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 .