James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

932 Chapter 14 Partial Derivatives
In particular,
f s97, 72d < 125 1 3.75s1d 1 0.9s2d − 130.55
Therefore, when T − 97°F and H − 72%, the heat index is
I < 1318F
■
Differentials
For a differentiable function of one variable, y − f sxd, we define the differential dx to be
an independent variable; that is, dx can be given the value of any real number. The differential
of y is then defined as
9 dy − f 9sxd dx
y
y=ƒ
dy
dx=Îx
Îy
(See Section 3.10.) Figure 6 shows the relationship between the increment Dy and the
differential dy : Dy represents the change in height of the curve y − f sxd and dy represents
the change in height of the tangent line when x changes by an amount dx − Dx.
For a differentiable function of two variables, z − f sx, yd, we define the differentials
dx and dy to be independent variables; that is, they can be given any values. Then the
differential dz, also called the total differential, is defined by
0
a a+Îx
tangent line
y=f(a)+fª(a)(x-a)
FIGURE 6
7et140406
05/03/10
MasterID: 01590
x
10 dz − f x sx, yd dx 1 f y sx, yd dy − −z −z
dx 1
−x −y dy
(Compare with Equation 9.) Sometimes the notation df is used in place of dz.
If we take dx − Dx − x 2 a and dy − Dy − y 2 b in Equation 10, then the differential
of z is
dz − f x sa, bdsx 2 ad 1 f y sa, bdsy 2 bd
So, in the notation of differentials, the linear approximation (4) can be written as
f sx, yd < f sa, bd 1 dz
Figure 7 is the three-dimensional counterpart of Figure 6 and shows the geometric interpretation
of the differential dz and the increment Dz: dz represents the change in height
of the tangent plane, whereas Dz represents the change in height of the surface z − f sx, yd
when sx, yd changes from sa, bd to sa 1 Dx, b 1 Dyd.
z
surface z=f(x, y)
{a+Îx, b+Îy, f(a+Îx, b+Îy)}
{a, b, f(a, b)}
dz
Îz
0
f(a,b)
FIGURE 7
x
f(a, b)
(a, b, 0)
Îx=dx
y
(a+Îx, b+Îy, 0)
Îy=dy
tangent plane
z-f(a, b)=f x (a, b)(x-a)+f y (a, b)(y-b)
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