45 The temperature at a point (x, y) on a flat metal plate...

45 The temperature at a point \((x, y)\) on a flat metal plate is given by \(T(x, y) = \frac{45}{13 + x^2 + y^2}\), where \(T\) is measured in \(^\circ C\) and \(x, y\) in meters. Find the rate of change of temperature (in \(^\circ C/m\)) with respect to distance at the point \((2, 1)\) in the x-direction and the y-direction. (a) the x-direction \(\text{ }^\circ C/m\)

Transcript

00:01 Hi there, so for this problem we are given the temperature function that depends on x and y.

00:07 So the temperature that depends on x and is equal to 1 plus x squared plus y square.

00:29 So the question for this is to find the rate of change of this temperature with respect to the distance at the point.

00:39 So the point that we are given is 3 and 1.

00:42 So the first value corresponds to the value of x and the second value corresponds to the value of y.

00:49 So the rate of this, let's call this eta x, the rate in the x disruption, which is the question for part a of this problem.

00:58 It's just the partial derivative of the temperature with respect to x and then evaluated that at the point 3 .1.

01:09 And something similar for the rate of change in the y direction.

01:14 That will be the partial derivative of the temperature with respect to why this evaluated at 3 .1.

01:22 Now, with that set, let's start by calculating the partial derivative with respect to x of the expression that we are given.

01:33 Now, first of all, we know that when we have the inverse of a function, that is going to be minus 1 divided by the square of that.

01:44 So in this case, that will be minus 50 divided by 1 plus x squared plus y square.

01:54 Then we need to multiply this by the internal derivative, which is the derivative of 1 plus x squared plus y square.

02:04 Now, since we are derivative partially with respect to x, we will have that that is just simple.

02:10 2 times x in here.

02:13 So we can simplify this order and we will find that this is minus 100 times x this divided by 1 plus x square, oh sorry if we got the square in here plus y square and all of this to the square.

02:32 And then we just need to evaluate this at the values that we are given so that will be minus 100 times 3, this divided by 1 plus 3 to the square, plus 1 to the square, and all of that to the square.

03:00 So using our calculator, we obtain a value of.

03:16 So the value that we obtained from this, i'm going to give you this with a three at decimal places, so that will be minus 2 .479...

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