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AP Calculus AB Chapter 2, Section 6

AP Calculus AB Chapter 2, Section 6. Related Rates 2013 - 2014. Related rates. We have used the chain rule to find implicitly. Another important use of the chain rule is to find the rates of change of two or more related variables that are changing with respect to time . . Related Rates.

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AP Calculus AB Chapter 2, Section 6

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  1. AP Calculus ABChapter 2, Section 6 Related Rates 2013 - 2014

  2. Related rates • We have used the chain rule to find implicitly. Another important use of the chain rule is to find the rates of change of two or more related variables that are changing with respect to time.

  3. Related Rates • For example: If water is draining out of a conical tank, what are the variables that are changing with respect to time?

  4. Conical Tank continued • Use implicit differentiation for the volume of a cone and describe what each piece means.

  5. Two Rates That Are Related • Suppose x and y are both differentiable functions of t and are related by the equation . Find when , given that when .

  6. Ripples in a Pond • A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

  7. Guidelines for Solving Related-Rate Problems • Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. • Write an equation involving the variables. • Use the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. • After completing step 3, substitute into the resulting equation all known values for the variables and their rates of change. Solve if necessary.

  8. Examples of mathematical models involving rates of change

  9. An Inflating Balloon • Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.

  10. The Speed of an Airplane Tracked by Radar • An airplane is flying on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when s = 10 miles, what is the speed of the plane?

  11. A Changing Angle of Elevation • A television camera at ground level is filming the lift-off of a space shuttle that is rising vertically according to the position equation , where s is measured in feet and t is measured in seconds. The camera is 2000 ft from the launch pad. Find the rate of change in the angle of elevation of the camera shown at 10 seconds after lift-off.

  12. The Velocity of a Piston • In the engine shown on page 153, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when .

  13. 2.6 Homework • Pg. 154 – 156, #’s: 3, 7, 15, 19, 27, 31, 43 • 7 problems

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