Applications Derivatives

Applications Derivatives - Contents 4 Related Rates 63 4.1...

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Unformatted text preview: Contents 4 Related Rates 63 4.1 General Guide for Solving Related Rate Problems . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Economic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2 CONTENTS Chapter 4 Related Rates Problem If a tree trunk adds 1 / 4 of an inch to its diameter and 1 foot to its height each year, how rapidly is its volume changing when its diameter is 3 feet and its height is 50 feet (assume that the tree trunk is a circular cylinder). Solution r h Let r be the radius and h be the height of the trunk. Then, dr dt = 1 2 1 4 = 1 8 in/yr = 1 96 ft/yr dh dt = 1 ft/yr and the volume V is V = πr 2 h . Since r and h are functions of time, t , dV dt = 2 πrh dr dt + πr 2 dh dt . Thus, when r = 3 2 ft. and h = 50 ft., dV dt = 2 π 3 2 (50) 1 96 + π 3 2 2 (1) = 61 π 16 ft 3 /yr . 64 Related Rates Observation The solution illustrates the minimum amount of information and explanation required for full marks . 4.1 General Guide for Solving Related Rate Problems Step 1. Draw a large diagram to illustrate the geometry of the problem at some arbitrary time t . Label any numerical quantities which remain fixed throughout the problem. Label all quantities which change with time by letters. Step 2. Find one or more relations among the quantities which vary. These relations must hold for all time. Usually the geometry or trigonometry of the diagram suggest an appropriate relation. Step 3. Differentiate these relations implicitly with respect to time. Step 4. Substitute in all given values such as rates and distances to enable you to solve the resulting equation for the required unknown. The diagram should be large and neat so that you may see clearly the geometry of the situation. Use one third of a page or more for the diagram. If you have a good diagram to stare at your chances of completing the question are also good. In problems involving ships, airplanes and autos, etc., travelling in different directions use a coordinate axis. If the object is moving to the right D t x is positive. If the object is moving to the left D t x is negative. If the object is moving upward in the diagram then D t y is positive. If the object is moving downward then D t y is negative. The above choice is the most natural one. If you are consistent in its use you won’t get things increasing when they are really decreasing or decreasing when they are supposed to be increasing. 4.2 Problems 1. A point moves on a hyperbola y 2- 2 x 2 = 2, where x and y are differentiable functions of time. What is the rate of change of y with respect to time when D t x = 1 2 , x = 1, and y =- 2?...
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Applications Derivatives - Contents 4 Related Rates 63 4.1...

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