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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 instantaneous rate of production at t=6. (D) The number of cars produced over the first 6 hours.

The function c(t) c(t) gives the number of cars produced in a factory by time t t (in hours) on a given day.\newlineWhat does ∫06c′(t)dt \int_{0}^{6} c^{\prime}(t) d t represent?\newlineChoose 11 answer:\newline(A) The average rate of change of the car production over the first 66 hours.\newline(B) The time it takes to produce 66 cars.\newline(C) The instantaneous rate of production at t=6 t=6 .\newline(D) The number of cars produced over the first 66 hours.

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Q. The function c(t) c(t) gives the number of cars produced in a factory by time t t (in hours) on a given day.\newlineWhat does ∫06c′(t)dt \int_{0}^{6} c^{\prime}(t) d t represent?\newlineChoose 11 answer:\newline(A) The average rate of change of the car production over the first 66 hours.\newline(B) The time it takes to produce 66 cars.\newline(C) The instantaneous rate of production at t=6 t=6 .\newline(D) The number of cars produced over the first 66 hours.
  1. Understand Integral Meaning: 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. In this case, the integral from 00 to 66 of c′(t) dtc'(t) \, dt represents the net change in the number of cars produced from time t=0t = 0 to t=6t = 6.
  2. Relate to Answer Choices: Relate the integral to the given answer choices.\newlineThe net change in the number of cars produced over the first 66 hours is the total number of cars produced during that time. This matches with one of the given answer choices.
  3. Eliminate Incorrect Choices: Eliminate incorrect answer choices based on the integral's meaning.\newline(A) The average rate of change of the car production over the first 66 hours would be represented by the total change divided by the time interval, which is not directly given by the integral.\newline(B) The time it takes to produce 66 cars is not related to the integral of the rate of production over time.\newline(C) The instantaneous rate of production at t=6t=6 is given by c′(6)c'(6), not the integral of c′(t)c'(t) from 00 to 66.\newline(D) The number of cars produced over the first 66 hours is the net change in the car production, which is what the integral of c′(t)c'(t) from 00 to 66 represents.
  4. Choose Correct Answer: Choose the correct answer based on the analysis.\newlineBased on the previous steps, the correct answer is (D) The number of cars produced over the first 66 hours.

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