5.3 The Fundamental Theorem of Calculus
1. Let , where
is the function whose graph is shown.
(a)Evaluate and
.
(b)Estimate ,
and
.
(c) On what interval is increasing?
(d) Where does have a maximum value?
(e) Sketch a rough graph of .
(f) Use the graph in part to sketch the graph of . Compare with the graph of
.
(a).
(b) .
(c) .
(d) .
(e)
(f) .
2-3. Draw the area represented by . Then find
in two ways:
(a)by using Part 1 of the FTC and
(b)by evaluating the integral using Part 2 and then differentiating.
2. .
(a) .
(b) .
3. .
(a) .
(b) .
4-7. Find the derivative of the function using part 1 of the FTC :
4. .
.
5. .
[Hint: ]
.
6. .
.
7. .
.
8-10. Evaluate the integral using Part 2 of the FTC.
8. .
.
9. .
10. .
.
11. Let . Use Part 1 of the FTC to find
.
so
.
12. Give a non-polynomial function (
) such that
and
.
For any function
set
.
Then clearly and
so
For example .
13. Let and
. Find
.
and
.
Hence , so
.
14. Let defined on
. Find
.
. So
. Note that
.
and for
. Hence
should be
.
Therefore .
15. Let .
Find.
Differentiate both sides to get
.
.
Hence,
.
16-17. Evaluate the integral and interpret it as a difference of areas.
16. .
.
17. .
.
18. If , where
, find
.
Therefore, .
19. Find the value of if
,
is continuous, and
.
.
20. If is continuous and
and
are differentiable functions, find a formula for
.