Definition of convex and concave functions.. 6.1.6.1 The geometrical meaning of convexity is that all points of the graph curve between two graph points lie below or on the rectilinear s
Trang 13 If there exist lim
x→a f (x) and lim x→a g (x), then
lim
x→a
f (x) g (x)
= lim
x→a f (x) x→alimg (x);
lim
x→a cf (x) = c lim x→a f (x) (c = const);
lim
x→a f (x)⋅g (x) = lim
x→a f (x)⋅ lim
x→a g (x);
lim
x→a
f (x)
g (x) =
lim
x→a f (x)
lim
x→a g (x)
if g(x)≠ 0, lim
x→a g (x)≠ 0
4 Let f (x)≤g (x) in a neighborhood of a point a (x≠a) Then lim
x→a f (x)≤ lim
x→a g (x),
provided that these limits exist
5 If f (x)≤g (x)≤h (x) in a neighborhood of a point a and lim
x→a f (x) = lim x→a h (x) = b,
then lim
x→a g (x) = b.
These properties hold also for one-sided limits
6.1.3-3 Limits of some functions
First noteworthy limit: lim
x→0
sin x
x =1
Second noteworthy limit: lim
x→∞
1+ 1
x
x
= e.
Some other frequently used limits:
lim
x→0
(1+ x) n–1
x→∞
a n x n + a n–1x n–1+· · · + a1 x + a0
b n x n + b n–1x n–1+· · · + b1 x + b0 =
a n
b n,
lim
x→0
1– cos x
x2 =
1
2, x→lim0
tan x
x =1, lim
x→0
arcsin x
x =1, lim
x→0
arctan x
x =1, lim
x→0
e x–1
x =1, lim
x→0
a x–1
x = ln a, lim
x→0
ln(1+ x)
x =1, lim
x→0
loga(1+ x)
x = loga e, lim
x→0
sinh x
x =1, lim
x→0
tanh x
x =1, lim
x→0
arcsinh x
x =1, lim
x→0
arctanh x
x =1, lim
x→+0x
a ln x =0, lim
x→+∞ x
–aln x =0, lim
x→+∞ x
a e–x=0, lim
x→+0x
x=1,
where a >0and b n≠ 0
See Paragraph 6.2.3-2, where L’Hospital rules for calculating limits with the help of derivatives are given
6.1.3-4 Asymptotes of the graph of a function
An asymptote of the graph of a function y = f (x) is a straight line whose distance from the point (x, y) on the graph of y = f (x) tends to zero if at least one of the coordinates (x, y)
tends to zero
Trang 2The line x = a is a vertical asymptote of the graph of the function y = f (x) if at least one of the one-sided limits of f (x) as x → a 0is equal to +∞ or –∞.
The line y = kx + b is an oblique asymptote of the graph of y = f (x) if at least one of
the limit relations holds: lim
x→+∞ [f (x) – kx – b] =0or lim
x→–∞ [f (x) – kx – b] =0
If there exist finite limits
lim
x→+∞
f (x)
x = k, lim
x→+∞ [f (x) – kx] = b, (6.1.3.2)
then the line y = kx + b is an oblique asymptote of the graph for x → +∞ (in a similar way,
one defines an asymptote for x → –∞).
Example Let us find the asymptotes of the graph of the function y = x
2
x– 1.
1◦ The graph has a vertical asymptote x =1 , since lim
x→1
x2
x– 1 =∞.
2◦ Moreover, for x → ∞, there is an oblique asymptote y = kx + b whose coefficients are determined by
the formulas (6.1.3.2):
k= lim
x→∞
x
x– 1 =1, b= limx→∞
x2
x– 1– x
= lim
x→∞
x
x– 1 =1.
Thus, the equation of the oblique asymptote has the form y = x +1 Fig 6.1 shows the graph of the function under consideration and its asymptotes.
O
x
y
y
x
1 2 3 4 5 6
7
y=x 1 x2
1 2 3
Figure 6.1 The graph and asymptotes of the function y = x
2
x– 1.
