Derivative as a rate of change.pptx - The Derivative as a Rate of Change 1 Example 1 It is estimated that x months form now the population of a certain

# Derivative as a rate of change.pptx - The Derivative as a...

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1 The Derivative as a Rate of Change

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2 It is estimated that x months form now, the population of a certain community will be Example 1 8000 20 ) ( 2 x x x p a. At what rate will the population be changing with respect to time 15 months from now? b. By how much will the population actually change during the 16 th month?
3 Solution: a. The rate of change of the population with respect to time is the derivative of the population function. That is 20 2 ) ( change of Rate x x p The rate of change of the population 15 months from now will be month per people 50 20 ) 15 ( 2 ) 15 ( p b. The actual change in the population during the 16 th month is people 51 8525 8576 ) 15 ( ) 16 ( p p Note : Since the rate varies during the month, the actual change in population during 16 th month differs from the monthly rate of the change at the beginning of the month.

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4 Relative and Percentage Rates of Change : The relative rate of change of a quantity Q(x) with respect to x is given by the ratio The corresponding percentage rate of change of Q(x) with respect to x is Q dx dQ x Q x Q Q(x) / ) ( ) ( of change of rate Relative ) ( ) ( 100 ) ( of change of rate Percentage x Q x Q x Q Relative and Percentage Rates of Change
5 Example 2 The gross domestic product (GDP) of a certain country was billion dollars t years after 1995. a. At what rate was the GDP changing with respect to time in 2005? b. At what percentage rate was the GDP changing with respect to time in 2005? 106 5 ) ( 2 t t t N Solution: a. The rate of change of the GDP is the derivative N’(t)=2t+5.

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