M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesChapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesLearning module LM 14.1: Functions of 2 or 3 variables:Learning module LM 14.3: Partial derivatives:Learning module LM 14.4: Tangent planes and linear approximations:Tangent planesLinearization Quadratic approximations and concavity Learning module LM 14.5: Differentiability and the chain rule:Learning module LM 14.6: Gradients and directional derivatives:Learning module LM 14.7: Local maxima and minima:Learning module LM 14.8: Absolute maxima and Lagrange multipliers:Chapter 15: Multiple Integrals |
Quadratic approximations and concavitySecond order partial derivatives are connected with
'concavity', but the relationship is more subtle than in the one
variable case. If the tangent plane at a point P(a,b,f(a,b))
on the graph of z=f(x,y) gives the best Linear
Approximation
L(x,y) = f(a,b)+fx|(a,b)(x−a)+fy|(a,b)(y−b) to f near P, then we
can expect that some Quadric surface will give the best Quadratic
Approximation to f near P. Just as the best Linear
Approximation is the degree 1 Taylor polynomial centered at (a,b)
for f, so this best Quadratic Approximation is the degree 2 Taylor
polynomial. For brevity we'll speak of these degree one and degree two
Taylor polynomials as simply the respective Linear and Quadratic
Approximations to z=f(x,y) at (a,b).
But what does all this mean graphically for a function z=f(x,y)? Well, the graph of a linear equation Ax+by+Cz=D is a plane, while the graph of a quadratic equation is a quadric surface. So near a point (a,b) the Linear Approximation L(x,y) at (a,b) approximates the graph of z=f(x,y) by a plane - the Tangent Plane - while the Quadratic Approximation Q(x,y) at (a,b) approximates the graph of z=f(x,y) by a quadric surface such as a paraboloid, hyperbolic paraboloid, or a hyperboloid. To illustrate these ideas, let's compute some more Quadratic Approximations. Set
Do these seem reasonable given the graph of f shown above? For a function y=f(x) of one variable, the degree two approximating polynomial in x would be a parabola, opening up or down, but in two variables things are more subtle because there are several possible approximating quadric surfaces. Question: all the definitions and calculations so far have been for functions z=f(x,y) of two variables. Is it clear how to extend the definitions to a function w=f(x,y,z) of three variables? Just as in one dimension, we can use higher derivatives to get an even more accurate approximation, and to express functions as power series.
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