CALCULUS - Ed Kornberg



CALCULUS

WORKSHEET 2 ON FUNDAMENTAL THEOREM OF CALCULUS

Work these on notebook paper. Use your calculator on problems 3, 8, and 13.

1. If [pic] what is the value of [pic]

2. If [pic]

3. Water is pumped out of a holding tank at a rate of [pic] liters/minute, where t is in

minutes since the pump is started. If the holding tank contains 1000 liters of water when

the pump is started, how much water does it hold one hour later?

4. Given the values of the derivative [pic] in the table and that [pic]

estimate [pic] for x = 2, 4, 6. Use a right Riemann sum.

|x |0 |2 |4 |6 |

|[pic] |10 |18 |23 |25 |

5. Consider the function f that is continuous on the interval [pic] and for which

[pic] Evaluate:

(a) [pic] (c) [pic]

(b) [pic] (d) [pic]

6. Use the figure on the right and the

fact that [pic] to find values

of P when t = 1, 2, 3, 4, and 5.

7. Using the figure on the right, sketch

a graph of an antiderivatives [pic]

satisfying [pic] Label each critical

point of [pic] with its coordinates.

TURN->>>

8. Find the value of [pic]

[pic]

10. A bowl of soup is placed on the kitchen counter to cool. The temperature of the soup is

given in the table below.

|Time t (minutes) |0 |5 |8 |12 |

|Temperature [pic] (°F) |105 |99 |97 |93 |

(a) Find [pic].

(b) Find the average rate of change of [pic] over the time interval t = 5 to t = 8 minutes.

11. The graph of [pic] which consists of a line

segment and a semicircle, is shown on the

right. Given that [pic] find:

(a) [pic]

(b) [pic]

12. (Multiple Choice) If [pic] and [pic] are continuous functions such that [pic] for all [pic],

then [pic]

(A) [pic] (B) [pic] (C) [pic]

(D) [pic] (E) [pic]

13. (Multiple Choice) If the function[pic]is defined by [pic] and [pic]is an antiderivatives

of [pic] such that [pic], then [pic]

(A) -3.268 (B) -1.585 (C) 1.732 (D) 6.585 (E) 11.585

14. (Multiple Choice) The graph of [pic] is shown in the figure at right.

If [pic] and [pic], then [pic]

(A) 0.3 (B) 1.3 (C) 3.3 (D) 4.3 (E) 5.3

Answers

1. 29

2. 4

3. 741.636 liters

4. 136, 182, 232

5. (a) 14 (b) 4 (c) 8 (d) 0

6. 1, 0, [pic], 0, 1

7. 21, 13, 15

8. 2.747

9. [pic]

10. (a)[pic] (b) [pic]

11. (a) 7 (b) [pic]

12. C

13. B

14. D

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