Chapter 4: Partial Differentiation
Section 4.7: Approximations
Essentials
For a sufficiently well behaved function fx,y, the change Δf≡fx+dx,y+dy−fx,y is approximated by the differential df≡fxx,y dx+fyx,y dy. Such a differential is often called either a total or an exact differential. The generalization to functions of more than two variables is obvious.
Contrary to the logic in single-variable calculus where the derivative is defined first, and the differential is defined in terms of the derivative as the derivative times an increment, for functions of several variables the differential must be defined first. If a function has a differential, then it is called differentiable!
A sufficient condition for the existence of a differential, and hence for the function of several variables to be differentiable, is the continuity of the first partial derivatives.
As for the function of one variable, a differentiable function is necessarily continuous.
Functions of several variables can be approximated by Taylor polynomials. Table 4.7.1 lists Taylor polynomials of degrees one and two for the function fx,y. The expansion point is taken as a,b.
Degree
Taylor Polynomial
1
fa+h,b+k=fa,b+fxa,bh+fya,bk
2
fa+h,b+k=fa,b+fxa,bh+fya,bk +12fxxa,bh2+2 fxya,bh k+fyya,bk2
Table 4.7.1 Taylor polynomials of degrees one and two
The equality of the mixed partial derivatives fxy and fyx is assumed in Table 4.7.1. Hence, instead of having two separate terms containing the product h k, both are combined into twice the one. It is sometimes useful to write the second-degree terms as half the quadratic form
hk fxxfxyfxyfyyhk
where the row vector hk is the transpose of the column vector hk and the intervening symmetric matrix of second partial derivatives is called a Hessian.
A general representation of the nth-degree term is 1n!h ∂∂x+k ∂∂ynfa,b, but unfortunately, no form of this expression is a valid Maple operator.
The role existence and continuity of partial derivatives plays in differentiability of a function of several variables is examined at length in Section 4.11.
Examples
Example 4.7.1
If x changes from 3 to 3.04 and y changes from −2 to −2.2, compare Δf and df for the function fx,y=3 x2+4 x y−5 y2+7.
Example 4.7.2
Approximate 43.033+56.22 by using the total differential for some appropriate function fx,y.
Example 4.7.3
Approximate 12.03+13.02+14.97 by using the total differential for some appropriate function fx,y,z.
Example 4.7.4
Use the total differential to approximate the value of fx,y=x2e2 xy2 at x,y=1.03,0.96.
Example 4.7.5
Use the total differential to approximate the value of fx,y,z=x lny+ zx2+y z at x,y,z=2.03,3.12,1.48.
Example 4.7.6
Use differentials to estimate the maximum error in determining the area of a rectangle measured (in cm) to be 4×5, if each measurement is accurate to .1 cm.
Example 4.7.7
Use differentials to estimate the maximum error in determining the surface area of a closed rectangular box measured (in inches) to be 15×6×21, if each measurement is accurate to .2 in.
Example 4.7.8
A closed box is constructed from lumber that is 3/4 in thick. The outside measurements are 9×7×5. Use differentials to estimate the interior volume of the box.
Example 4.7.9
Show that the plane tangent to fx,y=3 x2+5 x y−7 y2+4 at x,y=2,3 is the first-degree Taylor polynomial constructed at the point of contact. Show further that approximating Δf by the total differential df amounts to a tangent-plane approximation.
Example 4.7.10
At x,y=4,1, construct the second-degree Taylor polynomial for fx,y=3 x2+y3.
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