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The temperature at & polnt (X, Y, 2) (5 glven Tx Y, 2) T002 where /Is meusured PC and meter Find tte Fate charon temmprature at the point Pr4, ~clmFanmn (owjrd&...

Question

The temperature at & polnt (X, Y, 2) (5 glven Tx Y, 2) T002 where /Is meusured PC and meter Find tte Fate charon temmprature at the point Pr4, ~clmFanmn (owjrd'Pqinl (5,Gatch durocuon coc" tha tomperoturd Incrcjze (aetost ol P?rclFndInaximum Iin 04 Increasc

The temperature at & polnt (X, Y, 2) (5 glven Tx Y, 2) T002 where /Is meusured PC and meter Find tte Fate charon temmprature at the point Pr4, ~clm Fanmn (owjrd' Pqinl (5, Gatch durocuon coc" tha tomperoturd Incrcjze (aetost ol P? rclFnd Inaximum Iin 04 Increasc



Answers

Temperature Change The temperature at any point $(x, y)$ of a rectangular plate lying in the $x y$ -plane is given by $T=[x \sin (2 y)]^{\circ} \mathrm{C} .$ Find the rate of change of temperature at the point $\left(1, \frac{\pi}{4}\right)$ in the direction making an angle of $\frac{\pi}{6}$ with the positive $x$ -axis.

Okay. Yeah. For x Y and meters the temperature is given as T equals 500 minus 5000.6 X squared minus 1.5 Y squared. Given this information, what we want to determine is the following. At 23 or the 230.23? For X. Y. What is the rate of change of the temperature with respect the distances moved along the plate in the X and Y directions. Rate of change the function T with respect to two variables X and Y. Is alluding to partial derivatives or differentiation of multi varied functions. And this problem we want to use partial derivatives so we have to find them and you single variable differentiation techniques with respect to differentiate variable treating other variables constants. The example here demonstrates how to do this. That's in this problem. We want to solve for the partial derivatives DT DX DT Dy. And then plug into each of the 0.23 So you go ahead and do that G. D. Except by the power rule zero minus 00.6 and two X equals negative 1.2 X. The G d Y is them simply similarly negative three Y. Thus we have D t d x. Of 23 equals negative 2.4. D t d. Y. Up to three equals negative nine.

Todavia going to start problem number 54? Yeah. The function peak was X cubed plus Roy Square Bless X full first you two different shit with Just do it. We will get like 38 to square. Best work next review Full the Yanks kind upwind. Work calm. I was three into one square on the floor, Therefore which is a cordial for bigly SAGES Park Cindy similarly, we need to find the bike for the point x comma but forward. Do you believe it to be Fulbright? The way I had to learn a lot because for into group which is because it so a way that one gonna go is a global paid bigotry Sentient put

For this question we have f of X comity is equal to a tee to the power of negative 1/2 multiplied by E to the power of negative X squared, divided by teeth. So, um, if for part A, we needed to say What did the traces tell us about the way that heat is diffused to the bar and we have to sketch the vertical traces at Time T is equal to 12 and three. So a Time T is equal to three. We have f of X comma, one f of X number two and f of X comma three. These are all equal to this is one to the power of negative 1/2. So this is just e to the power of negative X squared. This is one over the square root of two times E to the power of negative X squared, divided by two one over the square it with three tens e to the power of negative X squared, divided by three. Finally, for part B for C 0.1 point two country and 20.4, it's just flipped. So we have f off zero plus or minus 0.2 common teeth B t to the power of negative 1/2 times eat the power of plus or minus zero point to do but a teak and so on and so forth for 0.4 and 0.6.

Okay, so this program we want Thio find the extreme values for T equals to 25 x square plus 25 y square Inside the region X squared plus y squared minus X one prospects y less so equals to one. Okay, so first we consider the imperial this region So, uh, passionate narrative off key our 50 x in the 50 wide. So the only critical points X equals two. Why put zero, which is inside the region. So at this point, p equals zero. Now we can see the boundary case. So on the boundary we considered a constraint equation geek Also x squared plus y squared plus x y minus one. We use the ground multiplier method Breaking off t equals to land that greater G So we have 50 X equals two Lambda times two x plus y And 50 Why it goes to learn that time to white plastics. Okay, so we use equation one. Subtract equation too. We have 50 x minus y It was two Lambda times X minus one. Okay, so we have two different cases. Case one is X equals toe. Why? So if X equals two why we parted this result into the constraint equation. Then we have three X squared equals toe one X equals two plus minus. Root off 3/3. So we have two points p one in close to 3/3 through two or 3/3 P two equals thio minus root on 3/3 minus root on 3/3 on the for both points for both points. Okay, for both points, P equals two ah, 50 times winter. So it's 50 over three and we have another case kiss to is if X is no equals to what that means, we can cancel all these two terms that gives us Lambda goes to 50. Yes, Ive lambda equals to 50. You plug it back to equation one. So we have 50 x equals toe 100 X plus 50 wide. That means X equals minus one. Then we use this and then the constraint equation. We have X equals two plus minus one. So we have two more points. So p three equals to one hour minus one before equals toe one minus 11 And for both points, the function t close to 50. So our conclusion is Um, yeah. This T courses here is the minimum value, and the T close to 50 is the maximum value. And this one, it could be a set of point set of point because it's not a maximum amendment touch, Yeah.


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