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 .