The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So t

The quantity of oxygen that can dissolve in water depends on...

Key Concepts

Units of the Derivative

The units of the derivative are given by the ratio of the units of the function to the units of the independent variable. In this problem, since oxygen solubility is measured in units of concentration (for example, milligrams per liter) and temperature in degrees (typically degrees Celsius), the derivative has units such as milligrams per liter per degree Celsius.

Derivative as Rate of Change

The derivative of a function represents the instantaneous rate of change of the function with respect to its independent variable. In this context, it quantifies how quickly the oxygen solubility in water changes as the water temperature changes, capturing the local slope of the solubility graph at any given temperature.

Interpretation of the Derivative at a Specific Point

Evaluating the derivative at a specific temperature, such as 16°C, provides an estimate of how much the oxygen solubility is expected to change for a very small change in temperature around that point. This value indicates the sensitivity of oxygen solubility to temperature variations at 16°C, which is important for understanding effects like thermal pollution on aquatic environments.

Transcript

00:02 In this problem, we have an amount of oxygen that can dissolve in water, which is dependent on the water's temperature.

00:10 So in this case, if we look at a graph, here we have the solubility of oxygen.

00:15 So i'm just going to call it s and the temperature t.

00:19 Our graph looks something like this, which means that as the temperature is increasing, the amount of oxygen that is able to be dissolved is decreasing.

00:28 So we have an inverse relationship.

00:30 This kind of reminds me of baking.

00:32 I love to bake.

00:34 And when you are dissolving yeast into water, the higher the temperature of the water, the lower amount of yeast is dissolved because it actually kills the yeast.

00:44 So it's kind of a similar relationship.

00:46 So here we're going to let s represent the solubility of the oxygen.

00:50 So let me just write that out.

00:51 So s is going to be the solubility, which just means the amount of oxygen that's able to be dissolved.

01:06 And those units are in milligrams per liter.

01:10 And we know that based off of the graph.

01:13 And then we're going to say that t is going to be the temperature of the water.

01:18 So i'm just going to write temp of water.

01:22 And the temperature of the water is in degrees celsius.

01:26 And again, we get that from the graph.

01:28 So in part a, they want us to figure out what s prime of t is.

01:34 And interpret that.

01:35 So s prime of t, well, that's just the derivative.

01:39 So s prime of, oops, t.

01:43 So that is the derivative.

01:45 And when we think about derivatives, we need to remember that a derivative is a rate of change.

01:49 So anytime that we're talking about derivatives and trying to explain what a derivative is, we should be using the word rate or rate of change.

01:58 Something, they go hand in hand, so something like that.

02:01 So in this case, it would be the rate at which, well, in general, the rate of change is the rate at which one quantity changes with respect to another.

02:12 So here we have two variables, right? we have s and we have t.

02:15 So s is our solubility of oxygen, and t at the temperature, is what it's in respect to.

02:25 So if we wanted to write this out as an actual answer, we can say that s prime of t is the rate.

02:36 At which the solubility of oxygen changes.

02:59 So there we have rate of change.

03:03 Those words are in there already.

03:05 Changes with respect to the temperature of the water.

03:32 And that if we want to say that that's our t, we could say you know, put s up here just so you know we're clear in our answer.

03:41 So s prime of t is the rate at which the solubility of oxygen changes with respect to the temperature of the water.

03:48 And again, those key words that we have rate of change.

03:52 Here we have the rate, oops, of change.

03:56 So we're still using that idea.

03:58 Then we have to figure out the units for this...

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