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Related Rates Pumping air into a balloon

Volume ( ) and Radius ( ) increase with respect to time ( ).

By the chain rule, the rates are related by

Measuring is easier, using the related rates above, we can obtain .

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Two quantities and are increase or decrease as time change. They are related via an explicit function

or an implicit function

By the chain rule, the rates and are then related by a relation.

If at a certain instance, a rate (or ) is given, then one can calculate the other rate using the above related rate.

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Strategy

1. Read the problem and draw diagram. 2. Introduce notation and quantities. 3. Express the given information and

the required rate in terms of derivatives.

4. Write an equation that relates the various quantities of the problem.

5. Use the chain rule.

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EX (Inflating a balloon) Air is being pumped into a spherical balloon so that its volume increases at a rate of cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

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EX (The sliding ladder problem) A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder is slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

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EX (Filling a tank) A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep.

(ANS: 0.283)

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Indeterminate Forms (IF) IF A limit

where

is called indeterminate form type .

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IF A limit

where

is called indeterminate form type .

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L’Hospital’s rule If

is IF or and exist, then

Remark L’Hospital can be used to find one-sided limits and limits at infinity.

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Proof We use the linear approximation:

But , we get

The approximation is ``=” when taking . So

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EX Find

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EX Use the L’Hopital’s rule to show that

and

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EX Evaluate the limit

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EX Evaluate the limit

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EX (L’H twice) Find