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.

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

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given that a car's velocity is modeled by VFT is equal to 60T divided by T + 3 in meters per second. Where T is greater than or equal to 0 is the time in seconds. Find the time in square bracket 0.102 when the instantaneous rate of change of velocity matches its average rate of change. Awesome. So it appears for this particular problem, we're trying to determine ultimately, we're trying to figure out the time within this specific interval of square bracket 0.10 square bracket where the instantaneous rate of change of velocity. Matches its average rate of change. So that is our final answer that we're ultimately trying to solve for is when or what is this time where the instantaneous rate of change of velocity matches its average rate of change at the specific interval given to us by the prom itself. Fantastic. So with that said, our first step in order to solve for this particular problem is we need to define all of our known variables and the target variable that we're trying to solve for, which is our unknown variable that we're trying to solve for. So we need to note that we are told that VAT, so we know the value of VAT, and VAT once again is equal to 60T divided by T + 3. Fantastic. And our target variable, the variable that we're trying to solve for ultimately is the time T. We're trying to figure out the time T, and this is when the instantaneous rate of change of VAT will be equal to the average rate of change. Over the interval, square bracket 0.10 square bracket, which, as you should recall, the square bracket means that it is a closed interval. If it was parentheses, then it would be an open interval, but it is a closed interval because it's in square brackets. So with that said, We need to note now, as we should recall a note, that this particular prong can be solved using the mean value theorem, or MVT for short, or derivatives. So we can use the mean value theorem for derivatives. So as we should recall and note, the mean value theorem states that if a function VAT is continuous on the closed interval, Of square bracket A B. And the differentiable on the open variable or open interval I should say, parenthesis A B parenthesis. Then there exists at least one C in parentheses A B, such that The prime of C is equal to V B minus V A. All divided by B minus A. So let's say that again. So the mean value theorem and so this is MVT, so the mean value theorem states that if a function. If a function, which in this case, VT. Is continuous on the closed interval square bracket Abra and differentiable on the open intervals, then there will exist at least 1 C in parentheses AB such that V C will equal the following expression of V B minus V of A divided by B minus A. So now we need to focus our attention on continuity. So as we should recall a note, our function for VFT. Is going to be once again is equal to 60T divided by T + 3 will be defined, so let's make a note of this. I'll be defined at T is greater than or equal to 0, and this denominator of T + 3 never becomes 0, or T is greater than or equal to 0, which will ensure that VFT will always be valid. Thus, the given function is continuous on square bracket 0.102 bracket. So, let's make a note of this. So it will be continuous. For these conditions. Awesome. So with that said, our next step now is we need to figure out the differentiability. So we need to find the derivative of the function of VFT by recalling and using the quotient rule. So when we use the quotient rule to find the first derivative of VAT. You will find that V T, where this prime symbol means where that's the first derivative of VAT. So V of T is equal to, and once again we're recalling and using the quotient rule, it's going to be equal to T + 3 multiplied by 60 minus 60T multiplied by 1, all divided by T + 3 to the power of 2. Which when we simplify this expression, it will be equal to in its most simple form, 180 divided by T + 3 to the power of 2. So once again, I, I skipped a few steps, but ultimately what we're gonna do is we're gonna take our expression for V T, and we're going to write it in its most simple form, and when we do, we will get 180 divided by T + 3 squared. So the denominator, as we should note, T + 3 squad will never become 0 at T is greater than or equal to 0, which ensures that this value of V of T, this function of V T to be precise. So V T will always be valid since this V of T will be defined for T is greater than or equal to 0. So the function will be differentiable on. This interval. Awesome. So I'll put a blue check mark to to indicate to remind us that it is differentiable within this interval. So since the conditions are met, we can apply and use the mean value theorem. So that will mean our next step is we need to find V of 0 and V of 10. So we need to note that V of 0 is equal to 60 multiplied by 0, divided by 0 + 3, because we're plugging in a value of 0 in for T, which will mean that this is equal to 0, and we need to note that V of 10. It is going to be equal to 60 multiplied by 10 divided by 10 + 3, which will mean it'll be equal to 600 divided by 13, and it's units of measurement are gonna be meters per second, which we can make sure we, even though it's 0, you can still say it's 0 m per second, because that is the units of measurement. So now, our next step, and once again just to reiterate, we just plugged in a value for 0 in for T and a value of 10 in for T into our initial expression for VFT. Fantastic. So now, our next step is we need to calculate the average rate of change of VAT over our specific interval of square brackets, 0.10. So the average rate of change, as we should recall, states that V of C is equal to 600 divided by 13 minus 0, divided by 10 minus 0, which will be all equal to when we simplify, 60 divided by 13 and it's units of measurement now are gonna be meters per second squared. So now, our next step is we need to set the instantaneous rate of change of T. Equal to the average rate of change in solve for T. So we determined that V C is equal to 60 divided by 13 m per second squared. So we need to say that V of T is equal to V of C. So that will mean that we can easily go ahead and write that 180. Divided by T + 3 to the power of 2 is equal to 60 divided by 13. So now, what we need to do is we need to take this expression and we need to rearrange it to isolate and solve for T all by itself. And when we do that, we will find that T is equal to + or minus the square root of 39 minus 3. Fantastic. So now, Our next step is we need to consider the positive value for tea. So when T. In this case, so we're considering the positive value of T, so that will be the condition where T is equal to square root of 39 minus 3, and don't forget that it's units are going to be seconds because time is in seconds, and we need to verify that T is within our interval of square bracket 0.10. So when we do that, we will find that T is equal to when we plug in the value of the square root of 39 minus 3 into our calculator, we will find that T. is approximately equal to 3.24 seconds, and once again, that's when we take the square root of 39 minus 3 and plug it into our calculator. And when we do that, we will find that the square root of 39. -3 is within our interval of square bracket 0.10, so therefore our solution will be valid. So that will mean, without a doubt the time T when the instantaneous rate of change of the. is equal to the average rate of change will be our final answer has to be without a doubt. T is equal to the square root of 39 minus 3 seconds. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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?

Key Concepts

Mean Value Theorem

Instantaneous Rate of Change

Average Rate of Change

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