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Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures.

If $ W(x) $ is the weight gain at temperature $ x $, construct a table of estimated values for $ W' $ and sketch its graph. What are the units for $ W'(x) $?

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03:24

Daniel Jaimes

01:19

Carson Merrill

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Oregon State University

University of Nottingham

Boston College

Lectures

04:40

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.

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Water temperature affects …

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The graph shows the influe…

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Changing Water Levels The …

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01:39

The graph shows the depth …

So in this problem, we're given this table of data for the weight gained by brook trout as a As a function of the temperature and degree c. And these are on 24 hour days. We're asked to find W prime for each of these. Make a table of it and draw the graph of it. And we're also asked what the units on W prime our hair. Well, since W prime is going to be the difference in the weight gain divided by the difference in the temperature, this is going to be in grams for degrees C. Because weight gain is in grams, the temperature is in degrees C. Okay, so we begin Trying to figure out w prime of 15.5. Okay, so that's going to be W of 17.7 -W of 15.5 over. Oops, this is upside down. Let's start over here. Yeah, That's right -W of 15.5. That's right over 17.7 -15 five. Okay, so this is 31.0 -37.2 Over 2.2, which is Approximately negative 2.8. So since 15.5 is the beginning of our table, we only have one interval to use to figure out The derivative of the average rate of change at 15.5. So we can only use that one interval there to do that with. So we have our first entry here now to do it. W prime for 17.7, we're going to do the average rate of change in both directions Coming into 17.7 and then averaged them. So w prime of 17.7. We'll say going going down the table, It's gonna be w of 17.7 -W. 15.5 over 17.7 -15.5. Which is a negative 2.8. Actually, let's leave this one is a fraction. So this is a negative 6.2 over 2.2. Yeah, or a negative 31/11. We were to simplify it. Then W prime at 17.7. Going up the table, It's going to be w at 20 -W at 17.7, divided by 20 point oh -17.7. Again, we're just using the average rates of change off the table, they're going both directions. So this is 19.8 minus 31 over 2.3. And so this is a negative uh what is that? 11.2 Over 2.3. So that would be a negative 1, 12/23. So then mm the average of these then is one half of -31/11 minus 1. 12/23 Which is approximately a negative 3.8. Now we have our second entry in the table, so negative 3.8. Okay, now We'll do the same thing for 20. Right, we'll do, going down the table and up the table, we already have down the table right From 1772 20. We have it given here. So what we need is w. prime at 20 Is w at 22.4 w at 20 over 22.4 -20. And so this is 9.7 -19.8 over 2.4. Which 9.7 months 19 points. So that's a negative uh 10 point 1/2 10.0.4. So that's 10. 1/24 for now to do the averages between these two, We'll have the w of 20 is 1/2 times negative 1 12/23 -1. No 1/24. And that is approximately A- 4.5. So we now have our third entry in the table negative 4.5. Okay, So then of course do 22.4 again, we're gonna go down the table and up the table. Okay? We already have down the table given right here. So let's do the other one from the other direction, which is W. of 24.4 on this w. 24 over. Well let's see is this This is 22.4. Sorry? So hey this is supposed to be 22.4. Sorry about that. Okay, So that's let's see over 22.4 -24. Okay, so then this is C- 28. That's supposed to be This is supposed to be 24.4. There's supposed to be 24.4. Okay So this is -9.8 -9.7 Over two Which is a negative 39 over for Okay now then That means that w prime at 22.4 again is the average of the store there and here here in here so that's a minus one oh 1/24 minus 39/4. Which When you go through the math here you get approximately a negative 7.0 So there's a 3rd and 4th entry -7.0 and the last entry we've already done we did it right here. The negative 39/4 which is Approximately a negative 9.75. Okay so this is negative 9.75. All right, so now we just need to graph these, we have the table and we have the units. We just need to graph these now. So what go into our graphing calculator over here and if I pulled by dez mose and I start putting these in right? So I'll have yes 15.5 Yes 17.7 money. 22.4 And 24.4. And then from for the values we just calculated -2.8 um -3.8 -4.5 minus seven and minus 9.75. Great. So now we have these points all graft here. So let's just adjust our graph a little bit here so we can see them and there they are graft right? Therefore us on the graph. And that gives us all of the points.

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