Download the App!
Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.
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.
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Official textbook answer
Video by Leon Druch
This textbook answer is only visible when subscribed! Please subscribe to view the answer
Calculus 1 / AB
Limits and Derivatives
Derivatives and Rates of Change
Missouri State University
Harvey Mudd College
University of Michigan - Ann Arbor
University of Nottingham
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
The number of bacteria aft…
Bacteria population The nu…
Suppose that in a certain …
The number, $N$, of bacter…
Bacteria Population The to…
Approximately 10,000 bacte…
The number $N$ of bacteria…
Bacteria Population Assume…
Growth of Bacteria The num…
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.
View More Answers From This Book
Find Another Textbook
draw a pentagon,a hexagon,a heptagon and an octagon what is the sum of the e…
a can finish a job in 12 hours & b in 14 hours. a & b works alternat…
Divide 32 into two parts such that 5 times one part added to 6 times the oth…
HR calls 15 software and 10 marketing professionals for an interview. Three …
If x, 15,25 are in continued proportion, find the value x.
7. In a A ABC, if 3 ZA = 4ZB = 6ZC, calculate the angles.
I divide into 100 with no remainder i am then thousand digit in 4,759,231 wh…
The average height of a family of five is 150 cm. If the heights of 4 family…
the volume of a soap cake is 160CM cube. its length and white are 10 cm and …
The breadth of a rectangular garden is two third of its length.if the perime…