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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

2.1.1 Introduction

2.1.2 Linear Functions

2.1.3 Domain, Range & Function Graphs

2.1.4 Slice Curves

2.1.5 Level Sets & Contour Lines

2.1.6 Continuity

2.1.7 Normal Vectors for Linear Function Graphs

2.2 Differentiation

2.3 More Derivatives

2.4 Integration

2.5 Vector Analysis

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Continuity (Page: 1 | 2 | 3 | 4 )

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Given a positive ε we can form the ε interval about z₀ = (0,0,f(x(t₀),y(t₀)) on the z-axis, and show the two planes at levels z₀ ± ε.

We can then find a ρ such that the graph of f(x,y) over the disc of radius ρ centered at (x(t₀),y(t₀),0) will lie between the two horizontal planes.

Finally, we can find a &delta so small that if | t - t₀ | < &delta, then the image of the parametrics curve (x(t),y(t)) will lie inside the disc of radius ρ and the curve (0,0,f(x(t),y(t))) will lie between the two horizontal planes.

Demos

Continuity of Parametric Curves
	The value of t0 is chosen on a slider bar, as well as the position of the &delta interval. The value of ρ is chosen so that the portion of the graph above the disc is between the planes, and then &delta is chosen so that the image of the parametric curve lies in the disc of radius ρ.