2,f(x,y,z)is a function, and f(9,5,7)=89, and the gradient f(9,5,7)=-6j+2j+3k a:

Question

2,f(x,y,z)is a function, and f(9,5,7)=89, and the gradient f(9,5,7)=-6j+2j+3k

a: find the maximum rate of change of f(x,y,z)at the point (x,y,z)=(9,5,7)

b: find the unit vector that points in the direction of maxium rate of change of f(x,y,z) at the point (x,y,z)=(9,5,7)

c: fing an equation of the tangent plane to the surface f(x,y,z)=89at the point (9,5,7)

3,Let f(x,y)=x^2(lny)+3x+2y+7, find the directional derivative of this functionf(x,y) at the point (x,y)=(2,1)in the direction of the vector <4,3>

4,There is a rectangular box whose length x, width y and height z are changing at the following rates,

the length x is decreasing at a rate of 2 ft/hour

the width y is increasing at a rate of 1ft/hour

the height z is increasing at a rate of 3ft/hour

find the rate at which the surface area A of the box is changing when x=y=3 ft and z=1 ft

z=f(x,y) is a function, and f(5,-3) -30, f (5,-3)-6, fy(5. 3)=10. Find the equation of the tangent plane z - L(x,y) to the graph of f(x,y) where x - 5 and y = - 3. z=L(x,y) = your result from part (a) to find the linear approximation of f(4.5, -3.1) f(x,y,z) is a function, and f(9, 5, 7) = 89, and the gradient f(9, 5, 7) =-6i + 2j + 3k. Find the maximum rate of change of f(x,y,z) at the point (x,y,z) = (9, 5, 7). Find the unit vector that points in the direction of maximum rate of change of f(x,y,z) at the point (x,y,z) = (9, 5, 7). Let f(x,y) = x2(lny)+3x+2y+7 Find the directional derivative of this function f(x,y) at the point (x, y)=(2,1) in the direction of the vector . There is a rectangular box whose length x. width y and height z are changing at the following rates: The length x is decreasing at a rate of 2 ft/hour, The width y is increasing at a rate of 1 ft/hour. The height z is increasing at a rate of 3 ft/hour. Find the rate at which the surface area A of the box is changing when x = y = 3 ft and z = 1 ft. (Use correct units in stating your answer.)

Explanation / Answer

a) the maximum rate of change of f(x,yx,z) is mangnitude of gradient of f(x,y,z) is sqrt(36+9+4)=7

b) the unit vector is (6/7)i+(2/7)j+(3/7)k

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