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Researchers measured the average blood alcohol concentration $ C(t) $ of eight men starting one hour after consumption of 30 mL of ethanol (corresponding to two alcoholic drinks).

(a) Find the average rate of change of $ C $ with respect to $ t $ over each time interval: (i) $ [1.0, 2.0] $ (ii) $ [1.5, 2.0] $ (iii) $ [2.0, 2.5] $ (iv) $ [2.0, 3.0] $ In each case, include the units.

(b) Estimate the instantaneous rate of change at $ t = 2 $ and interpret your result. What are the units?

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a) (i) $-0.15 \mathrm{mg} / \mathrm{mL} \cdot \mathrm{hr}$ $\\$(ii) $-0.12 \mathrm{mg} / \mathrm{mL} \cdot \mathrm{hr},$$\\$(iii) $-0.12 \mathrm{mg} / \mathrm{mL} \cdot \mathrm{hr}$$\\$(iv) $-0.11 \mathrm{mg} / \mathrm{mL} \cdot \mathrm{hr}$$\\$b) -0.12

06:09

Daniel Jaimes

01:12

Carson Merrill

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Raisa B.

September 17, 2019

how did u determined the table

University of Michigan - Ann Arbor

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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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Researchers measured the a…

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The average blood alcohol …

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So in this problem we are given that researchers measured blood alcohol concentration of eight men Uh starting one hour after consumption of a couple of drinks. And we're giving a table rooty isn't ours. One 1.5, 2.2.5 and 3.0 and the concentration C of T in milligrams per millimeter 0.31 0.24, 0.18, 0.12 And 0.08. First of all, we're asked to find the average change of C with respect to T over each time interval. So going from 1.02, we have 0.18 -0.3, 1 Over 2 -1 Is a -0.18. This is milligrams per millimeter and time is in ours. Okay. And from 1.5 two, We have 0.18 -0.24 over 2 -1.5. So that a negative 0.06 over 0.5. So that's a negative 0.12 again, milligrams per mil later our Okay, The next one from 2.0 To 2.5. Well, from our table This will be 0.1, two -0.18. We're just using our average rate of change equation here between these two timeframes, keeping everything coordinated minus two. and so I have a negative 0.06 over 0.5, Which again is a negative 0.12 milligrams per million. Later our Okay. And the last one here, we're going from 2.0 23 oh. And so that's 0.08 minus 0.18 over 3.0 -2. And so that's a negative 0.10 over one. That's a negative 0.10 milligrams per million later. Power. All right. In part b says to estimate the instantaneous rate of change, estimate the instantaneous right of change. Ah T equals two. Okay, so then at T equals two. What we can do is come up here and think about what happens as we approach two from each direction left and right. Well from 1.5 to 2 we did right here A- -12. And from right here we did it from the right And got a negative 0.12. So we have the same rate coming in from both sides. So normally we would average these two. But since they're the same, that means we're going to have the same instantaneous rate as we have a constant rate coming in from both sides left and right.

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