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The table gives the number $ N(t) $, measured in thousands, of minimally invasive cosmetic surgery procedures performed in the United States for various years $ t $.

(a) What is the meaning of $ N'(t) $? What are its units?

(b) Construct a table of estimated values for $ N'(t) $.

(c) Graph $ N $ and $ N' $.

(d) How would it be possible to get more accurate values for $ N'(t) $?

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a) Doneb) table has been constructed.c) $\mathrm{N}(\mathrm{t}), N^{\prime}(\mathrm{t})$ plottedd) If we further reduce the time interval, we will get more accurate $N^{\prime}(t)$

04:09

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Cara C.

September 16, 2019

Can you show an example equation for how you filled in the table? What you're saying doesn't make any sense.

Missouri State University

Harvey Mudd College

University of Nottingham

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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in this problem. We're given this table of data of T and N. F. T. Where N. A. T. Is measured in thousands. The minimally invasive cosmetic surgery procedures performed in the United States on the years from 2000 and 2012. And the first thing we're asked is what is the meaning of in prime of T. So in prime of T. Is the rate at which minimally invasive cosmetic surgery procedures. Amazing. Nurture three procedures performed in the US is changing right? It's the rate of change of these over time or with respect to time. Okay. And the units on this of course is going to be in 1000 surgeries her here. This is gonna be the changing in over the change in T. Okay. They were asked to construct a table of in prime of T. So we're going to have to calculate these 1234567 entries. Okay so to start off with For 2000 we only have one interval here to do the rate of change over. Does that rate of change is going to be The in prime. So in prime of 2000 Will be in of 2002- in of 2000 Over 2002 -2000. Which is going to be 48 97 -55 10 from the table divided by two. And that's going to be a negative 30 6.5. So there's our first entry -306.5. Now to do for 2002 we're going to do down the table and up the table and take the average of these. Okay, so we already have down the table here. So now we need is up the table I'll do it like that and that's in of 2004 Minour end of 2002 over 2004 -2002. Which will be 70 500 48 97. Mhm mm Which is which is 13 oh 1 1/2. And so then In prime at 2002 it's gonna be the average of these two entries -306.5 plus 1301.5. And so we're going to get 4 97.5 positive number there. Okay so we go 4 97.5. Okay, we do the same thing For 2004. We already have going down the table, we just need up the table now. So I have going down the table right here, I just need in prime at 2004 the average rate of change in 2004 going up the table which is in of 2006- in of 2004 over 2006 2004, which is 91 58 minus 7500 over to Okay, which is 16 5 58 divided about two. So this is 8 29. And so then in prime at 2004 Is going to be the average of these two. So that's 13:01.5 Plus 8 29 is 1065.25 1065.25 Or a 3rd entry. Then for 2006. No, down the table and up the table, we already have down the table done accomplished it right here. So in prime at 2006 going up the table Is in of 2008- in of 2006 over 2,800s 2006 years on average rate of change equation, This is over too. And so in of 1008. Well that's 10 917 -91 58. So that is 1759 divided by two. So this is 8 79.5. Well then doing the average of these two, that's one half times eight, plus 879 and a half is me 8 54.25. Look back up here to my table. 8 54.25. Okay, Again, for 2008. Go down the table and up the table and average the two and we already have going down the table right here. And so to figure out the other one, This is n of 2010 -10. 2008 over 2010 -2008. Okay. And of 2010 Will be 11581 off of our table minus 10917 divided by two because 2010 months, 2000 and just to so that is 6 64 divided by two is 332. So then average these two together. I have one half times 8 79.5 Plus 3 32 gives me 605.75. Some back up to the table 60575. Okay, no, 2010 again. We'll go down the table and up the table. On average, the two we already have down the table done right here, we just started to go up the table. So in of 2010 going up the table is in of 2012 Minour end of 2010, divided by two. Right? Because we did 2012 minus 2010, This can be divided by two. So in of 2012 is 13 0 35 miles 11. 81. And so that is 7 27. So then in prime at 2010 we average these two together. 332-plus 7, 27 Is 5, 29 and a half. So I have 5:29.5. And then for 2012 there's only one direction to go from 2010 down to 2012 while we already have that one right here, don't we? At 7 27. So this is 727. Now we're asked to graph this data wrapped asta graph in and in prime. So on our on our graph graphing here and dez most right, we can pull up the table and We can start entering 2000 To 2004. 2006. 2010. Mi 12 12. Okay. And then in the first one, just a graph in We have 55 10. We have 48 97. Yeah, 7500 Have 91. 58. May have 10917. And then we have 11 5 81. Then we have 13. Okay. So on our graph, right, we're gonna get this data to show up out here me to my ex from 2000 to 2012. And we'll have the y Go from 4500. Yeah, 18,000. And there's our data out there right there for us. Okay. Now for in prime I believe yours in the book has a lot has lines drawn through there. But you can see pretty quickly which graph it is out there from that data. That's all the data points. All right, so then to do in prime, let's see do this to another one. Another table again Acceptable up 2000 two for six. Okay. Yeah, 2012 then or are our data off of our table? Remember we had -306.5. So that's negative 306.5. Then we had 497 and a half or 97.5 1065 65 points 25. And then before point five And then 605.75. And then 5 29.5. Mhm. Okay. And the end we had 7 27. 7 27. Okay. And so this data shows up down here as the black points on our graph right here. Okay. So that gives you the graph for the derivative event graph of in prime. All right. And then the last question says, how would it be possible to obtain more accurate values of in prime? Well, the way to gain more accurate values. More accurate values of in prime 50 right would be to obtain more values of t between 2000 and 2012. Because we did by doing that we could we could calculate the average rate of change over smaller values. So we would have more accuracy in our in our derivative in our in private

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