Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.2 Differentiation 2.2.1 Partial Derivatives 2.2.2 Critical Points 2.2.3 Tangent Planes and Normal Vectors 2.2.4 Chain Rule 2.2.5 Differentiability 2.4 Integration Search

Tangent Planes and Normal Vectors (Page: 1 | 2 )

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If both partial derivatives of f exist at a point (x0,y0), then the equation of the tangent plane of f at (x0,y0) is L(x,y) = f(x0,y0) + fx(x0,y0)(x - x0) + fy(x0,y0)(y - y0).

The general form of a plane through the point (x0,y0,z0) is L(x,y) = z0 + p(x - x0) + q(y - y0) for appropriate choice of p and q. In order for the tangent lines to the slice curves at (x0,y0) to lie in the tangent plane, we must have p = fx(x0,y0) and q = fy(x0,y0).

The tangent line to the slice curve y = y0 is given by z = z0 + fx(x0,y0)(x - x0). The vector (1, 0, fx(x0,y0)) lies in the direction of the tangent line to the y = y0 slice curve.

The unit vector in the direction of (-fx(x0,y0),-fy(x0,y0),1) is called the unit normal to the graph of f at (x0,y0,z0), written N(x0,y0). Thus N(x0,y0) = (-fx(x0,y0),-fy(x0,y0),1)/√(fx2(x0,y0) + fy2(x0,y0) + 1).

Demos

 Tangent Plane For this demonstration, choose a point (x0,y0) in the domain using the white hotspot in the window labeled "Domain: f(x,y)". The second window graphs the function f(x,y) and the x- and y-slice curves at the point f(x0,y0). The tangent lines to each of the slice curves at f(x0,y0) are also drawn, and it is clear that together they determine a plane. This plane, which is shown in yellow, is tangent to the graph at f(x0,y0).

 Intersection Set of Tangent Plane This demo shows the tangent planes for the monkey saddle Ax3 - 3Bxy2. Note that the intersection set of the tangent plane with the function graph allows us to classify the points of the surface into two categories: for some points in the domain, the tangent plane meets a neighborhood of the point (x0,y0,f(x0,y0) on the graph in a single point, whereas for other points, the tangent plane meets the graph in a pair of curves, or occasionally in a more complicated set of points.

 Normal Vectors This demo shows the unit normal vector to the tangent plane for a given point (x0,y0) in the domain. Note that this vector is perpendicular to each of the normal lines drawn to the slice curves.

Exercises

• 1. Try entering f(x,y) = abs(abs(x) - abs(y)) - abs(x) - abs(y). Using the limit definition of the partial derivatives, show that fx(0,0) = fy(0,0) = 0. Are fx(0,0) and fy(0,0) well-defined? Using the definintion of the tangent plane, show that the tangent plane is z(x,y) = 0. Move the hot spot to the origin. Is the plane z = 0 tangent to the surface? Note that to be tangent to a surface means to touch it at only one point.
• 2. Which is a stronger condition: the tangent plane exists at a point (x0,y0); or, the partial derivatives are well-defined at a point (x0,y0)?
• 3. Use the second demo to classify the points of the graphs of the function f(x,y) = Ax2 + 2Bxy + Cy2 for different values of A, B and C.
• 4. Now change the function f(x,y) in the demo and classify the points of the functions f(x,y) = x2 + y2, g(x,y) = x2 - y2, and h(x,y) = -x4 + 2x2 - y2.