6.1.4 Infinitely Small and Infinitely Large Functions
6.1.4-1 Definitions
A function f (x) is called infinitely small for x → a if lim
x→a f (x) =0
Trang 3A function f (x) is said to be infinitely large for x → a if for any K >0the inequality
|f(x)|> K holds for all x≠a in a small neighborhood of the point a In this case, one writes
f (x) → ∞ as x → a or lim
x→a f (x) = ∞ (In these definitions, a is a finite number or any of
the symbols∞, +∞, –∞.) If f(x) is infinitely large for x → a and f(x) >0(f (x) <0) in
a neighborhood of a (for x≠a), one writes lim
x→a f (x) = + ∞ (resp., lim
x→a f (x) = – ∞).
6.1.4-2 Properties of infinitely small and infinitely large functions
1 The sum and the product of finitely many infinitely small functions for x → a is an
infinitely small function
2 The product of an infinitely small function f (x) for x → a and a function g(x) which
is bounded in a neighborhood U of the point a (i.e.,|g(x)|< M for all xU , where M >0
is a constant) is an infinitely small function
3 lim
x→a f (x) = b if and only if f (x) = b + g(x), where g(x) is infinitely small for x → a.
4 A function f (x) is infinitely large at some point if and only if the function g(x) = 1
f (x)
is infinitely small at the same point
6.1.4-3 Comparison of infinitely large quantities Symbols of the order: O and o Functions f (x) and g(x) that are infinitely small for x → a are called equivalent near a if
lim
x→a
f (x)
g (x) =1 In this case one writes f (x) ∼ g(x).
Examples of equivalent infinitely small functions:
(1+ ε) n–1∼ nε, a ε–1∼ ε ln a, loga(1+ ε) ∼ ε log a e,
sin ε ∼ ε, tan ε ∼ ε, 1– cos ε ∼ 12ε2, arcsin ε ∼ ε, arctan ε ∼ ε,
where ε = ε(x) is infinitely small for x → a.
Functions f (x) and g(x) are said to be of the same order for x → a, and one writes
f (x) = O g (x)
if lim
x→a
f (x)
g (x) = K, 0<|K|<∞.*
A functionf (x) is of a higher order of smallness compared with g(x) for x → a if
lim
x→a
f (x)
g (x) =0, and in this case, one writes f (x) = o g (x)
6.1.5 Continuous Functions Discontinuities of the First and the
Second Kind
6.1.5-1 Continuous functions
A function f (x) is called continuous at a point x = a if it is defined in that point and its
neighborhood and lim
x→a f (x) = f (a).
For continuous functions, a small variation of their argumentΔx = x – a corresponds to
a small variation of the functionΔy = f(x) – f(a), i.e., Δy →0asΔx →0 (This property
is often used as a definition of continuity.)
* There is another definition of the symbol O Namely, f (x) = O g(x)
for x → a if the inequality
|f(x)| ≤K|g(x)|, K = const, holds in some neighborhood of the point a (for x≠a).
Trang 4A function f (x) is called right-continuous at a point x = a if it is defined in that point
(and to its right) and lim
x→a+0f (x) = f (a) A function f (x) is called left-continuous at a point
x = a if it is defined in that point (and to its left) and lim
x→a–0f (x) = f (a).
6.1.5-2 Properties of continuous functions
1 Suppose that functions f (x) and g(x) are continuous at some point a Then the functions f (x) g (x), cf (x), f (x)g(x), f (x)
g (x) (g(a)≠ 0) are also continuous at a.
2 Suppose that a function f (x) is continuous on the segment [a, b] and takes values of different signs at its endpoints, i.e., f (a)f (b) <0 Then there is a point c between a and b
at which f (x) vanishes:
f (c) =0 (a < c < b).
3 If f (x) is continuous at a point a and f (a) >0(resp., f (a) <0), then there is δ >0
such that f (x) >0(resp., f (x) <0) for all x(a – δ, a + δ).
4 Any function f (x) that is continuous at each point of a segment [a, b] attains its largest and its smallest values, M and m, on that segment.
