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 .
(e) ( is blue, is red)
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.
4-7. Find the derivative of the function using part 1 of the FTC.
5. . [Hint: ]
8-10. Evaluate the integral using Part 2 of the FTC.
= = (64 + 32 + 8) - (0 + 0 + 0) = = 104.
11. Let . Use Part 1 of the FTC to find .
12. Give a non-polynomial function () such that and .
For any function set .
Then clearly and so
For example .
13. Let and . Find .
Hence , so .
14. Let defined on . Find .
. So . Note that .
and for . Hence should be .
15. Let . Find.
Differentiate both sides to get .
16-17. Evaluate the integral and interpret it as a difference of areas.
= since = 0.
18. If , where , find .
19. Find the value of if , is continuous, and .
20. If is continuous and and are differentiable functions, find a formula for .