Reference no: EM13351266

1. Evaluate the derivatives of the following:

a. f (x) = x/ sin(cos(x²))

b. f (x) =(x^3/2 - x) / tan(x)

c. f (x) = tan²(x²)dx

2. A square piece of cardboard is made into an open top box by cutting out squares on each corner. If the length of the sides of the cardboard is L, what could the dimensions be for the cutout so the volume is a maximum? Minimum?

3. A lighthouse that is 1 kilometer from shore lamp rotates 8 times per minute.

a. What speed is the lamp's spot moving as it passes the point where it is perpendicular to shore?

b. How fast is the spot moving along the shore when the spot is 1 km from the point at which it is perpendicular?

c. Where is it moving the fastest and why?

4. If you have 300 meters of fence and you desire to fence along a river (requiring no fence along it) and you have 4 fence posts (you don't have to use them all), assuming all segments of the fence have equal lengths, how should you configure the fence to get the maximum area enclosed?

5.Suppose a Ferris wheel with radius of 12 meters is rotating at a rate of 2 rotations per minute.

a. How fast is a person rising when the person is 3 meters above the horizontal line running through the center (3 or 9 o'clock position)?

b. How fast is a person rising when the person is 3 meters above the base of the Ferris wheel?

c. Suppose a person riding the Ferris when drops a ball while going up when they are 6 meters from the bottom of the Ferris wheel.

i. When will it hit the ground?

ii. How fast will it be going when it hits the ground?

d. At which point should it be dropped so it is going fastest when it hits the ground? Explain your answer

7. For each of the following pairs of functions over the designated intervals, graph each one together, fully describe all the features of each function (that is as in problem 6) of your graph, and find the area between them bounded by the given values of x.

8. Volume of a sphere using both volumes by both revolution methods

9. Volume of a cylinder with  cone cut out by both revolution methods

10. Volume of a torus (donut)

 

11. Sphere with a rectangular cut out around (as given in the below figure) by both revolution methods

12. Using the methods from this class, evaluate the volume of a triangular pyramid whose base is an equilateral triangle with 8 cm sides and height 19 cm.

13. A 45 degree wedge is cut from a cylinder with 1 meter radius as given. What is its volume? Solve it two ways.

14. Evaluate the volume remaining of a sphere of radius 5 with a hole of radius 3 drilled through its center.