Related Rates and the Chain Rule

Volume of a Sphere

• V = 4/3 πr³

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• What is DV/ Dr?

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• If we want the rate of change of volume with respect to time, what do we need?

Volume of a Sphere

• V = 4/3 πr³

DV/ Dr = 4πr^{2 just took derivative�}

• If we want the rate of of change of volume with respect to time, what do we need? �

If I give you the rate of change of the radius is 2 inches per minute, what does that represent?

Using the Chain Rule

• DV/Dt = DV/Dr (Dr/Dt)��Can you get the rate of the sphere’s volume change when the radius is 6 inches?

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• Did you include units?

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Using the Chain Rule

• DV/Dt = DV/Dr (Dr/Dt) given dr/dt��Can you get the rate of the sphere’s volume change when the radius is 6 inches?�

4 π r² (2) plug in r = 6� 4 π 6² (2) �

• 288π in³/min What if DV/ Dt was negative?

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Triangles…

• Use Pythagorean’s Theorem

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• List x = dx/dt=

y = dy/dt =

z= dz/dt = ��Remember Fixed length’s and direction

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Now try #49 on page 137�

And #50 � How is this different? � How is it the same?