MA35 Demos


Demo1-1 Demo1-2 Demo1-3 Demo1-4 Demo2-1 Demo2-2 Demo2-3 Demo3-1 Demo3-2 Demo3-3 Demo3-4 Demo3-5 Demo4-1 Demo4-2 Demo5-1 Demo5-2 Demo5-3 Demo5-4 Demo6-1 Demo6-2 Demo7-1 Demo7-2 Demo7-3 Demo7-4 Demo8-1 Demo8-2 Demo8-3 Demo8-4

Demo Definitions

Demo 1
For possible insertion into a page 4 of section 2.1.3 on slice curves: Text: We can describe a slice curve with x = x0 as a parametric curve x(t) = x0, y(t) = t, so f(x(t),y(t)) = f(x0,t). Similarly the slice curve with y = y0 can be given as (x(t),y(t),f(x(t),y(t))) = (t, y0,f(t,y0)). The slice curve through (x0,y0) with slope m can be described as (x0 + t. y0 + mt, f((x0 + t. y0 + mt). In polar coordinates, this can be written (x0 + tcos(θ0), y0 + tsin(θ0),f(x0 + tcos(θ0), y0 + tsin(θ0), where m = tan(θ0). In general the slice curve over the parametric curve (x(t),y(t)) in the domain of a function f is the curve (x(t),y(t),f(x(t),y(t))).


Demo 2
For possible insertion in a second page of the continuity section 2.1.5: Text: 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 δ such that (x(t),y(t)) is within the disc of radius ε about (x(t0),y(t0)) whenever |t - t0| is less than δ. We achieve this by choosing δ 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| < δ.

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 δ, 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 δ small enough will make the δ strips lie in the respective ε/2 intervals, so the image of the parametric curve will lie in the square therefore in the disc.

Demo 3

For possible insertion in a third page in the continuity section: Text: 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 {plusminus} 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 δ so small that if |t-t0| {lt} δ, 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.


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