(a). Explanation Proof: The given logistic growth function is, Q ( t ) = A 1 + B e − k t (1) It is known that, the inflection point is obtained after putting the double derivative equal to 0. Differentiate both the sides of equation (1) with respect to t , Q ′ ( t ) = − A ( B e − k t ( − k ) ) ( 1 + B e − k t ) 2 Q ′ ( t ) = A k ( B e − k t ) ( 1 + B e − k t ) 2 (2) Use equation (1) to determine an expression for B e − k t , Q = A 1 + B e − k t 1 + B e − k t = A Q B e − k t = A Q − 1 Substitute A Q − 1 for B e − k t in the equation (2), Q ′ ( t ) = A k ( A Q − 1 ) ( 1 + A Q − 1 ) 2 = A k ( A Q − 1 ) A 2 Q 2 = Q 2 k ( A − Q Q ) A = Q k ( A − Q ) A Further simplify the above equation, Q ′ ( t ) = Q k ( A − Q ) A Q ′ ( t ) = Q k ( 1 − Q A ) Further, differentiate the above equation with respect to t , Q ″ ( t ) = Q ′ k ( 1 − Q A ) + Q k ( − Q ′ A ) = Q ′ k − Q ′ k ( Q A ) − Q k ( Q ′ A ) = Q ′ k − Q ′ k Q A − Q k Q ′ A = Q ′ k − 2 k Q ′ Q A Substitute Q k ( 1 − Q A ) for Q ′ in the above equation, Q ″ ( t ) = Q (b). To determine To interpret: The result from part (a).

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 5.6, Problem 35E

(a).

To determine

To show: The inflection point of the graph of Q is Q=A2 and this is obtained when t=(lnB)/k .

(b).

To determine

To interpret: The result from part (a).

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Chapter 5.6, Problem 34E

Chapter 5.6, Problem 36E

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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The inflection point of the graph of Q is Q = A 2 and this is obtained when t = ( ln B ) / k .

Solution Summary: The author explains that the inflection point of the graph of Q is Q=A2 and this is obtained when t = (mathrmln