Growth rate of bamboo Bamboo belongs to the grass family and i...

Question

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

b. Based on the Mean Value Theorem, what can you conclude about the instantaneous growth rate of bamboo measured in millimeters per second between 10:00 A.M. and 3:00 P.M.?

Accepted Answer

Hello there. Today we're gonna 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. A company's stock price increased from $100 to $150 over a 10 day period. According to the mean value theorem, what can be concluded about the stock price change rate during this period? Awesome. So it appears that for this particular problem, we're asked to figure out by using the mean value theorem. So we're trying to use the mean value theorem. We're gonna use that to help us figure out what kind of statement, based off of our multiple choice answers, which of our multiple choice answer statements can be concluded about the stock price change rate during this specific 10 day period. So, with that said, now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is the stock price change rate was always $5 per day. B is the stock price change rate was never $5 per day. C is at some point, the stock price change rate was exactly $5 per day, and D is the stock price change rate was 0 at some point. OK. So first off, in order to help us solve for this problem, let us define all of our known variables. So let us note that our initial stock price, which we'll call F of A, and our initial stock price was equal to $100 and our final stock price, which we'll call F of B and F of B, our final stock price will be equal to $150. And the time period, as we should recall, is B minus A. And B minus A will be equal to 10 days. And now that was our first step. Our first step was to define all of our known variables. Now our second step is we need to state the mean value theorem, which we're going to call MVT for short. So the mean value theorem, as we should recall, states that if a function F is continuous, On a closed interval, where a closed interval means that square bracket A B square bracket, since these square brackets indicate a close interval. So the mean value theorem once again states that if a function F is continuous on the closed interval square bracket AB2 bracket. And differentiable on an open interval, which is parenthesis A B, then there will exist at least one C in parenthesis A, B, such that F C is equal to F of B minus F of A, all divided by B minus A. And now, with this information, our third step is we need to apply the MVT, the mean value theorem, to our given problem. So that means we need to substitute in all of our known variables into our MVT equation. So when we do that, we will find that F of C, where this prime symbol indicates the first derivative. And this will be equal to $150 minus $100. And then we need to divide everything by 10 days. So that means we can simplify this expression and we could say that it's equal to $50 divided by 10 days, which will simplify to F of C. So you take F C, so this function of F a C, the first derivative of FAC. And when we simplify down to our most simple form, you'll find that it's $5 per day will be our final answer. Hooray, we did it. So, in conclusion, according to the mean value theorem, there exists at least 1. C in the interval, 0.10 such that the rate of change of the stock price F is $5 per day. And this corresponds with multiple choice answer letter C, which once again states that at some point, the stock price change rate was exactly $5 per day. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

b. Based on the Mean Value Theorem, what can you conclude about the instantaneous growth rate of bamboo measured in millimeters per second between 10:00 A.M. and 3:00 P.M.?

Key Concepts

Mean Value Theorem

Instantaneous Growth Rate

Average Growth Rate

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