Properties of functions of bounded variation.. Here, all functions are considered on a finite segment [a, b].. The sum, difference, or product of finitely many functions of bounded varia
Trang 1248 LIMITS ANDDERIVATIVES
3 Let f (x) be a function on a finite segment [a, b] satisfying the Lipschitz condition
f (x1) – f (x2)≤L|x1– x2|,
for any x1 and x2 in [a, b], where L is a constant Then f (x) has bounded variation and
b
Va f (x)≤L (b – a).
4 Let f (x) be a function on a finite segment [a, b] with a bounded derivative|f (x)| ≤L,
where L = const Then, f (x) is of bounded variation and Va b f (x)≤L (b – a).
5 Let f (x) be a function on [a, b] or [a, ∞) and suppose that f(x) can be represented
as an integral with variable upper limit,
f (x) = c + x
a ϕ (t) dt,
where ϕ(t) is an absolutely continuous function on the interval under consideration Then
f (x) has bounded variation and
b
Va f (x) = b
a |ϕ (x)|dx
COROLLARY Suppose that ϕ(t) on a finite segment [a, b] or [a, ∞) is integrable, but
not absolutely integrable Then the total variation of f (x) is infinite.
6.1.7-3 Properties of functions of bounded variation
Here, all functions are considered on a finite segment [a, b].
1 Any function of bounded variation is bounded
2 The sum, difference, or product of finitely many functions of bounded variation is a function of bounded variation
3 Let f (x) and g(x) be two functions of bounded variation and|g (x)| ≥K >0 Then
the ratio f (x)/g(x) is a function of bounded variation.
4 Let a < c < b If f (x) has bounded variation on the segment [a, b], then it has bounded variation on each segment [a, c] and [c, b]; and the converse statement is true In this case,
the following additivity condition holds:
b
Va f (x) =
c
Va f (x) +
b
Vc f (x).
5 Let f (x) be a function of bounded variation of the segment [a, b] Then, for a≤x≤b,
the variation of f (x) with variable upper limit
F (x) =
x
Va f (x)
is a monotonically increasing bounded function of x.
6 Any function f (x) of bounded variation on the segment [a, b] has a left-hand limit
lim
x→x0 – 0f (x) and a right-hand limit lim x→x0 + 0f (x) at any point x0 [a, b].
Trang 26.1.7-4 Criteria for functions to have bounded variation.
1 A function f (x) has bounded variation on a finite segment [a, b] if and only if there is
a monotonically increasing bounded functionΦ(x) such that for all x1, x2 [a, b] (x1< x2), the following inequality holds:
|f (x2) – f (x1)| ≤Φ(x2) –Φ(x1).
2 A function f (x) has bounded variation on a finite segment [a, b] if and only if f (x)
can be represented as the difference of two monotonically increasing bounded functions on
that segment: f (x) = g2(x) – g1(x).
Remark The above criteria are valid also for infinite intervals (–∞, a], [a, ∞), and (–∞, ∞).
6.1.7-5 Properties of continuous functions of bounded variation
1 Let f (x) be a function of bounded variation on the segment [a, b] If f (x) is continuous at a point x0(a < x0 < b), then the function F (x) =Vx
a f (x) is also continuous
at that point
2 A continuous function of bounded variation can be represented as the difference of two continuous increasing functions
3 Let f (x) be a continuous function on the segment [a, b] Consider a partition of the
segment
a = x0< x1< x2 <· · · < x n–1< x n = b
and the sum v = n–1
k=0
f (x k+1) – f (x k) Letting λ = max|x k+1– x k|and passing to the limit
as λ →0, we get
lim
λ→0v=
b
Va f (x).
