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How do you find the instantaneous rate of change of revenue when 1000 units are produced if the revenue (in thousands of dollars) from producing x units of an item is #R(x) = 12x - .005x^2#?

Answer 1

Just take the derivative of #R# and plug in #x=1000#. Doing so gives #R'(x)=12-0.01x# so that #R'(1000)=12-10=2# thousand dollars per unit produced.

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Answer 2

Evelyn Agard

To find the instantaneous rate of change of revenue when 1000 units are produced, you need to find the derivative of the revenue function ( R(x) = 12x - 0.005x^2 ) with respect to ( x ), and then evaluate it at ( x = 1000 ).

Find the derivative of ( R(x) ) with respect to ( x ): ( R'(x) = \frac{dR}{dx} = 12 - 0.01x ).
Evaluate the derivative at ( x = 1000 ): ( R'(1000) = 12 - 0.01(1000) ).
Calculate ( R'(1000) ) to find the instantaneous rate of change of revenue when 1000 units are produced.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.