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MA35 Labs

1 Single Variable Calculus

2 » Two Variable Calculus

3 Three Variable Calculus

Polar Coordinate Labs

Two Variable Functions

Contents

2.1 Functions of Two Variables

2.2 Differentiation

2.2.1 Partial Derivatives

2.2.2 Critical Points

2.2.3 Tangent Planes and Normal Vectors

2.2.4 Chain Rule

2.2.5 Differentiability

2.3 More Derivatives

2.4 Integration

2.5 Vector Analysis

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Differentiability

Text

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 f_x and f_y such that

lim_{h --> 0} |f(p+h) - f(p) - f_x(p)h₁ - f_y(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) - f_x(p)h₁ - f_y(p)h₂|/|h| < ε or

-ε|h| + f_x(p)h₁ + f_y(p)h₂ < (f(p+h) - f(p)) < ε|h| + f_x(p)h₁ + f_y(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.

Demos

Geometric Interpretation of Differentiability

Start by choosing a point (x0,y0) in the domain window. This corresponds to a point (x0,y0,f(x0,y0)) on the function graph. Next choose an epsilon using the tapedeck in the control panel. The challenge is to find a small enough &delta-disc neighborhood of (x0,y0) in the domain so that the surface over this disc domain lies between the two cones. Use the close-up window to check whether the condition of differentiability can be met.

Exercises