5 A function f (x) that is continuous on a segment [a, b] takes any value c[m, M ] on that segment, where m and M are, respectively, its smallest and its largest values on [a, b].
6 If f (x) is continuous and increasing (resp., decreasing) on a segment [a, b], then on
the segment
f (a), f (b)
(resp.,
f (b), f (a)
) the inverse function x = g(y) exists, and is
continuous and increasing (resp., decreasing)
7 If u(x) is continuous at a point a and f (u) is continuous at b = u(a), then the composite function f u (x)
is continuous at a.
Remark Any elementary function is continuous at each point of its domain.
6.1.5-3 Points of discontinuity of a function
A point a is called a point of discontinuity of the first kind for a function f (x) if there exist finite one-sided limits f (a +0) and f (a –0), but the relations lim
x→a+0f (x) = lim x→a–0f (x) = f (a)
do not hold The value|f(a +0) – f (a –0)|is called the jump of the function at the point a.
In particular, if f (a +0) = f (a –0)≠f (a), then a is called a point of removable discontinuity.
Examples of function with discontinuities of the first kind.
1 The function f (x) =
0 for x <0,
1 for x≥ 0 has a jump equal to1at the discontinuity point x =0.
2 The function f (x) =0 for x≠ 0,
1 for x =0 has a removable discontinuity at the point x =0.
A point a is called a point of discontinuity of the second kind if at least one of the one-sided limits f (a +0) or f (a –0) does not exist or is equal to infinity
Examples of functions with discontinuities of the second kind.
1 The function f (x) = sin1
x has a second-kind discontinuity at the point x =0 (since this function has
no one-sided limits as x → 0 ).
2 The function f (x) =1/x has an infinite jump at the point x =0
Trang 56.1.5-4 Properties of monotone functions at points of discontinuity.
Any monotone function f (x) always has a left-hand limit and a right-hand limit at its discontinuity point x = x0; moreover, if f (x) is a nonincreasing function, then
f (x0–0)≥f (x0)≥f (x0+0);
if f (x) is a nondecreasing function, then
f (x0–0)≤f (x0)≤f (x0+0)
6.1.6 Convex and Concave Functions
6.1.6-1 Definition of convex and concave functions
1◦ A function f (x) defined and continuous on a segment [a, b] is called convex (or convex downward) if for any x1, x2in [a, b], the Jensen inequality holds:
f
x1+ x2 2
≤ f (x1) + f (x2 2) (6.1.6.1) The geometrical meaning of convexity is that all points of the graph curve between two graph points lie below or on the rectilinear segment joining the two graph points (see
Fig 6.2 a).
y=f x( )
y=f x( )
1
f x
2
f x
1
f x
f(x + x12 2)
2
2
1
f x ( ) + f x( )
2
f x
f(x + x12 2)
2 2 1
f x ( ) + f x( )
2 2 1
x + x
2 2 1
x + x
Figure 6.2 Graphs of convex (a) and concave (b) functions.
If for x1 ≠x2, condition (6.1.6.1) holds with < instead of≤, then the function f (x) is called strictly convex.
2◦ A function f (x) defined and continuous on a segment [a, b] is called concave (or convex upward) if for any x1, x2in [a, b] the following inequality holds:
f
x1+ x2
2
≥ f (x1) + f (x2)
The geometrical meaning of concavity is that all points of the graph curve between two graph points lie above or on the rectilinear segment joining the two graph points (see
Fig 6.2 b).
If for x1 ≠x2, condition (6.1.6.2) holds with > instead of≥, then the function f (x) is called strictly concave.
Trang 66.1.6-2 Generalized Jensen inequalities.
The inequalities (6.1.6.1) and (6.1.6.2) admit the following generalizations:
f (q1x1+· · · + q n x n)≤q1f (x1) +· · · + q n x n for a convex function,
f (q1x1+· · · + q n x n)≥q1f (x1) +· · · + q n x n for a concave function,
where q1, , q n are arbitrary positive numbers such that q1+· · · + q n=1, and x1, , x n are arbitrary points of the segment [a, b].
