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Related Rates Chapter 3.7

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Related Rates The Chain Rule can be used to find the rate of change of quantities that are related to each other The important point to remember about related rates problems is that changes are occurring with respect to time For example, when water is draining out of a conical tank, the volume of the tank is related to both the height of the water and the radius at the surface at a particular time t Also, the radius and height are related to each other

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Related Rates

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Finding Related Rates

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Example 1: Two Rates That Are Related

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Problem Solving With Related Rates Most related rates problems are verbal descriptions You will usually have to create an equation (sometimes equations) to model the situation The rest of the examples require you to model from a verbal description

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Example 2: Ripples in a Pond

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Guidelines For Solving Related Rate Problems 1.Identify all given quantities and quantities to be determined. Make a sketch and label the quantities 2.Write an equation involving the variables whose rates of change either are given or are to be determined 3.Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time 4. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change. Include units in your final answer (if appropriate)

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Variables and Rates of Change Verbal StatementMathematical Model The velocity of a car after traveling for 1 hour is 50 miles per hour Water is being pumped into a swimming pool at a rate of 10 cubic meters per hour

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Example 3: An Inflating Balloon

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Example 4: The Speed of an Airplane Tracked by Radar

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Example 5: A Changing Angle of Elevation

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Example 6: The Velocity of a Piston

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Exercise 3.6a Page 187, #1-8, 13-25

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Exercise 3.6b Page 188, #27-53 odds

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