Mean Value Theorem The population of a culture of cells grows ...

Question

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.

a. What is the average rate of change in the population over the interval [0, 8]?

Accepted Answer

Hello. In this video, we are going to be finding the average rate of change over a defined interval. We are told that a function C of T is equal to 80T divided by T + 4 in PPM. This represents the concentration of a solution where T greater than or equal to 0, is the time in minutes. We want to determine the average rate at which the concentration changes over the interval from 0 to 5. Now, since this problem is asking us to find an average rate of change over a defined interval, this tells us that we can go ahead and apply the mean value theorem in order to find this average rate of change. In order to use the mean value theorem, we need to satisfy the first two conditions of the theorem. So, the function given to us is C of T equal to 80T divided by T + 4. The first condition of the mean value theorem asks us, is CFT. Continuous. On the closed interval from 0 to 5. In order to answer this question, we will go ahead and check the domain of the given function. Now, the function C of T is in the form of a rational function. What we want to do is we want to find the restrictions of the domain by taking the denominator and setting it equal to 0. This will give us T + 4 equal to 0, and by subtracting 4 to both sides of this equation, we get T equal to -4. What this tells us is that if T is equal to -4, we will get an undefined value. But since -4 is not included in the domain from 0 to 5, that means that C of T is continuous on this interval. So the first condition is satisfied. Now we need to check the 2nd condition. The second condition asks us, is C of T differentiable? On the open interval from 0 to 5. In order to check this condition, we will calculate the derivative of CFT and check its domain. So, in order to find the derivative of CFT, we will use the quotient rule. C T is equal to the denominator T + 4 multiplied by the derivative of the numerator, which is 80 minus the numerator 80T multiplied by the derivative of the denominator which is 1, and this will be divided by the original denominator squared, which will be T + 4 quantity squared. Let's go ahead and simplify. If we distribute 80 into T + 4, we get 80T plus 240. Minus 80T divided by T + 4 quantity squared. And by combining like terms together, C of T is equal to 240 divided by the quantity T + 4 squared. Now notice that this rational function is going to have the same restriction as the original function. T cannot equal to -4, otherwise we will have an undefined value from the derivative. Since the derivative is defined on the open interval from 0 to 5, that tells us that CFT is differentiable on this interval, so the second condition is satisfied. Now that the first two conditions are satisfied, we can use the 3rd condition to find the average rate of change of CFT. The third condition states that there is a value c such that C of C. is equal to the average rate of change of the function. That will be C of 5 minus C of 0, divided by 5 minus 0. What we will go ahead and do is we will solve the right-hand side of this equation in order to find the average rate of change of the original function. So The first part tells us to take the value of 5 and plug it into the original function. That will give us 80 multiplied by 5, divided by 5 + 4. Then we will subtract 0 plugged into the original function. This will give us 80, multiplied by 0, divided by 0 + 4. And this will all be divided by 5 minus 0. This will further simplify to be 400 divided by 9 minus 0, all divided by 5. The numerator will simplify to be 400 divided by 9, and if we combine both of the denominators by multiplying them together, we get 400 divided by 45. This will further simplify to give us the value of 80 divided by 9. And 80 divided by 9 in terms of PPT is going to be the average rate of change of the original function C of T. And with that being said, the solution to this problem is going to be a. So, I hope this video helps you in understanding on how to use the mean value theorem, and I will go ahead and see you all in the next video.

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.

a. What is the average rate of change in the population over the interval [0, 8]?

Key Concepts

Average Rate of Change

Function Evaluation

Continuous Functions

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