The function b(t) gives the number of books sold by a store by time t (in days) of a given year. What does int_(45)^(50)b^(')(t)dt represent? Choose 1 answer: (A) The number of books sold between day 45 and day 50 (B) The number of days it takes to sell 50 books (C) The change in the rate of selling books between t=45 and t=50 (D) The total number of books sold by day 50

Question

The function  b(t) gives the number of books sold by a store by time  t (in days) of a given year. What does  int_(45)^(50)b^(')(t)dt represent? Choose 1 answer: (A) The number of books sold between day 45 and day 50 (B) The number of days it takes to sell 50 books (C) The change in the rate of selling books between  t=45 and  t=50 (D) The total number of books sold by day 50

Bytelearn · Accepted Answer

Understand Integral Meaning: Understand the meaning of the integral of the derivative of a function. The integral of the derivative of a function over an interval gives the net change in the function over that interval. This is based on the Fundamental Theorem of Calculus, which states that if F is an antiderivative of f on an interval [a, b], then the integral from a to b of f(x) dx is equal to F(b) - F(a).Apply to Function b(t): Apply the understanding to the given function b(t). The integral from 45 to 50 of b'(t) dt represents the net change in the number of books sold by the store from day 45 to day 50. This is because b'(t) is the rate of change of the number of books sold with respect to time, and integrating this rate of change over the interval from 45 to 50 gives the total change in the number of books sold over that interval.Match to Answer Choices: Match the understanding to the given answer choices. The net change in the number of books sold between day 45 and day 50 corresponds to answer choice (A) The number of books sold between day 45 and day 50. This is because the integral of the rate of change of the number of books (b'(t)) gives the total change in the number of books (b(t)), not the rate of change itself or the total number of books sold by a specific day.

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