Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

J88 Functions of Several Variables [Ch. 6
Solution. f(x+Ax, y + Ay) = (x -f Ax) 2
-f (x + A*) (y -f- Ay) (y -f Ay) 2 ;
= 2x Ax + AA 2 + x - Ay + y Ax + Ax Ay 2y Ay Ay a =
= [(2x + y) A* + (x 2y) A^l + (A* 2
+ AA> Ay Ay
Here, the expression d/ = (2x + y) A* -{-(* 2y) Ay is the total differential of
the function, while (A* 2
-f AJC* AyAy2 ) is an infinitesimal of higher order
comared with VAx 2
+Ay 2 .
compared with .
2. Find the total differential of the function
Example
Solution.
3. Applying the total differential of a function to approximate calculations.
For sufficiently small |AA:| and |Ay| and, hence, for sufficiently small
Q= y Av 2 -f Ay 2 , we have for a differentiate function z = f(x t y) the approximate
equality Az^dz or
. dz
Example 3. The altitude of a cone is // = 30cm, the radius of the base
fl = 10cm. How will the volume of the cone change, if we increase H by
3mm and diminish R by 1 mm?
Solution, The volume of the cone is V = -~-nR 2 H. The change in volume
we replace approximately by
AV =^ dV = -j n (2RH dR + R* dH) =
the differential
o
= lji( 2.10.30.0.1 + 100.0.3) = lOjiss 31. 4 cm*.
3
Example 4. s -01
Compute 1.02 approximately.
Solution. We consider the function z^x^. The desired number may be
considered the increased value of this function when jc=l, y = 3, Ajc = 0.02,
Ay = 0.01. The initial value of the function z = s
l =l,
Hence, 1.02 8 -01 ^ 1+0.06=1.06.
In x Ay = 3-1.0.02+ 1- In 1-0.01 =0.06.
1831. For the function f(x,y) = x*y find the total increment
and the total differential at the point (1, 2); compare them if
a) Ax=l, A//-2; b) A* = 0.1 f Ay = 0.2.
1832. Show that for the functions u and v of several (for
example, two) variables the ordinary rules of differentiation holcb
v du udv