Chain Rule with partial derivatives

In summary, the homework Equations state that the temperature is increasing at a constant rate of 4 units/sec. TheAttempt at a Solution states that the temperature is at its maximum at t=0 and its minimum at t=2pi.
  • #1
widmoybc
8
0

Homework Statement



Let T= g(x,y) be the temperature at the point (x,y) on the ellipse x=2sqrt2 cos(t) and y= sqrt2 sin(t), t is from 0 to 2pi. suppose that partial derivative of T with respect to x is equal to y and partial derivative of T with respect to y is equal to x. Locate the max and min temperatures by examining dT/dt and the derivative of dT/dt.



Homework Equations



chain rule with partial derivatives



The Attempt at a Solution



I got an answer of 4 for dT/dt. this means that the temperature is increasing at a constant rate of 4 units/sec. meaning that the min is at t=0 and the max at t=2pi. do i need to find the area under the dT/dt graph to find the max and min temps? or am i way off?
 
Physics news on Phys.org
  • #2
You haven't shown your work for dT/dt but I'm pretty sure you have a sign mistake. If you think about this problem physically, it represents the temperature in an ellipse shaped wire. The temperature can't be increasing all the way around because when t = 0 and t = 2pi you are at the same point of the wire.
 
  • #3
Ok, this is what i have
dT/dt= ∂T/∂x (dx/dt) + ∂T/∂y (dy/dt)
=y(-2sqrt2 sint(t)) + x(sqrt 2 cos(t))
then plug in y and x from the given and i got...
(-4sint*sint) + (4 cost*cost)
when i graphed that, i got a straight line at 4.
Where did i go wrong?
 
  • #4
Ok nevermind, i just graphed in degrees instead of radians. it's been a long week, obviously :( ok, i think i know what I'm doing now. sorry for the confusion :P
 
  • #5
Your derivative is OK, but you don't get a straight line. Sure, T(t) = 4 at t = 0 and t = 2pi. Did you try any other points? Like t = pi/2? Trig functions don't generally give you straight lines.
 
  • #6
I forgot to graph in radians. i switched modes and got a curve with maximums at t=0, pi, and 2 pi. minimums at pi/2 and 3pi/2. how do i find which is the actual max and min if I'm given just the parametric? do i need the second derivative? for the second derivative i got -8sint*cost + 8cost*sint. but when i try and graph that, i get nothing in my window. I'm so lost.
 
  • #7
Ok, found my problem, it should be -16costsint for the second derivative. do i need the second derivative at all?
 
  • #8
"do i need the second derivative at all? "

I guess it couldn't hurt. What do you know about the extrema of a continuous function on a closed interval?
 
  • #9
Nevermind, i got it all figured out. the zeros of the first derivative give you the maxs and mins and you just plug back in. apparently i can't remember calc 1 anymore. whoops. but thank you for your help!
 

1. What is the chain rule with partial derivatives?

The chain rule with partial derivatives is a mathematical concept used in multivariable calculus to calculate the derivative of a composite function. It states that the partial derivative of a composite function is equal to the product of the partial derivatives of each individual function.

2. Why is the chain rule with partial derivatives important?

The chain rule with partial derivatives is important because it allows us to calculate derivatives of complex functions that are composed of smaller, simpler functions. It is also a fundamental concept in many fields of science and engineering, such as physics, chemistry, and economics.

3. How do you apply the chain rule with partial derivatives?

To apply the chain rule with partial derivatives, you first need to identify the composite function and its individual functions. Then, take the partial derivatives of each individual function with respect to their respective variables. Finally, multiply these partial derivatives together to get the overall partial derivative of the composite function.

4. Can the chain rule with partial derivatives be extended to higher dimensions?

Yes, the chain rule with partial derivatives can be extended to higher dimensions. In fact, it is a fundamental concept in multivariable calculus and is used to calculate derivatives in three or more dimensions.

5. How is the chain rule with partial derivatives related to the chain rule in single-variable calculus?

The chain rule with partial derivatives is an extension of the chain rule in single-variable calculus. While the single-variable chain rule deals with derivatives of a function with one independent variable, the chain rule with partial derivatives deals with derivatives of a function with multiple independent variables.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
543
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
832
  • Calculus and Beyond Homework Help
Replies
28
Views
2K
Replies
9
Views
661
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
961
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Back
Top