Functions of Several Variables, Temperature?

[CalleighMay]

Hello, my name is Calleigh and i am new to the forum! I am in Calculus II and have a few questions on some problems. I am using the textbook Calculus 8th edition by Larson, Hostetler and Edwards. Could someone please help me?

 

The problem is on pg 942 in chapter 13.6 in the text, number 76. It reads:

The temperature at point (x,y) on a metal plate is modeled by:

T(x,y)=400e^-((x^2+y)/2) where x>=0 and y>=0.

 

It asks to find the directions of no change in heat on the plate from the point (3,5).

It also asks to find the direction of greatest increase in heat from the point (3,5).

 

 

Does anyone know what this problem is talking about? Usually it helps if i can picture it in my head but i'm lost... My professor suggested drawing a picture, but i haven't the slightest clue even where to begin.

 

Any help would be greatly appreciated! Thanks guyssss

[timo]

The question asks for derivatives of a R²->R function.

 

EDIT: Here's a colored plot of the function. Not sure if that helps you understanding; you'll have to solve the question analytically, anyways.

Edited August 2, 2008 by timo

[CalleighMay]

Thanks for the response Atheist!

 

Do you know what i need to take the derivative of? To be honest i don't even know what T(x,y) means- i've never heard of having more than one variable in the parenthesis. Do i just disregard that?

 

Thanks

 

Would you think i was stupid if i didn't understand those rainbow graphs? =/ I need to solve this mathematically so i guess we should start there... umm where do i begin? lol

[timo]

No I wouldn't. I didn't add an explanation, after all.

 

For the graphs:

The two axes represent possible choices for the two variables x and y. Usually, then you draw a graph for an f(x) type function, you draw possible values of x on the horizontal and the corresponding f(x) on the vertical. Since in the plot I have two variables I have already used up the horizontal and the vertical axis for possible values of the variables. So if you want to draw a picture of the function you must use some other approach, the common ones being

- Drawing a fake 3D plot with the value of the function at the f(x,y) as z-coordinate. Often looks nice but is usually not particularly helpful for any other thing than having drawn a nice picture.

- Fixing one variable to a number and then drawing something like e.g. f(x,5), the values of the function when y is 5, in the conventional way. Perhaps just draw several of those graphs e.g. one for y=1, one for y=2, ... .

- Using colors to indicate the value. Different colors then represent different values. That's pretty bad for reading off exact values but sometimes nice to get a feeling for the function. The (approximate) translation from the colors to the actual numbers is given on the right-hand side. In this case, you can see that the values tend to decrease as the values of x and y increase, for example (do not mistake equal colors for equal values, that might just be because the number of colors is limited!).

 

Anyways:

Don't take the pictures too seriously. They are just supposed to visualize something and not particularly important for anything else. Look around for a calculus tutorial covering functions with more than one variable - either in a book or on the internet, the stuff is so folklore that you can probably even trust the internet here (except if you are a mathematician). THE keyword you are looking for for this question is the gradient but seeing you already opened other threads on related (i.e. f(x,y) stuff) topics you are probably best adviced to read up a few basics somewhere.

[CalleighMay]

Those graphs are certainly colorful though =P

 

Someone suggested a different method and i gave it a shot, could someone tell me if this is right?

 

We now have:

 

T(x,y)= 400e^-((x^2+y)/2) [-xi-(1/2)j]

T(3,5)=400e^(-7[-3i-(1/2)j)]

 

So there will not be change in directions perpendicular to the gradient + or - (i-6j)

 

and the largest increase will be in the direction of the gradient -3i-(1/2)j

 

does this sound right? thanks

Edited August 4, 2008 by CalleighMay
multiple post merged