Explanation Given: The expression for f ( t ) is, f ( t ) = t e − t 45 Formula used: The integration of the function f ′ ( x ) is, f ( x ) = ∫ f ′ ( x ) d x The formula for integration by parts is, I = ∫ u d v = u v − ∫ v d u Here, u and v are two independent functions. The integration of function e k x is performed as, I = ∫ e k x d x = 1 k ⋅ e k x + c Here, k is a constant and c is the integration constant. Calculation: The derivative of the function for total number of customers F ( t ) gives the rate at which customers enter the store. This means that the function for the rate at which customers enter the store f ( t ) , is the derivative of F ( x ) . Write the expression for total number of customers F ( t ) . F ( t ) = ∫ f ( t ) d t Substitute t e − t 45 for s ′ ( t ) . s ( t ) = ∫ t e − t 45 d t …… (1) Consider u = t and d v = e − t 45 d t . Calculate d u d t . d u d t = d ( t ) d t d u d t = 1 d u = d t Calculate v from the expression for dv . ∫ d v = ∫ e − t 45 d t v = 1 ( − 1 45 ) e − t 45 v = − 45 e − t 45 Understand that it is not necessary to write a constant value in the expression for v as, this is an intermediate step and the integration constant will be added to the final expression obtained for F ( t ) . Substitute u for t and dv for e − t 45 d t in equation (1)

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Chapter 13.3, Problem 6CP

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To calculate: The number of customers that would have entered by 2:00 p.m. if the function is f(t)=te−t45.

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Chapter 13.3, Problem 5CP

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The number of customers that would have entered by 2:00 p.m. if the function is f ( t ) = t e − t 45 .

Solution Summary: The author explains how the derivative of the function for total number of customers F(t) gives the rate at which customers enter the store.

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To calculate: The number of customers that would have entered by 2:00 p.m. if the function is f(t)=te−t45.

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