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Oxygen solubility The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.)The graph shows how oxygen solubility $S$ varies as a function of the water temperature $T .$(a) What is the meaning of the derivative $S^{\prime}(T) ?$ What are its units?(b) Estimate the value of $S^{\prime}(16)$ and interpret it.

(a) The derivative $S^{\prime}(T)$ measures the change in oxygen solubility per unit change in temperature. Its units are milligrams per liter per degree Celsius.(b) An estimate of $S^{\prime}(T) : \overline{S^{\prime}(T)}=\frac{\delta S}{\delta T}=\frac{8-12}{24-8}=-4 / 16=-0.25$ milligrams per Liter per degree CelsiusThis means that at 16 $\mathrm{Cr}$ , for a degree rise in temperature, the oxygen solubility decreases by 0.25 milligrams per liter.

Calculus 1 / AB

Chapter 3

Derivatives

Section 1

Derivatives and Rates of Change

Harvey Mudd College

Baylor University

Boston College

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

0:00

The quantity of oxygen tha…

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(a) Explain why the concen…

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At sea level and $25^{\cir…

we're given this chart that tells us the maximum amount of oxygen that can dissolve in to the water at a specific temperature. What we want to do is first determine the meeting of the derivative s prime of tea and what its units are. Whoa, let's first look at the humans of it. And then maybe from that we can determine what it beats. All right, so s prime of tea. Well, this is going to be approximately equal to the change in s over. The change in T our units for s are going to be milligrams her leader. Our temperature is in Celsius, so we could rewrite this as milligrams per liter her degree Celsius. So this here is going to be our units. And well, what this is telling us is how the oxygen So your ability small you ability changes as temperature also changes. Right now, they want us to best mate as prime of 16 and interpret that well, that's prime of 16 is he could Well, we already said it's the change of s and the change in time so we could just go ahead and find two points and so maybe we use this point at this point here, and then we could connect those before line like so And depending on how good your eyesight is, I think these were going to be approximately so. At 24 degrees Celsius, it looks like it's about nine. So nine and 24 blindness and the other point looks like it is okay, 12.5 now, depending on where you think it is, you might get something slightly different than me but a cz, long as it's in about the same ballpark, then it will still be a good estimation. But this gives around negative seven over 30 two and the units with this, our milligram, her leader perfectly Celsius. So the way we would interpret this is at 16 degrees Celsius oxygen, so you ability goes down 7/32 milligrams per liter.

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