First exam practice sheet



Date of test Thursday, Oct. 16 in class

Material covered Chapter 6, sections 1, 2, 3, 4, 5

Ch 1 section 7

Ch 3 section 5

Allowable materials Calculator (no TI-89 or 92)

3”(5” index card of notes

Trigonometry formulas handout

Sample problems

1. Eliminate the parameter to express the equation in Cartesian form and sketch the curve. [pic] [pic]

2. Eliminate the parameter to express the equation in Cartesian form and state the domain and range of the Cartesian equation.

a. [pic] [pic]

b. [pic] [pic]

c. [pic] [pic]

3. For the parametric curve defined by the equation [pic]

a. For what values of t is the tangent line to the curve vertical?

b. What are the equations of the tangent lines from part (a)?

4. The parametric equation for a cycloid is [pic].

a. Find an equation of the tangent line to the cycloid at [pic]

b. Find the area under one period of the cycloid

5. Find the length of the parametric curve [pic] for [pic]

6. Set up an integral for the length of the curve [pic] for [pic]. Do not evaluate the integral.

7. Sketch the region enclose between the curves [pic] and [pic] and find its area.

8. Sketch the region in the first quadrant enclosed between the curves [pic], [pic], and the y-axis and find its area.

9. Find the area enclosed by one loop of the parametric curve [pic]

(hint: After setting up the integral, use the identity [pic])

[pic]

10. Use an integral to find the volume of the pyramid drawn:

[pic]

11. Sketch the region enclosed by the curves [pic] and [pic], and then find the volume of the solid generated by revolving this region about the y-axis.

12. Use an integral to find the volume of the frustrum of a cone drawn below.

[pic]

13. For the function [pic], whose graph is drawn below,

[pic]

a. Find the average value of the function over the interval [pic]

b. Estimate from the graph the value of x at which the function f takes on the average value.

14. A spring with natural length 5 cm is stretched to a length of 7 cm, requiring a force of 10 Newtons. How much work is done in this stretching?

15. A 1000 lb piano is raised up to the window of an apartment building in Chicago, a height of 120 ft. The cable that lifts it weighs 2 lb per foot. How much work is done?

16. Calculate the total hydrostatic force on the dam drawn below:

[pic]

17. Find the moments and center of mass of the lamina drawn below, bounded by the curves [pic], [pic], and the x-axis, with density [pic].

[pic]

1.

Answers:

2. [pic] [pic]

[pic]

3.

a. [pic] [pic]

b. [pic] [pic]

c. [pic] [pic]

4.

a. [pic] for any integer k

b. [pic]

5.

a. [pic]

b. [pic]

6. [pic]

7. [pic]

8.

[pic]

Area:[pic]

9.

[pic]

Area: [pic]

10. [pic]

11. 12

12.

[pic]

[pic]

13. Volume: [pic]

14.

a. [pic]

b. [pic]

15. 0.1 J

16. 134,400 ft. lb.

17. [pic]

18. [pic] [pic]

[pic]

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