Page 82 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 4] DERIVATIVES 73

0
lim f ðxÞ ¼ lim f ðxÞ ð19Þ
0
x!x 0 gðxÞ
x!x 0 g ðxÞ
whenever the limit on the right can be found. In case f ðxÞ and g ðxÞ satisfy the same conditions
0
0
as f ðxÞ and gðxÞ given above, the process can be repeated.
2. If lim f ðxÞ¼1 and lim gðxÞ¼1, the result (19)is also valid.
x!x 0 x!x 0
These can be extended to cases where x !1 or 1, and to cases where x 0 ¼ a or x 0 ¼ b in which
only one sided limits, such as x ! aþ or x ! b , are involved.
0
0
Limits represented by the so-called indeterminate forms 0 1, 1 ,0 ,1 ; and 1 1 can be
1
evaluated on replacing them by equivalent limits for which the above rules are applicable (see Problem
4.29).
APPLICATIONS
1. Relative Extrema and Points of Inflection
See Chapter 3 where relative extrema and points of inflection were described and a diagram is
presented. In this chapter such points are characterized by the variation of the tangent line, and
then by the derivative, which represents the slope of that line.
Assume that f has a derivative at each point of an open interval and that P 1 is a point of the graph of
f associated with this interval. Let a varying tangent line to the graph move from left to right through
P 1 . If the point is a relative minimum, then the tangent line rotates counterclockwise. The slope is
negative to the left of P 1 and positive to the right. At P 1 the slope is zero. At a relative maximum a
similar analysis can be made except that the rotation is clockwise and the slope varies from positive to
negative. Because f 00 designates the change of f ,we can state the following theorem. (See Fig. 4-6.)
0

Fig. 4-6

Theorem. Assume that x 1 is a number in an open set of the domain of f at which f is continuous and
0
f 00 is defined. If f ðx 1 Þ¼ 0 and f ðx 1 Þ 6¼ 0, then f ðx 1 Þ is a relative extreme of f . Specifically:
0
00
(a) If f ðx 1 Þ > 0, then f ðx 1 Þ is a relative minimum,
00
(b) If f ðx 1 Þ < 0; then f ðx 1 Þ is a relative maximum.
00
(The domain value x 1 is called a critical value.)
This theorem may be generalized in the following way. Assume existence and continuity of
derivatives as needed and suppose that f ðx 1 Þ¼ f ðx 1 Þ¼ f ð2p 1Þ ðx 1 Þ¼ 0and f ð2pÞ ðx 1 Þ 6¼ 0( p a posi-
00
0
tive integer). Then:
(a) f has a relative minimum at x 1 if f ð2pÞ ðx 1 Þ > 0,
(b) f has a relative maximum at x 1 if f ð2pÞ ðx 1 Þ < 0.