Linear approximation & Total differentials

Back when the Ancient Greeks didn't have calculators, they relied on a method called linear approximation to help estimate the values of functions. Ever wonder how our modern calculators do it? Probably the same way! By constructing a tangent line, we can approximate the value of an unknown point when given the value of a known point. The formula we've all seen last year is:





Now when we extend this to a 3-dimensional space, we construct a plane tangent to the function. This can be summarized by:

f(x+Δx,y+Δy) = f(x,y) + fx (x,y)Δx + fy (x,y)Δy

One example we did in class is:

f(x,y) = x^2+3xy-y^2
Find: f(2.05, 2.96) Known: f(2,3)
Therefore, Δx = 0.05 and Δy = .04

fx = 2x+3y f(x+Δx,y+Δy) = f(x,y) + fx (x,y)Δx + fy (x,y)Δy
fy = 3x-2y 13+0.05(13)-0.04(0) = 13.65



The total differential of a function is the sum of all the partial derivatives. Total differentials can help estimate the maximum error (Δf) in using the differential. It can be summarized by:

z = f(x,y)
dz = fx (x,y)dx + fy (x,y)dy OR Δz = fx (x,y)Δx + fy (x,y)Δy