1. Bonaventura Francesco Cavalieri 1598 – 1647 Bonaventura Francesco Cavalieri 1598 – 1647 Bonaventura Cavalieri was an Italian mathematician who developed a method of indivisibles which became a factor in the development of the integral calculus. Bonaventura Cavalieri was an Italian mathematician who developed a method of indivisibles which became a factor in the development of the integral calculus.

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6. Steps for Related Rates Problems: II Known Derivative Express the given rates in terms of derivatives. IV Equation Write an equation that relates the various quantities of the problem. If necessary, use geometry of the situation to eliminate one of the variables by substitution. I Introduction -Read the problem carefully. -Draw a picture (sketch), if possible. -Introduce notation. Assign symbols to all quantities that are functions of time. III Unknown Derivative Express the unknown rates in terms of derivatives.

7. Steps for Related Rates Problems: V Differential Equation Use the Chain Rule to differentiate both sides of the equation with respect to t, also known as implicit differentiation to find a differential equation. VI Snap Shot Substitute the given information into the resulting equation and solve for the unknown derivative. VII Conclusion State your conclusion to the problem.

8. A ladder 15ft in length leans against a vertical wall, with the bottom of the ladder 5ft from the wall on a horizontal floor. If at that time the bottom end of the ladder is being pulled away at the rate of 2 ft/sec, at what rate does the top of the ladder slip down the wall? Example

9. Suppose a particle is moving on the curve At the point the particle is moving left at 3 feet per second. How is the particle moving vertically at that point? Example

10. Suppose a liquid is to be cleared of sediment by pouring it through a cone-shaped filter. Assume the height of the cone is 16 inches and the radius at the base of the cone is 4 inches. If the liquid is flowing out of the cone at a constant rate of how fast is the is the depth of the liquid decreasing when the level is 8 inches deep? Example

11. A camera is mounted at a point 3000 feet from the base of a rocket launching pad. Let us assume that the rocket rises vertically and the camera is to take a series of photographs of the rocket. Because the rocket will be rising, the angle of elevation of the camera will have to vary at just the right rate to keep the rocket in sight. If the rocket is rising vertically at 800 ft/sec when it is 4000 feet up, how fast must the angle of elevation of the camera change at that instant to keep the rocket in sight? Example

12. Runners stand at first and second base in a baseball game. At the moment a ball is hit, the runner at first base runs to second base at 18 ft/s; simultaneously the runner on second runs to third base at 20ft/s. How fast is the distance between the runners changing 1 second after the ball is hit? (Hint: The distance between consecutive bases is 90 ft and the bases lie at the corners of a square. Example