The temperature at any point on a metal plate is given by the function . T(x,y)=

Question

The temperature at any point on a metal plate is given by the function . T(x,y)=sqrt(x^2+y^2-1) a) (4 pts) Find the domain of and state using set notation, . b) (4 pts) Make a sketch of the domain. c) (4 pts) Give the equations of the level curves/contours corresponding to T=0,1,2,3 Sketch the level curves on one set of axis and clearly label each curve. Hint: square both sides of the equation for each level curve and then determine the type of resulting curve and sketch that curve. d) (4 pts) Suppose you are at the point and you can move either parallel with the x-axis or the y-axis. In which direction will you experience the greatest rate of change in T and what is that rate of change?

Explanation / Answer

a.) domain points= (x^2+y^2>=1)

c.) T=0

x^2+y^2=1

T=1

x^2+y^2=2

T=2

x^2+y^2=5

T=3

x^2+y^2=10

d.) when parallel to xor y only y or x will be changing

so T'(xory)= 1/2(sqrt(x^2+y^2-1)*(2x)= x/sqrt(x^2+y^2-1)

so for max T rate change T''(xor y)= 0

T"(xory)=(y^2-1)/(x^2+y^2-1)^(3/2)

so direction is y=1 and rate of change= 1

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