Problem 54 Hard Difficulty

The number of bacteria after $ t $ hours in a controlled laboratory experiment is $ n = f(t) $.

(a) What is the meaning of the derivative $ f'(5) $? What are its units?

(b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, $ f'(5) $ or $ f'(10) $? If the supply of the nutrients is limited, would that affect your conclusion? Explain.

Answer

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03:26

Daniel J.

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Video Transcript

we have we have a function F F. T. Uh Where f. Is the number of bacteria and t. Is the number of hours as the number of hours go by? Uh bacteria could be increasing. Uh If uh they have a good supply of nutrients. However if their supply of nutrients that they feed on is limited. Uh then the number of bacteria could actually be uh decreasing as time goes on. So whether bacteria increase or decrease as time goes on as the hours uh increase depends on the nutrients good nutrients, good supply nutrients uh dan F. Is going to be increasing um insufficient nutrients limited supply nutrients not enough to feed the bacteria. Uh Then the number of bacteria are going to be decreasing now if a function is increasing that means it's first derivative is going to be positive. If a function is decreasing uh that means it's first derivative is going to be negative. Now what do we mean by F. Prime of five? Well F prime of five. Okay. The derivative of F. With respect to T. Is the rate at which the number of bacteria is changing with respect to a change in time. With respect to a change in the number of hours. So how does the first derivative F. Prime the derivative of F. With respect to T. Means what is the rate in the change in the number of bacteria with respect to a change in the number of hours? So f prime of five means the rate at which the number of bacteria is changing with respect to a change in time at five hours. So I'll type that out for you. So F. Prime of five. The derivative of F. With respect to t. When T. Is five uh Can be interpreted as the rate at which the number of bacteria is changing as the time changes when five hours have elapsed. So at the five hour mark how is the rate of which number of bacteria is changing with respect to time changing? Now if there is an unlimited supply of nutrients meeting bacteria are going to do very well. Uh they're going to keep growing and growing and growing. Uh So which do we think would be larger? Um The rate at which uh the bacteria are increasing At the five hour mark. Or the rate at which the bacteria are increasing at the 10 hour mark. If there is an unlimited supply nutrients which one of these values would be greater. The rate at which bacteria are increasing after five hours. Or the rate at which the bacteria, the number of bacteria is increasing after 10 hours. Well if there is an unlimited supply of nutrients to bacteria are going to do well and they're gonna multiply and multiply and multiply. So at the 10 hour mark they're going to be growing a lot more and a lot faster than they did at the five hour mark. So if there is an unlimited supply of nutrients we would expect the rate at which the bacteria are growing at the 10 hour mark to be more than to five hour marks. At a rate at the five hour mark Would be less than the rate at which the bacteria are growing at the 10 hour mark. Now if the supply of nutrients was limited we would change this sign um to be a greater than sign. If the supply of nutrients was limited, bacteria are not going to do well okay and slow and so they're not going to grow quickly. So the rate at which they would be growing at five hours Would be much more than the rate that they are growing at 10 hours. If the supply nutrients was limited because they'll do well in the beginning but because the nutrient supply is limited uh they're not going to do well as the hours go by. But right now because there is an unlimited supply. Uh the rate at which they grow at 10 hours is going to be more than the rate at which uh the bacteria are growing at five hours.