The temperature in degrees Celsius on the surface of a flat metal plate is given

Question

The temperature in degrees Celsius on the surface of a flat metal plate is given T(x, y) = 4xy + xsin(x) + cos(y). a.?) What is the rate of change of temperature with respect to distance (measured in feet) if we start moving from (2, 3) towards the positive y-axis moving parallel to the x-axis? b.) 1f we are dropped at the point (2,2) on the plate, and assuming we can only move in four directions, parallel to they-axis or parallel to the x-axis, describe the direction we would move to obtain a cooler position on the plate quicker. Justify your answer using partial derivatives. c.) If we are dropped at the point (3, 2) on the plate, and assuming we can move in any direction, what is the direction of motion that will lead to the hottest temperatures the quickest, and what is the direction of motion that will lead to the coolest temperatures the quickest? d.) If we are dropped at the point (2,2) on the plate, and assuming we can move in any direction, what is the rate of change in the temperature on the late if we move in the direction parallel to the line y: 3/4x.

Explanation / Answer

a)y is constant and xi changing if particle moved parallel to x_axis y=3

then,rate of change of temperature is:T ' (x,y)=4y+sinx+xcosx=12+sinx+xcosx

b)D = fxx(a,b) fyy(a,b) - fxy2(a,b)

fxx(x,y)=cosx+cosx-xsinx

fyy(x,y)=-cosy, fxy(x,y)=4

D at (2,2,)

D=-16-cosy(2cosx-xsinx)<0

hence (2,2) is a saddle point

c)at(3,2)

D=-16-cos2(2cos3-3cos2)<0

(3,2) is a saddle point

d)gradient of F:

<4y+sinx+xcosx,4x-siny> at y=3/4x

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