2) The temperature distribution in a room is given by T (x, y, z) = 75 + cos(xyz

Question

2) The temperature distribution in a room is given by

T (x, y, z) = 75 + cos(xyz),

where T is temperature in degrees Fahrenheit and x, y, and z are measured in feet.

a) If an insect is hovering at the point (1,2,3) and begins to fly in the

direction of the vector < - 6 , 2, 3 > at what rate is the temperature changing at

that moment? Give the exact answer and an appropriate decimal estimate.

b) In what direction is the rate of change of temperature the greatest?

c) What is the value of this maximum rate of change of temperature? Give the exact answer and an appropriate decimal estimate.

Explanation / Answer

initial point (1,2,3) and begins to fly in the

direction of the vector < - 6 , 2, 3 >

so, direction vector is = <-7, 0, 0>

(a)
T (x, y, z) = 75 + cos(xyz),

T -7, 0, 0) = 75 + cos(0) = 75


(b)
T (x, y, z) = 75 + cos(xyz),

dT/dx = -sin(xyz) (yz)

dT/dy = -sin(xyz) (xz)

dT/dz = -sin(xyz) (xy)

direction is the rate of change of temperature the greatest = 75

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