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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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If the coordinate functions x(t) and y(t) are continuous functions of the parameter t, then the function that sends t to the point (x(t),y(t)) is continuous. This means that for any t₀ in the domain, and any positive ε, there is a &delta such that (x(t),y(t)) is within the disc of radius ε about (x(t₀),y(t₀)) whenever | t - t₀ | is less than &delta. We achieve this by choosing &delta so small that | x(t) - x(t₀) | < ε/2 and | y(t) - y(t₀) | < ε/2, by virtue of the continuity of x(t) and y(t) at t₀. Then √((x(t)-x(t₀))² + (y(t)-y(t₀))²) < √(ε²/4 + ε²/4)) = ε/2 < ε if | t - t₀ | < &delta.

Demos

Continuity of Parametric Functions

Define an interval for t, and two functions x(t), y(t). In a three-dimensional graph, show (t, x(t),0), (t, 0, y(t)), and (0,x(t),y(t)). For a given t0 and interval on the t-axis determined by a &delta, show the strip above this interval in the first plane and in the second plane, and show the ε/2 strip about x(t0) in the third plane and the ε/2 strip about y(t0 in the third plane, intersecting in a square region completely contained in ε disc about (x{t0),y(t0)) in the third plane. Choosing &delta small enough will make the &delta strips lie in the respective ε/2 intervals, so the image of the parametric curve will lie in the square therefore in the disc.