6.1.8 Convergence of Functions
6.1.8-1 Pointwise, uniform, and nonuniform convergence of functions
Let{f n (x)}be a sequence of functions defined on a set X ⊂ R The sequence{f n (x)}is said
to be pointwise convergent to f (x) as n → ∞ if for any fixed xX, the numerical sequence {f n (x)}converges to f (x) The sequence{f n (x)}is said to be uniformly convergent to a function f (x) on X as n → ∞ if for any ε >0there is an integer N = N (ε) and such that for all n > N and all xX, the following inequality holds:
|f n (x) – f (x)|< ε. (6.1.8.1)
Note that in this definition, N is independent of x For a sequence {f n (x)} pointwise
convergent to f (x) as n → ∞, by definition, for any ε > 0 and any x X, there is
N = N (ε, x) such that (6.1.8.1) holds for all n > N (ε, x) If one cannot find such N independent of x and depending only on ε (i.e., one cannot ensure (6.1.8.1) uniformly; to be more precise, there is δ >0such that for any N >0there is k N > N and x N Xsuch that
|f k N (x N ) – f (x N)| ≥ δ), then one says that the sequence{f n (x)}converges nonuniformly
to f (x) on the set X.
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6.1.8-2 Basic theorems
Let X be an interval on the real axis.
THEOREM Let f n (x)be a sequence of continuous functions uniformly convergent to
f (x) on X Then f (x) is continuous on X.
COROLLARY If the limit function f (x) of a pointwise convergent sequence of
contin-uous functions{f n (x)}is discontinuous, then the convergence of the sequence{f n (x)}is nonuniform
Example The sequence{f (x)}= {x n}converges to f (x)≡ 0as n → ∞ uniformly on each segment
[0, a], 0< a <1 However, on the segment [0, 1] this sequence converges nonuniformly to the discontinuous
function f (x) =0 for 0 ≤
x< 1,
1 for x =1.
CAUCHY CRITERION.A sequence of functions{f n (x)}defined on a set XR uniformly
converges to f (x) as n → ∞ if and only if for any ε >0there is an integer N = N (ε) >0
such that for all n > N and m > N , the inequality|f n (x) – f m (x)|< ε holds for all xX
6.1.8-3 Geometrical meaning of uniform convergence
Let f n (x) be continuous functions on the segment [a, b] and suppose that{f n (x)}uniformly
converges to a continuous function f (x) as n → ∞ Then all curves y =f n (x), for sufficiently large n > N , belong to the strip between the two curves y = f (x) – ε and y = f (x) + ε (see
Fig 6.3)
O
x y
b a
y=f x( )
y=f x( )+ ε
y=f x( )-ε
y=f x n( )
Figure 6.3 Geometrical meaning of uniform convergence of a sequence of functions{f (x)}to a continuous
function f (x).
6.2 Differential Calculus for Functions of a Single
Variable
6.2.1 Derivative and Differential, Their Geometrical and Physical
Meaning
6.2.1-1 Definition of derivative and differential
The derivative of a function y = f (x) at a point x is the limit of the ratio
y = lim
Δx→0
Δy
Δx = limΔx→0
f (x + Δx) – f(x)
whereΔy = f(x+Δx)–f(x) is the increment of the function corresponding to the increment
of the argumentΔx The derivative y is also denoted by y
x, ˙y, dy dx , f (x), df dx (x).
Trang 4Example 1 Let us calculate the derivative of the function f (x) = x2.
By definition, we have
f (x) = lim
Δx→0
(x + Δx)2– x2
Δx = limΔx→0 (2x+Δx) =2x. The incrementΔx is also called the differential of the independent variable x and is
denoted by dx.
A function f (x) that has a derivative at a point x is called differentiable at that point The differentiability of f (x) at a point x is equivalent to the condition that the increment
of the function, Δy = f(x + dx) – f(x), at that point can be represented in the form
Δy = f (x) dx + o(dx) (the second term is an infinitely small quantity compared with dx as
dx →0)
A function differentiable at some point x is continuous at that point The converse is
not true, in general; continuity does not always imply differentiability
A function f (x) is called differentiable on a set D (interval, segment, etc.) if for any
xD there exists the derivative f (x) A function f (x) is called continuously differentiable
on D if it has the derivative f (x) at each point xD and f (x) is a continuous function on
D
The differential dy of a function y = f (x) is the principal part of its increment Δy at the
point x, so that dy = f (x)dx, Δy = dy + o(dx).
