The function c(t) gives the number of cars produced in a factory by time t (in hours) on a given day. What does int_(0)^(6)c^(')(t)dt represent? Choose 1 answer: (A) The average rate of change of the car production over the first 6 hours. B The time it takes to produce 6 cars. (C) The number of cars produced over the first 6 hours. (D) The instantaneous rate of production at t=6.

Question

The function  c(t) gives the number of cars produced in a factory by time  t (in hours) on a given day. What does  int_(0)^(6)c^(')(t)dt represent? Choose 1 answer: (A) The average rate of change of the car production over the first 6 hours. B The time it takes to produce 6 cars. (C) The number of cars produced over the first 6 hours. (D) The instantaneous rate of production at  t=6.

Bytelearn · Accepted Answer

Understand the concept: Understand the meaning of the integral of a derivative. The integral of a derivative function over an interval gives the net change of the original function over that interval. This is based on the Fundamental Theorem of Calculus.Apply Fundamental Theorem: Apply the Fundamental Theorem of Calculus to the problem. The integral from 0 to 6 of c'(t) dt represents the net change in the number of cars produced from time t = 0 to time t = 6.Interpret the result: Interpret the result in the context of the problem. Since c(t) represents the number of cars produced, the net change in c(t) from time 0 to time 6 is the total number of cars produced during the first 6 hours.Match interpretation to answer: Match the interpretation to the given answer choices. The interpretation from Step 3 directly corresponds to answer choice (C) The number of cars produced over the first 6 hours.

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