Calculus-Sec-5-3-Solution
5.3 The Fundamental Theorem of Calculus by SGLee - HSKim- SWSun
1. Let 
, where 
is the function whose graph is shown below.
(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) (
is blue, 
is red)
(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. 
= 
= (64 + 32 + 8) - (0 + 0 + 0) = 
= 104.
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. 
= 
since 
= 0.
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 
.
