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M408M Learning Module Pages

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Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space


Chapter 13: Vector Functions


Chapter 14: Partial Derivatives


Learning 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 planes
      Linearization
      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 concavity

Quadratic Approximations and Taylor's Theorem

Second 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)(xa)+fy|(a,b)(yb) 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).


  The Quadratic Approximation to a function z=f(x,y) at (a,b) is given by Q(x,y) = f(a,b)+fx|(a,b)(xa)+fy|(a,b)(yb) +12fxx|(a,b)(xa)2+fxy|(a,b)(xa)(yb)+12fyy|(a,b)(yb)2. At (x,y)=(a,b), the value, first derivatives and second derivatives of Q(x,y) equal the value, first derivatives and second derivatives of f(x,y). In fact, Q(x,y) is the unique quadratic polynomial with this property.



Example: Find the linear and quadratic approximations to f(x,y)=2xy near (1,2).

Solution: We first compute derivatives at (1,2): f(1,2)=1,fx(1,2)=1,fy(1,2)=12, fxx(1,2)=2,fxy(1,2)=12,fyy(1,2)=12, so the Linear Approximation to f(x,y)=2xy near (1,2) is given by
2xy  L(x,y)=1(x1)12(y2)=3x12y, while the Quadratic Approximation is 2xy  Q(x,y)=1(x1)12(y2)+(x1)2+12(x1)(y2)+14(y2)2, which after some algebra becomes 2xy  64x2y+x2+12xy+14y2.


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
z = f(x,y) = sinxsiny,πx,yπ; the graph of f is shown to the right in interactive form - try grabbing and moving it! Then fx=cosxsiny,fy=sinxcosy, while 2fx2=sinxsiny,  2fy2=sinxsiny. and 2fxy=cosxcosy,
   Thus: when f(x,y)=sinxsiny,

  near (0,0):f(x,y)    xy, a hyperbolic paraboloid;

  near (π2, π2):f(x,y)    112(xπ2)212(yπ2)2, a paraboloid opening downwards;

  near (π2, π2):f(x,y)    1+12(xπ2)2+12(y+π2)2, a paraboloid opening upwards.


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.
Taylor Series in Two Variables: If f(x,y) is an analytic function of two variables, then f(x,y)=n,m=0cn,m(xa)n(yb)m, where cn,m=1n!m!n+mfxnym(a,b).