## Precalculus 2 Practice Exam

Test 4 Time: 1.5 hours

1. Find the center, foci, and vertices of the ellipse Answer: The center is (-5,4), the foci are (-5,2), and (-5,6), and the vertices are (-5,8), (-5,0).

2. Find the magnitude and direction angle of the vector .

Answer: The magnitude is , and the direction angle is approximately .

3. Each cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 480 feet apart and 60 feet above the roadway. The cables touch the roadway midway between the towers. Find an equation for the parabolic shape of each cable.

Answer: The equation is , if the midpoint on the roadway is chosen as the origin of the coordinate system.

4. Convert the polar equation to rectangular form.

Answer: The rectangular equation is 5. Consider the equation • Suppose we wish to rotate the axis to eliminate the xy-term. Find the angle of rotation . (Use exact values.)

Answer: The angle of rotation .

• Using your result from above, express x and y in terms of x' and y'. 6. Find two sets of polar coordinates for the point (-4,3), where .

Answer: The two sets of polar coordinates are , and .

7. Let , and .
• Find the projection of u onto v.

Answer: The projection of u onto v is .

• Find the vector component of u orthogonal to v.

Answer: The vector component is .

8. Find an equation for the hyperbola with vertices at (-1,5) and (-1,1), and with asymptotes , and .

Answer: The equation for the hyperbola is 9. Follow these instructions carefully: Sketch the graph of by hand. Have a table of at least seven appropriate values for r and . Use exact values for (no decimals). Sketch your graph by plotting the ordered pairs from your table (use the graph paper below). The shape of the curve must be apparent from your table of values. If it isn't, you must choose more appropriate values for . You may use symmetry to avoid plotting too many points. If you would like to practice on the Final Exam, click on Next Exam.