(a) Explanation Given information: The concentration of a certain drug in patient’s bloodstream t hr after injection is, C ( t ) = 0.2 t t 2 + 1 (1) Formula used : The quotient rule of differentiation is, f ′ ( x ) = d d x [ f ( x ) g ( x ) ] = g ( x ) ⋅ d d x f ( x ) − f ( x ) ⋅ d d x g ( x ) [ g ( x ) ] 2 The above formula is defined when the denominator g ( x ) ≠ 0 . Calculation: The derivative of the equation (1) with respect to time gives the rate of change of concentration of the drug so use the quotient rule of differentiation for the given equation to get the derivative function C ′ ( t ) . C ′ ( t ) = d d t [ 0.2 t t 2 + 1 ] = ( t 2 + 1 ) ⋅ d d t ( 0.2 t ) − ( 0.2 t ) ⋅ d d t ( t 2 + 1 ) ( t 2 + 1 ) 2 = ( t 2 + 1 ) ⋅ ( 0.2 ) − ( 0.2 t ) ⋅ ( 2 t ) ( t 2 + 1 ) 2 = 0.2 t 2 + 0 (b) To determine To find: The value of rate of change of the concentration of drug at 1 2 hr , 1 hr and 2 hr after the injection.

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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Differential Equations

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Question

Chapter 3.2, Problem 52E

(a)

To determine

To find: The rate with respect to time at which the concentration of drug C(t)=0.2tt2+1 is change.

(b)

To determine

To find: The value of rate of change of the concentration of drug at 12 hr , 1 hr and 2 hr after the injection.

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Chapter 3.2, Problem 51E

Chapter 3.2, Problem 53E

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

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The rate with respect to time at which the concentration of drug C ( t ) = 0.2 t t 2 + 1 is change.

Solution Summary: The derivative of the equation (1) with respect to time gives the rate of change of concentration of a certain drug in patient’s bloodstream.