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

Chapter II
DIFFERENTIATION OF FUNCTIONS
Sec. 1. Calculating Derivatives Directly
1. Increment of the argument and increment of the function. If x and x l
are values of the argument x, and y = f(x) and t/ 1 = /(jc1 ) are corresponding
values of the function y = f(x), then
^x~x l
is called the increment of the argument x in the interval (x, xj, and
or
x
A0=0i y
/ = f (x,) -f (x) = f (x + A*) -
r
Fig. 11
is called the increment of the function y in the same interval (jc, *,) (Fig. 11,
where &x=MA and by AN). The ratio
5s the slope of the secant MN of the of
graph
the function y=*f(x) (Fig. 11)
and is called the mean rate of change of the function y over the interval
(x, *-f Ax).
Example t. For the function