CALCULUS II EXAM IV

CALCULUS II EXAM IV PRACTICE PROBLEMS

1. A jet is descending toward the ground along a path described by the upper branch of the hyperbola 16y² - x² = 400. When the jet reaches the vertex of this hyperbolic path it will begin gaining altitude again along the same hyperbolic path. The y-coordinate gives the jet's height above the ground in meters. How close to the ground does the jet come? Along the path of the jet, 100 meters beyond its lowest point, there is a building 25 meters high. Will the jet clear the building? If yes, by how much, and if no, by how much? Give your answer to the nearest 1/1000 of a meter. Click on the picture at the right to see an animation and click here to see an animation with scales.

2. Find the focus and the endpoints of the latus rectum of the parabola whose equation is

2x² - 8x + 24y + 104 = 0

3. Find the equations of the asymptotes of the hyperbola whose equation is

16x² - 9y² - 64x - 54y - 161 = 0

4. Give a parametric representation of the ellipse whose equation is given with the orientation being clockwise.

100x² + 25y² - 2500 = 0

5. Sketch the graph of the following polar equation and also write the equation in rectangular form.

6. Find the distance along the given curve from the point where t = 0 to the point where t = 2.

x = t³, y = (3t²) / 2

7. Find the area of the region enclosed by the polar graph of the following equation.

8. Approximate the surface area of the surface generated by revolving the polar curve given below about the x-axis (polar axis). Give your answer to the nearest 1/100.

9. Find an equation of the ellipse with a focus at (0,-4) and endpoints of the major axis at (0,-6) and (0,4).

10. Find a parametric representation of the hyperbola whose equation is

16x² - 9y² - 64x - 54y - 161 = 0

11. Translate the polar equation into rectangular coordinates and graph it.

12. A bird is perched at the top of a pole that is 10 feet high. It flies in an elliptical path to the top of a pole that is 30 feet high and 50 feet away from the 10 foot pole. The center of the elliptical path is directly above the 10 foot pole and at the same height as the 30 foot pole. It takes the bird 10 seconds to fly from the top of the 10 foot pole to the top of the 30 foot pole. Give a parametric representation of the path of the bird where t = 0 represents the time when the bird leaves the 10 foot pole, t = 10 represent the time when the bird arrives at the top of the 30 foot pole, and the coordinates of the top of the 10 foot pole are (0,-20). See the picture below. Find the bird's horizontal flight speed 20/3 seconds after it leaves the 10 foot pole. Click here or on the picture below to see an animation of the bird's flight from pole to pole.

This site contains links to other Internet sites. These links are not endorsements of any products or services in such sites, and no information

in such site has been endorsed or approved by this site.

Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu