Labware - MA35 Multivariable Calculus - Two Variable Calculus





The function f is differentiable at a point (x0,y0) if

lim(x,y) --> (x0,y0) |f(x,y) - L(x,y)|/d((x,y), (x0,y0)) = 0

where d((x,y), (x0,y0)) = √[(x-x0)^2 + (y - y0)^2] is the distance from (x,y) to (x0,y0) and L(x, y) is the equation for the tangent plane to the function f(x, y) at the point (x0, y0).

An alternative definition is that a function f is differentiable at a point p in its domain if there exist functions fx and fy such that

limh --> 0 |f(p+h) - f(p) - fx(p)h1 - fy(p)h2|/|h| = 0

where h = (h1,h2). This condition can be written in terms of epsilons and deltas as well. We say that f is differentiable at p if for any ε > 0 there exists a &delta > 0 such that

|f(p+h) - f(p) - fx(p)h1 - fy(p)h2|/|h| < ε or

-ε|h| + fx(p)h1 + fy(p)h2 < (f(p+h) - f(p)) < ε|h| + fx(p)h1 + fy(p)h2

whenever |h| < &delta. This definition of differentiabilty is similar in form to the definition of continuity. Recall that in the geometrical interpretation of continuity the challenge was to find a small enough &delta-disc domain at p such that the surface over the domain would lie between two horizontal plates a distance ε above and below f(p). In the geometrical interpretation of differentiability, as seen in the inequality above, the challenge is to find a &delta-disc domain, centered at p, that is small enough so that the surface over the domain lies between two cones, where the value of ε determines the slope of the cones.



  • Use the demonstrations for continuity and differentiability in two variables to give an argument for why differentiability at a point implies continuity at a point.