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Calculating Rate of Change of Area in a Growing Circle

Introducing a third variable, time, in determining the rate of change of the area of a circle as both the area and radius evolve over time. Utilizing implicit differentiation, find the derivative of the area with respect to time considering the changing radius.

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Calculating Rate of Change of Area in a Growing Circle

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  1. Let’s introduce a “third” variable, time, t The Area of a circle is given by the formula The parameters are A and r

  2. Drop a pebble into a pond Both the Area and radius grow with respect to time

  3. Find the rate of change of the Area with respect to time That is, find:

  4. Notice, the variables do not agree This will be “implicit differentiation” with respect to t. When taking the derivative of A and r we will need to multiply by dA/dt and dr/dt respectively

  5. To find the rate of change of the area with respect to time, we need to know 2 things the radius, r, and its rate of change, dr/dt. This becomes a “related rate” problem in the next section

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