By about how much will g(x, y, z) = x + x cos(z) − y sin(z) + y − 2 change if the point P(x, y, z) .........

Question

By about how much will g(x, y, z) = x + x cos(z) − y sin(z) + y − 2 change if the point P(x, y, z) moves from

P₀(2, −1, 0) a distance of ds = 0.3 unit toward the point P₁(0, 1, 2)?

Answer 1

Δg(x,y,z) ≈ dg(x,y,z) = ∂g/∂x·Δx + ∂g/∂y·Δy + ∂g/∂z·Δz = (1 + cosz)Δx + (1 - sinz)Δy + (- xsinz - ycosz)Δz

Partial derivatives we avoluate at point P₀(2, - 1, 0). So Δg(x, y, z) ≈ 2Δx + Δy + Δz.

Vector P₀P₁ = (- 2, 2, 2). AbsP₀P₁ = 2√3. And because P₀P = P₀P₁·0.3/(2√3) = (-1,1,1)·0.3/√3 =

= (- √3/10, √3/10, √3/10).

So Δx = - √3/10, Δy = √3/3 and Δz = √3/3 and Δg ≈ 2(- √3/10) + √3/10 + √3/10 = 0

Answer 2

Hi Mark T.,

You have submitted 6 problems at once requiring similar techniques to solve (derivatives in 3-space). That's "spamming" your problem set.

--Cheers, -- Mr. d.