Question

I conducted an experiment that measured the temperature of a cup of coffee over a 10 minute period of time and recorded the results. I was able to calculate the average rate of change in temperature, but I cannot figure out how to calculate the instantaneous rate of change at a given time during the experiment. Could someone please help me out? I can give you some sample data if you need it.

Answer 1

http://i.imgur.com/gBnxj.png

Answer 2

I tried plotting it in Excel and applying a number of different trendlines to get different equations. But none of the trend lines seemed very accurate.. And I don't really know what to do with the equation after getting one. Do I just derive it?

Answer 3

But if a linear equation is used, won't that mean the instantaneous rate of change will be the same for any point in time?

According to the questions I'm following, the instantaneous rate of change should vary over time.

Also, how you you use that equation to calculate the instantaneous rate of change?

Answer 4

That does look quite a bit better.. So, for example, to find the instantaneous rate of change at t=2, would it look like this:

0.02578671x^3 - 0.620922x^2 + 3.40938x - 1

0.02578671(2)^3 - 0.620922(2)^2 + 3.40938(2) - 1

= 3.54136568 degrees / minute

Answer 5

Is that how you would do it?

Answer 6

I think I screwed up the deriving, Wolfram says it's this: http://www.wolframalpha.com/input/?i=derive+f%28x%29+%3D+0.00859557x4%E2%88%920.206974x3%2B1.70469x2%E2%88%926.70212x%2B69.6503

Answer 7

Either way, I thought I was going to have to use the limit as x approaches zero thing.. I guess not?

Answer 8

My first question would be: What level is this class? Is it something like calc 1? Intro to stats? Or is it actually an advanced class in stats or similar?

Answer 9

The approach I would take varies depending on the level of class.

Answer 10

11th Grade Level 2 Mathematics, we're just starting calculus.

Answer 11

In that case, I would take the simplest approach possible. For example, to find an approximation of the instantaneous change at minute 2, Find the slope between minute 1 and 2, and the slope between minute 2 and 3. Now, just average the slopes. What you'll get is an approximation of the instantaneous change.

Answer 12

If this was advanced stats/linear algebra, I would go with making a model up to some accuracy, and then using that.

Answer 13

I'm not sure exactly what method we're supposed to use. But we have been working with limits and differentiation etc. So maybe that first approach may be a little too simple.

Answer 14

And the way I see it, they're introducing you to what a derivative represents. And that's what this exercise is supposed to do.

In fact, just a couple hours ago, I was doing some tutoring, and a very similar problem came up on the online hw for the calc 1 class. The way it was supposed to be solved (it gave the correct answer) was by the method I described. It's a rough approximation, but it get's the idea of what a derivative means graphically.

Answer 15

However, in most cases, the method I described will do better than making a linear approximation. I can't guarantee anything beyond linear, but it isn't a half-bad estimate as long as the function isn't jumping all over the place.

Answer 16

Okay I might have to use that method. But would deriving a quartic be another appropriate way?

Answer 17

You could do that. And at a quartic, you might very well have a more accurate estimate of what this actually might be. However, finding that quartic is difficult without lin. algebra or computing power.

This method is simple, decent, and a good introduction to the idea of a derivative. Hence, I would assume that this is how they want you to do it.

Answer 18

To rephrase, there's nothing wrong with doing a quartic, except that it adds needless complexity to the problem.

Answer 19

Ok thanks. This experiment also involved other experiments that gave different results when graphed. Some were almost perfectly linear, some were more like this. Would I need to use different methods for the different experiments?

Answer 20

I wouldn't, unless they ask you to.

Answer 21

Sounds good. I might keep this thread open for the rest of the day until I finish the assignment, in case I have any more questions.

Thanks for the help guys :)

Answer 22

You're welcome.