6.1.6-3 Properties of convex and concave functions
1 The product of a convex (concave) function and a positive constant is a convex (concave) function
2 The sum of two or more convex (concave) functions is a convex (concave) function
3 If ϕ(u) is a convex increasing function and u = f (x) is a convex function, then the composite function ϕ(f (x)) is convex Some other properties of composite functions:
ϕ (u) is convex and decreasing, u = f (x) is concave =⇒ ϕ(f(x)) is convex,
ϕ (u) is concave and increasing, u = f (x) is concave =⇒ ϕ(f(x)) is concave,
ϕ (u) is concave and decreasing, u = f (x) is convex =⇒ ϕ(f(x)) is concave.
4 A non-constant convex (resp., concave) function f (x) on a segment [a, b] cannot
attain its largest (resp., smallest) value inside the segment
5 If y = f (x) and x = g(y) are single-valued mutually inverse functions (on the
corre-sponding intervals), then the following properties hold:
f (x) is convex and increasing ⇐⇒ g(y) is concave and increasing,
f (x) is convex and decreasing ⇐⇒ g(y) is convex and decreasing,
f (x) is concave and increasing ⇐⇒ g(y) is convex and increasing,
f (x) is concave and decreasing ⇐⇒ g(y) is concave and decreasing.
6 A function f (x) that is continuous on a segment [a, b] and twice differentiable on the interval (a, b) is convex downward (resp., convex upward) if and only if f (x)≥ 0(resp.,
f (x)≤ 0) on that interval
7 Any convex function f (x) satisfying the condition f (x0) =0can be represented as the integral
f (x) =
x
x0
h (t) dt, where h(t) is a nondecreasing right-continuous function.
6.1.7 Functions of Bounded Variation
6.1.7-1 Definition of a function of bounded variation
1◦ Let f (x) be a function defined on a finite segment [a, b] Consider an arbitrary partition
of the segment by the points
a = x0< x1< x2 <· · · < x n–1< x n = b
Trang 7and construct the sum
v=
n–1
k=0
f (x k+1) – f (x k) (6.1.7.1)
whose terms are absolute values of the increments of f (x) on each segment of the partition.
If, for all partitions, the sums (6.1.7.1) are bounded by a constant independent of the
partition, one says that the function f (x) has bounded variation on the segment [a, b] The supremum of all such sums over all partitions is called the total variation of the function
f (x) on the segment [a, b] The total variation is denoted by
b
Va f (x) = sup{v}
A function f (x) is said to have bounded variation on the infinite interval [a, ∞) if it is
a function of bounded variation on any finite segment [a, b] and its total variation on [a, b]
is bounded by a constant independent of b By definition,
∞
Va f (x) = sup
b>a
b
Va f (x)
4
2◦ In the above definitions, the continuity of the function f (x) is not mentioned A
contin-uous function (without additional conditions) may have bounded or unbounded variation
Example Consider the continuous function
f(x) =
xcos 2π x if x≠ 0 ,
and the partition of the segment [ 0 , 1 ] by the points
0 < 1
2n < 1
2n– 1 <· · · < 13 < 12 <1.
Then the sums (6.1.7.1) corresponding to this partition have the form
v = 1 + 1
2+· · · +
1
n → ∞ as n → ∞.
Therefore,V10 f(x) = ∞.
6.1.7-2 Classes of functions of bounded variation
Next, we list some common classes of functions of bounded variation
1 Any bounded monotone function has bounded variation Its total variation on the
segment [a, b] is defined by Vb
a f (x) =|f(b) – f (a)| Remark The last statement is true for infinite intervals (–∞, a] and [a, ∞); in the latter case, the total
variation is equal to V∞ a f (x) =|f(∞) – f(a) |
2 Suppose that f (x) is a bounded function on [a, b] and this segment can be divided
into finitely many parts
[a k , a k+1] (k =0, 1, , m –1; a0= a, a m = b),
so that the function f (x) is monotone on each part Then f (x) has bounded variation on [a, b].
Remark This statement is also true for infinite segments.