Pick a specific point P 0 = ( a, b, f(a, b) ) on the surface z = f(x, y) , other

Question

Pick a specific point P0 = (a, b, f(a, b)) on the surface z = f(x, y), other than (0, 0, 0) or (3, 21, -3402).

Note: You may use whimsical values for a and b, such as the month and day of your birthday. For example, March 21st becomes a=3 and b=21, so the 3rd coordinate is f(3, 21) = -3402.

1. Calculate the directional derivative of the vector in the direction of greatest increase of the surface at P0. Use algebra and/or calculus techniques and justify all work.

2. Find a direction vector in which the directional derivative of f(x, y) at P0 is equal to zero. Use algebra and/or calculus techniques and justify all work.

Explanation / Answer

f(x, y) = x3y – x2y2

x=3, y =21

=>z=f(3,21)=33*21 – 32*212 =-3402

let a =1 , b=1

z=f(1, 1) = 131 – 1212

=>z =0

point on surface is (1,1,0)

1)

f(x, y) = x3y – x2y2

gradient,f= 3x2y – 2xy2 ,x3 – 2x2y

at point (x,y)=(1,1)

f= 3*12*1 – 2*1*12 ,13 – 2*12*1

f= 1, -1

directional derivative of the vector in the direction of greatest increase of the surface at (1,1,0) =|f|

directional derivative =[12+(-1)2]

directional derivative =2

2)

let the  direction vector in which the directional derivative of f(x, y) at P0 is equal to zero is c,d

f. c,d =0

=> 1,-1. c,d=0

=>(1*c)+(-1*d)=0

=>c-d =0

=>c = d

we can choose any value of c. let c =2

then

a direction vector in which the directional derivative of f(x, y) at P0 is equal to zero is 2,2

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