Question

Determine and state the maximum error of the function G(x,y,z)=20 ln⁡(xyz^2 ) using differentials when (x,y,z)=(2,3,4) and given that the error in x is ±0.10, in y is ±0.15 and in z is ±0.20.

Answer 1

Answer:

The maximum error in the function G, ΔG = ±4

Step-by-step explanation:

G(x,y,z) = 20 In (xyz²)

Total or maximum error for a multi-variable function is given by

ΔG = (∂G/∂x) (Δx) + (∂G/∂y) (Δy) + (∂G/∂z) (Δz)

(∂G/∂x) = 20yz²/xyz² = 20/x

(∂G/∂y) = 20xz²/xyz² = 20/y

(∂G/∂z) = 40xyz/xyz² = 40/z

Δx = ±0.10

Δy = ±0.15

Δz = ±0.20

ΔG = (∂G/∂x) (Δx) + (∂G/∂y) (Δy) + (∂G/∂z) (Δz)

ΔG = (20/x) (0.10) + (20/y) (0.15) + (40/z) (0.2)

At the point (x,y,z) = (2,3,4)

ΔG = (20/2) (0.10) + (20/3) (0.15) + (40/4) (0.2) = 10(0.10) + 20(0.05) + 10(0.2) = 1 + 1 + 2 = 4

ΔG = ±4