Explanation Given Information: The “For Geniuses” series of books that has been taking off since Duffin House first gained exclusive rights to the series 6 months ago. Revenues from Precalculus for Geniuses are expected to follow the curve R ( t ) = 10 , 000 + 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] dollars 0 ≤ t ≤ 72 , where t is the time in months from now and R ( t ) is the monthly revenue. Formula Used: If f ( x ) and g ( x ) are two functions then sum and difference rules for the differentiation is, d d x [ f ( x ) ± g ( x ) ] = d d x f ( x ) ± d d x g ( x ) If f is a differentiable function of u and v of function x, then the product rule is given by d d x f ( u ) ⋅ g ( u ) = g ( u ) d d x f ( u ) + f ( u ) d u d x g ( u ) . Constant multiple rules are d d x k ⋅ g ( x ) = k d d x g ( x ) where k is a constant. Calculation: Consider the function, R ( t ) = 10 , 000 + 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] . Now evaluate the derivative of the function R ( t ) = 10 , 000 + 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] where f ( u ) = 10 , 000 and g ( u ) = 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] Apply the sum rule, R ′ ( t ) = d d t 10 , 000 + d d t 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] Now differentiate the constant term with respect to t , R ′ ( t ) = d d t 10 , 000 R ′ ( t ) = 0 Then apply the constant multiple rules and product rule to evaluate the differentiation of the function d d t 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] , d d t 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] = 20 , 000 [ d d t e − 0.05 t sin [ π 6 ( t − 2 ) ] ] = 20 , 000 [ sin [ π 6 ( t − 2 ) ] d d t e − 0.05 t + e − 0.05 t d d t sin [ π 6 ( t − 2 ) ] ] Further solved as: d d t 20 , 000 e − 0.05 t sin [ π 6 ( t − 2 ) ] = 20 , 000 [ − 0

7th Edition

ISBN: 9781337291248

Author: Waner, Stefan.

Publisher: Cengage Learning,

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Chapter 9, Problem 27RE

To determine

To calculate: The rate at which revenue to the nearest dollar is changing 20 months from now for the “For Geniuses” series of books that has been taking off since Duffin House first gained exclusive rights to the series 6 months ago. Revenues from Precalculus for Geniuses are expected to follow the curve R(t)=10,000+20,000e−0.05tsin[π6(t−2)] dollars 0≤t≤72, where t is the time in months from now and R(t) is the monthly revenue.

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Chapter 9 Solutions

Applied Calculus

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Solution Summary: The author calculates the rate at which revenue to the nearest dollar is changing 20 months from now for the "For Geniuses" series of books.

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Chapter 9 Solutions