Math 129

Calculus I

Final Review

1. Determine the following limits algebraically.

a.

b.

2. Let

Is continuous at 1? Justify your answer in terms of limits!

3. Find (first derivative) for the following.

a.

b.

c.

d. , 0 < x < p

e.

4. Find (second derivative) for the following.

a.

b.

5. A right circular cone (point down) whose radius is twice its height is being filled with molten steel at the rate of 12 p cubic feet per minute. How fast is the depth of the molten steel changing when the cone is filled to a depth of 3 feet?

6. For the following function:

i) find the critical numbers ,

ii) determine the open intervals on which f is increasing/decreasing,

iii) identify and classify (max/min) all relative extrema,

.

7. A function has a second derivative given by

Find the intervals where is concave up and concave down.

8. Evaluate the following indefinite integrals.

a.

b.

9. Evaluate the following definite integrals using the Fundamental Theorem of Calculus; i.e., no calculator.

a.

b.

10. Find the area of the region bounded by , and the

x-axis.

13. Rainbow trout were stocked in Emerald Lake some years ago. The population began to grow at an exponential rate; i. e., for some constant k. (P(t) is the population after t years.) After 3 years there were 6,000 fish and after 5 years the population had risen to 10,000 fish. What was the initial population?

14. Write the exponential definition for

a. sinh x

b. cosh x

15. Use implicit differentiation and right triangle trigonometry to find Sketch a diagram to show how the right triangle was used. (y = arcsin x iff

x = sin y)

16. Find the area of the region bounded by.

17. Consider the upper half of the circle given by

a. Find the volume of the solid formed by rotating this semi-circle about the

x-axis to show that the volume of a sphere is

b. Find the surface area formed by rotating this semi-circle about the x-axis to show that the surface area of a sphere is .

18. Set up, but do not evaluate, a definite integral that would give the length of the curve y = ln x from (1, 0) to (4, ln 4).

19. Find the derivatives of the following.

a.

b. y =

c. y =

d. y = arccos

e. y =

20. Evaluate the following integrals.

a.

b.

c.

21. Use logarithmic differentiation to find , x > 0.

22. Find the work done in compressing a spring 5 cm if the force required to compress it to 5 cm is 80 Newtons.

23. Find the work done in emptying a tank of water by pumping the water out the top. The tank is a cylinder that is 3 meters in diameter and 5 meters high and is only half full. Use 9800 Newtons/cm³ as the weight of the water.

24. Find the center of mass for the lamina of constant density bounded by the graphs of .