Credit Card Losses in the United Kingdom The mail and the...

Credit Card Losses in the United Kingdom The mail and the Internet are major routes for fraud against merchants who sell and ship products. These merchants must often accept credit cards that are not physically present (called CNPs, "credit cards not present"). The amount of CNP fraud in the United Kingdom from 2008 through 2012 was $$f(t)=328.9-92.07 \ln (t+1) \quad(0 \leq t \leq 4)$$ million British pounds (GBP) in year $t,$ where $t=0$ corresponds to $2008 .$ What was the average amount of CNP fraud losses in the United Kingdom over the period from 2008 through 2012 ?

Credit Card Losses in the United Kingdom The mail and the Internet are major routes for fraud against merchants who sell and ship products. These merchants must often accept credit cards that are not physically present (called CNPs, "credit cards not present"). The amount of CNP fraud in the United Kingdom from 2008 through 2012 was
$$f(t)=328.9-92.07 \ln (t+1) \quad(0 \leq t \leq 4)$$
million British pounds (GBP) in year $t,$ where $t=0$ corresponds to $2008 .$ What was the average amount of CNP fraud losses in the United Kingdom over the period from 2008 through 2012 ?

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Key Concepts

Definite Integration

Definite integration is a fundamental concept in calculus used to find the total accumulation, such as area under a curve, over a specific interval. It is critical for calculating the average value of a function by summing up the contributions of the function's values across the interval.

Natural Logarithm and Its Properties

The natural logarithm is an essential mathematical function with base e, and it frequently appears in continuous models and integration problems. Understanding its properties, such as the ability to transform multiplicative relationships into additive ones and its integration rules, is crucial when dealing with functions that include logarithmic expressions.

Average Value of a Function

In calculus, the average value of a continuous function over a closed interval is determined by integrating the function over that interval and then dividing by the length of the interval. This concept provides a single number that represents the overall behavior of the function across that interval.

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