Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.1 Functions of Two Variables 2.1.1 Introduction 2.1.2 Linear Functions 2.1.4 Slice Curves 2.1.6 Continuity 2.4 Integration Search

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.