launched vertically upward from the ground at an initial velocity of
64ft/s. Then it's velocity t seconds after it was launched is v ( t ) = −32t + 64 (why?). Find the
height of the object at t = 3 seconds. [Hint: Use the fact that v ( t ) = h′ ( t ) and the
Fundamental Theorem of Calculus.]
13) ______________
14. Suppose that a car is traveling at 60mph when the driver spots a dog in the middle of the
freeway. She slams on the brakes so the car's acceleration is given by a ( t ) = −40e− t for
0 ≤ t ≤ 2 , where t is the number of seconds that have elapsed since she started braking. What
will the car's velocity be at t = 2 seconds?
14) ______________
15. Use the alternative form of the Fundamental Theorem of Calculus to find the following
derivatives.
(a)
d
dx
∫
x
3
t 4 − 7 dt
(b)
x4
d
dx 1
∫
tan t 3 dt (Hint: Let u = x 4 and use the Chain Rule.)
a) ____________
b) ____________