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The graph shows the influence of the temperature $ T $ on the maximum sustainable swimming speed $ S $ of Coho salmon.
(a) What is the meaning of the derivative $ S'(T) $? What are its units?
(b) Estimate the values of $ S'(15) $ and $ S'(25) $ and interpret them.
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Calculus 1 / AB
Limits and Derivatives
Derivatives and Rates of Change
David Base G.
October 23, 2020
Daniel, thanks this was super helpful.
Calculus: Early Transcendentalshas kept me up at night until I found this
Missouri State University
University of Michigan - Ann Arbor
Idaho State University
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 graph shows the influe…
Swimming speed of salmon T…
Tne graph shows the influe…
The quantity of oxygen tha…
Figure 1.37 shows the leng…
Fish length Assume the len…
All right. So today we're gonna look at E coli e coli doubles every 20 minutes in this example. That's fast. We're gonna work with ours. So um that's actually one third of an hour, 60 minutes. Right? Divided by 20 gives me one third. Okay. We know that we start off with 50 sales of E. coli and in general for exponential growth. We have this form um P where P is going to be our our number of sales of E. Coli and PFT is equal to P. S. Zero E. To the Katie. um we have already information that we started 50 so I can go ahead and plug that in but I still don't know. Okay. So I need to solve for K. So I'm going to use this info that E. Coli doubles every one third hour to solve for K. All right. So um so if we double that means we're at 100 cells and Um that is when T. is 1/3. So we want to solve for K. So let's go ahead and divide both sides by 50 and wolves. You know we're good. Okay. And then we'll do Ln of both sides. So Elena too. Ln defeated that one third. K. Is just one third. K. Because in verse composite functions unravel each other mathematically. And we just get our explain it out and then I can cross multiply or just multiply both sides by three K. is in three. Elena to going to switch sides. And if I want I can bring that explain and do a little log roll that exponent. Or that multiplier of three can become an exponent. So Kay is actually natural log of eight. Alright so that's gonna be helpful anyway that is our K. Value. And so that's part A. Okay so part B. Now we are going to um Find our general formula. So pft now that we know okay our general formula is 50 E. to the Ln of eight times T. And I can manipulate this. Um I have a team multiplier. So if I rewrite it as tea Ln of eight that tea can become an expert on it. So I get 50 E. To the L. N. Eight to the T. But again in verse composite functions out. Pops the argument. So Pft Can be rewritten as 50 times 8 to the T. And both of these are fine ways to write it. Just two different ways to write our formula. Great. Okay let's do part C part C. Now we are interested to know the population of E. Coli after six hours. So we're going to plug in 50 Times 8 to the six. And this is in units of cells. And when we plug that in we get a really huge number Of 13,107,200 sales. So that is really rapid growth. No thank you on the equal line. Okay so next we are going to find the derivative of our population with time. So um we could use either form, I'm gonna go ahead and just use the second form here. So 50 goes along for the ride when I take a derivative of an expert on it uh function. Uh that our exponential function, it looks like itself times the natural log of the base. Um So that is our our rate of change uh sales of E. Coli with time. If I want to know what it is at six hours I'm gonna go ahead and plug in six for tea. This by the way is in sales per hour. And we end up if we plug that in our calculator once again it sells per hour. Um And we get hold let me just fix this per hour. Whoops! Okay so if we plug that in our calculator we get this huge amount. Really big rate of growth. Really crazy big. Okay so that is already growth in sales per hour. So really really big growth. Okay so last part now we are going to let's change colors. Let's do part E. Um Now we are going to figure out how long it takes to get to a million cells. So we're going to set our function which is uh whoops, let's fix that. Our pft which is 50 E. To the L. N. A. Oh actually I'll use the second one. Never mind. We have two forms. Let's use the easier one. Okay so 50 times eight to the T. We need to make that equal to one million cells. And we're basically just going to solve for T. So we're going to divide through by 50 And that will give us 20,000 cells. Then we are going to um we want to do the inverse operation. So that is If I do log base eight um of both sides then I will get t to pop out. So T. Is um Log Base eight of 20,000. Which you can do. You use another log roll and do LNA 20,000 Over Ln of eight. Very useful log roll. And when we plugged that in our calculator we get 4.76 hours to reach a million cells. Okay. Hopefully you enjoyed that. Have a wonderful day.
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