Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.

b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?

검증된 단계별 안내

First, understand the Mean Value Theorem (MVT). It states that for a function continuous on [a, b] and differentiable on (a, b), there exists at least one point c in (a, b) such that the derivative at c is equal to the average rate of change over [a, b].

Calculate the average rate of change of the function P(t) = \( \frac{100t}{t+1} \) over the interval [0, 8]. This is given by \( \frac{P(8) - P(0)}{8 - 0} \).

Find P(0) and P(8) by substituting t = 0 and t = 8 into the function P(t). This will give you the values needed to compute the average rate of change.

Next, find the derivative of the function P(t) = \( \frac{100t}{t+1} \). Use the quotient rule, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \).

Set the derivative equal to the average rate of change calculated earlier and solve for t. This will give you the point c in the interval [0, 8] where the instantaneous rate of change equals the average rate of change.

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11m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the instantaneous rate of change (the derivative) equals the average rate of change over that interval. This theorem is fundamental in connecting the behavior of a function to its derivative.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a point is defined as the derivative of the function at that point. It represents how the function value changes at that specific moment, providing insight into the function's behavior. For the given function P(t), this would involve calculating P'(t) to find the rate of change at any time t.

Average Rate of Change

The average rate of change of a function over an interval [a, b] is calculated as the difference in the function values at the endpoints divided by the difference in the input values, expressed as (P(b) - P(a)) / (b - a). This concept helps in understanding how the function behaves over a specified range, and is essential for applying the Mean Value Theorem.

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?