1. warmup 1)

2. 5.4: Fundamental Theorem of Calculus

3. The Fundamental Theorem of Calculus, Part 1
If f is continuous on , then the function
has a derivative at every point in , and

4. First Fundamental Theorem:
1. Derivative of an integral.

5. First Fundamental Theorem:
1. Derivative of an integral.
2. Derivative matches upper limit of integration.

6. First Fundamental Theorem:
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

7. First Fundamental Theorem:
New variable.
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

8. The long way:
First Fundamental Theorem:
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

9. 1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

10. The upper limit of integration does not match the derivative, but we could use the chain rule.

11. The lower limit of integration is not a constant, but the upper limit is.
We can change the sign of the integral and reverse the limits.

13. Group Problem:

14. The graph above is g(t)

15. Neither limit of integration is a constant.
We split the integral into two parts.
It does not matter what constant we use!
(Limits are reversed.)
(Chain rule is used.)

16. The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of , and if F is any antiderivative of f on , then
(Also called the Integral Evaluation Theorem)

17. is a general antiderivative
so…

18. Remember, the definite integral gives us the net area
Net area counts area below the x-axis as negative
The net area, or if this were
a definite integral, would
=5-3+4=6
The area, or “total area”,
or area to the x-axis,
would be 5+3+4=12

20. Group Work

23. Find g(-5)
Find all values of x on the open interval (-5,4) where g
is decreasing. Justify your answer.
c) Write an equation for the line tangent to the graph of g at x = -1
d) Find the minimum value of g on the closed interval [-5,4].
Justify your answer.

24. Solution
Find g(-5)
Find all values of x on the open interval (-5,4) where g
is decreasing. Justify your answer.
c) Write an equation for the line tangent to the graph of g at x = -1
d) Find the minimum value of g on the closed interval [-5,4].
Justify your answer.

25. Group Problem

26. Group Problem

27. Using FTC with an initial condition:
IF the initial condition is given, it accumulates normally
and then adds the initial condition.

28. Ex. If oil fills a tank at a rate modeled by
and the tanker has 2,500 gallons to start.
How much oil is in the tank after 50 minutes pass?
f(a)
a is the lower limit

29. Ex.
Given

30. 1) 2)

31. 2) Where does is the particle at t=5 ?

35. the end