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

Sec. 6}_Differentials of First and Higher Orders_73
713. Without calculating the derivative, find
for x=\ and Ax = .
d(l-x')
714. The area of a square S with side x is given by S = x*.
Find the increment and the differential of this function and explain
the geometric significance of the latter.
715. Give a geometric interpretation of the increment and
differential of the following functions:
a) the area of a circle, S = nx*\
b) the volume of a cube, v=^x\
716. Show that when Ax *0, the increment in the function
// = 2 X , corresponding to an increment Ax in x, is, for any x,
equivalent to the expression 2* In 2 A*.
717. For what value of x is the differential of the function
y = x 2
not equivalent to the increment in this function as Ax >0?
718. Has the function y = \x\ a differential for x = 0?
719. Using the derivative, find the differential of the function
y cos x for x = y and Ax --= ~ .
720. Find the differential of the function
for x = 9 and Ax- 0.01.
721. Calculate the differential of the function
for x-^-J and Ax^.
In the following problems find the differentials of the given
functions for arbitrary values of the argument and its increment.
722. y^'-m- 727. y = x\nx x.
723. -f cosec (p.
725. //--=arctan~. 730. s = arc lane*.
726. y = e~ x \
731 Find d// if x* + 2xy y* = a*.
Solution. Taking advantage of the invariancy of the form of a differential,
we obtain 2x dx + 2 (y dx + x dy) 2y dy =
Whence