MA35 Demos 1. Parametric Curves in the Plane: A parametric curve in the plane is given by a pair of coordinate functions (x(t),y(t)) with the same domain α ≤ t ≤ β. If both of the coordinate functions are differentiable, then the velocity vector is defined to be (x'(t),y'(t)). The length of the velocity vector √(x'(t)2 + y'(t)2) is defined to be the speed at the point t, denoted s'(t). The distance along the curve from α to t is given by s(t) = ∫αts'(t)dt. Demo: The default curve is a unit circle x(t) = cos(t), y(t) = sin(t) traversed in a counter-clockwise direction. The demo shows the curve being traced out by a moving point, with the velocity vector indicated as a tangent vector at the point (x(t),y(t)) on the curve. Separate graphs indicates the speed at each point t and the distance s(t) along the curve. 2. Tangent Plane to a Smooth Function Graph: The tangent plane to a smooth function graph at a point (x0,y0,f(x0,y0) is the graph of the best linear approximation of the function in a neighborhood of the point (x0,y0). The slice curve (x,y0,f(x,y0)) is a plane curve dependent on x, with tangent vector (1, 0, fx(x0,y0 at the point (x0,y0). Similarly the slice curve (x0,y,f(x0,y)) is a plane curve dependent on y, with tangent vector (0, 1, fy(x0,y0 at the point (x0,y0). If the tangent plane exists, it will have to contain all linear combinations of these vectors, A plane through the point (x0,y0,z0) has the form A(x-x0) + B(y-y0) = z - z0. The tangent line in the plane y = y0 is then given by z = z0 + fx(x0,y0)(x-x0) so A = fx(x0,y0 and similarly B = fy(x0,y0. It follows that the equation for the tangent plane to the graph of z = f(x,y) at the point (x0,y0,f(x0,y0) is given by z(x,y) = f(x0,y0) + f(x0,y0)(x - x0) + f(y0,y0)(y - y0). Demo: We indicate the tangent vectors (1, 0, fx(x0,y0 and (0, 1, fy(x0,y0 as well as the parallelogram z(x,y) for x0 - 1 ≤ x ≤ x0 + 1 and y0 - 1 ≤ y ≤ y0 + 1. Exercise: Note that there are some points in the domain where the tangent plane meets a neighborhood of the point (x0,y0,f(x0,y0) on the graph in a single point, whereas in other cases, the tangent plane meets the graph in a pair of curves, or occasionally in a more complicated set of points. Classify the points of the graphs of the functions f(x,y) = x2 + y2, g(x,y) = x2 - y2, and h(x,y) = -x4 + 2*x2 - y2. What about the function f(x,y) = Ax2 + 2B*x*y + C*y2? 3. Chain Rule: Every smooth curve (x(t),y(t)) in the domain of a function f determines a curve (x(t),y(t),z(t)) on the graph of the function, where z(t) = f(x(t),y(t)). The velocity vector of this curve at t = t0 will be (x'(t0),y'(t0),z'(t0). If the function f(x,y) has a well-defined tangent plane at the point (x(t0),y(t0), f(x(t0),y(t0)), then the velocity vector of the curve (x(t),y(t),f(x(t),y(t)) at t = t0 will line in the tangent plane. The first two coordinates of this tangent vector will be x'(t0) and y'(t0) so the third coordinate will be z'(t0) = fx(x(t0),y(t0)x'(t0) + fy(x(t0),y(t0)y'(t0). This expression is known as the Chain Rule. Since the same rule will hold at all points of the domain, we can write the Chain Rule as as z'(t) = fx(x(t),y(t))x'(t) + fy(x(t),y(t))y'(t). Demo: We show the curve and its velocity vector in the horizontal plane and the vectors (x'(t0), 0, fx(x(t0),y(t0)x'(t0)), (0, y'(t0), fy(x(t0),y(t0)y'(t0), and ((x'(t0),y'(t0),fx(x(t0),y(t0)x'(t0) + fy(x(t0),y(t0)y'(t0)), with the appropriate color coding.