Functions of Several Variables, Temperature?

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CalleighMay

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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 ;)
 
CalleighMay said:
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 ;)
Click to expand...

Review your text about gradient (upside-down triangle) and tell us how you can use that here.
 
Thanks for the reply :)

I looked in the section for similar problems, but i canot even see any examples that relate to this one. I think my problem is i don't know what it's even asking for. The whole temperature thing is confusing me =/
 
Someone suggested a different method and i gave it a shot, could someone tell me if this is right?

-400xe^-((x^2+y)/2), p=-1200e^-7
400(-.5)e^-((x^2+y)/2), p=200e^-7
-200e^-7 (6,1)

Now, i have no idea what this means, i just followed the example and did the same thing. Can anyone understand this and tell me if its right? lol Thanks ;)
 
You are asked to find where the temperature does not change.

Are you familiar with the concept of the derivative of a function giving the slope at the points of the function? Are you familiar with the value of the slope of an horizontal line, and thus the slope of a one-variable function where the function is "horizontal at a point" (generally: where the function has a local maximum or minimum)?

This question is asking a two-variable-function analog to this.

Note: This all should have been covered extensively in class before this homework was assigned. Unfortunately, we cannot here provide the missing hours of classroom instruction, so please specify if you first need lesson links, so you can learn this material well enough to understand the help that tutors are trying here to provide to you.

Thank you! :D

Eliz.
 
Yeah i know that the derivative is also known as the slope, and the slop of a horiz line is 0 since the slope doesn't change (rise/run=0/infinity which is 0).

Oh this isn't a homework lol he just gave us some problems to look at so we know what it will be like next semester, but i REALLY want to understand these to impress him! All of my friends are confused so i want to stand out :)
 
CalleighMay said:
he just gave us some problems to look at so we know what it will be like next semester, but i REALLY want to understand these to impress him! All of my friends are confused...
Click to expand...
If this hasn't yet been covered in class, then it is only reasonable that everybody would be confused: you're all missing weeks of background material! :shock:

If you'd like to try to learn this material on your own, that's great! But unfortunately we really cannot teach courses within this environment. (The volunteers can help students with specific exercises, but this assistance requires that the student have at least a basic grasp of the underlying material, something which is obviously missing in this case. We cannot, sorry to say, provide that foundational underpinning; that's what your next calculus course is going to be for.)

For the lessons you seek, try here:

. . . . .Paul's Online Notes

. . . . .The Calculus Page

Have fun! :D

Eliz.
 
part b:

\(\displaystyle {\nabla}T(x,y)=400e^{\frac{-(x^{2}+y)}{2}}\left[(-x)i-\frac{1}{2}j\right]\)

\(\displaystyle {\nabla}T(3,5)=400e^{-7\left[-3i-\frac{1}{2}j\right]}\)

There will not be change in directions perp. to the gradient: \(\displaystyle \pm (i-6j)\)

part c: The largest increase will be in the direction of the gradient \(\displaystyle -3i-\frac{1}{2}j\)
 
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