Need help with d & e 1. Let f(x,y,z) = xy + yz + zx, and let P be the point (1,2

Question

Need help with d & e

1. Let f(x,y,z) = xy + yz + zx, and let P be the point (1,2,3). (a) Find a unit vector in the direction in which the function is increasing the most rapidly at the point P, and the corresponding rate of increase. (b) Find a unit vector in a direction in which the directional derivative of the function at P is 0. (e) Find the directional derivative of f in the direction of b = 2i + 4j-10k at P. (d) Find tile equations of the tangent plane and normal line to the surface f(x, y, z) = 11 at P. (e) Use differentials to estimate f(1.1, 1.8,3.2). Need help with d & e

Explanation / Answer

d)

f(x,y,z) = xy + yz + zx at Point, P = (1 , 2 , 3)

fx = partial derivative of f with x
fx = y + z
fx = 2 + 3
fx = 5

fy = partial derivative of f with y
fy = x + z
fy = 1 + 3
fy = 4

fz = y + x
fz = 2 + 1
fz = 3

The plane is given by :

fx(x - x0) + fy(y - y0) + fz(z - z0) = 0

5(x - 1) + 4(y - 2) + 3(z - 3) = 0

5x - 5 + 4y - 8 + 3z - 9 = 0

5x + 4y + 3z - 22 = 0 -----> ANSWER for tangent plane

Normal line :

Point = (1,2,3)

Grad F = (fx,fy,fz) = (5,4,3)

So, the line is given by parametric equations :

r(t) = Point + t*(gradF)

r(t) = (1,2,3) + t(5,4,3)

r(t) = (1,2,3) + (5t,4t,3t)

r(t) = (1 + 5t , 2 + 4t , 3 + 3t)

So, x(t) = 1 + 5t , y(t) = 2 + 4t , z(t) = 3 + 3t ------> ANSWER for normal line

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e)

f(1.1 , 1.8 , 3.2)

The linearization formula for f(x,y,z) is :

L(x,y,z) = f(x0,y0,z0) + fx(x - x0) + fy(y - y0) + fz(z - z0)

We know that f(x0,y0,z0) = f(1,2,3) = 11 given

L(x,y,z) = 11 + 5(x - 1) + 4(y - 2) + 3(z - 3)

L(x,y,z) = 11 + 5x + 4y + 3z - 22

L(x,y,z) = 5x + 4y + 3z - 11

Into this, plug in x = 1.1 , y = 1.8 and z = 3.2 :

L(1.1 , 1.8 , 3.2) = 5(1.1) + 4(1.8) + 3(3.2) - 11

L(1.1 , 1.8 , 3.2) = 11.3 ----> ANSWER to e

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