The approximate relationΔy≈dy or f (x + Δx) ≈f (x) + f (x) Δx (for small Δx) is
often used in numerical analysis
6.2.1-2 Physical and geometrical meaning of the derivative Tangent line
1◦ Let y = f (x) be the function describing the path y traversed by a body by the time x.
Then the derivative f (x) is the velocity of the body at the instant x.
2◦ The tangent line or simply the tangent to the graph of the function y = f (x) at a point
M (x0, y0), where y0= f (x0), is defined as the straight line determined by the limit position
of the secant M N as the point N tends to M along the graph If α is the angle between the x-axis and the tangent line, then f (x0) = tan α is the slope ratio of the tangential line
(Fig 6.4)
O
x
y
y
dy M
N
α
α
Δy
0
y=f x( )
x0 x + x0 Δ
Figure 6.4 The tangent to the graph of a function y = f (x) at a point (x0, y0).
Equation of the tangent line to the graph of a function y = f (x) at a point (x0, y0):
y – y0= f (x0)(x – x0)
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Equation of the normal to the graph of a function y = f (x) at a point (x0, y0):
y – y0= – 1
f (x0)(x – x0).
6.2.1-3 One-sided derivatives
One-sided derivatives are defined as follows:
f
+(x) = lim
Δx→+0
Δy
Δx = limΔx→+0
f (x + Δx) – f(x)
Δx right-hand derivative,
f
–(x) = lim
Δx→–0
Δy
Δx = limΔx→–0
f (x + Δx) – f(x)
Δx left-hand derivative.
Example 2 The function y =|x|at the point x =0has different one-sided derivatives: y +(0) = 1, y
– (0) = –1,
but has no derivative at that point Such points are called angular points.
Suppose that a function y = f (x) is continuous at x = x0 and has equal one-sided
derivatives at that point, y+ (x0) = y– (x0) = a Then this function has a derivative at x = x0 and y (x0) = a.
6.2.2 Table of Derivatives and Differentiation Rules
The derivative of any elementary function can be calculated with the help of derivatives of basic elementary functions and differentiation rules
6.2.2-1 Table of derivatives of basic elementary functions (a = const).
(a) =0, (x a) = ax a–1,
(e x) = e x, (a x) = a x ln a, (ln x) = 1
x ln a, (sin x) = cos x, (cos x) = – sin x, (tan x) = 1
cos2x, (cot x) = – 1
sin2x,
(arcsin x) = 1
√
1– x2, (arccos x)
= – 1
√
1– x2, (arctan x) = 1
1+ x2, (arccot x)
= – 1
1+ x2, (sinh x) = cosh x, (cosh x) = sinh x, (tanh x) = 1
cosh2x, (coth x) = – 1
sinh2x,
(arcsinh x) = 1
√
1+ x2, (arccosh x)
√
x2–1,
(arctanh x) = 1
1– x2, (arccoth x)
= 1
x2–1.
Trang 66.2.2-2 Differentiation rules.
1 Derivative of a sum (difference) of functions:
[u(x) v (x)] = u (x) v (x).
2 Derivative of the product of a function and a constant:
[au(x)] = au (x) (a = const).
3 Derivative of a product of functions:
[u(x)v(x)] = u (x)v(x) + u(x)v (x).
4 Derivative of a ratio of functions:
*u (x)
v (x)
+
= u (x)v(x) – u(x)v (x)
v2(x) .
5 Derivative of a composite function:
f (u(x))
= f u (u)u (x).
6 Derivative of a parametrically defined function x = x(t), y = y(t):
y
x =
y t
x t
7 Derivative of an implicit function defined by the equation F (x, y) =0:
y
x= –F F x
y (F x and F yare partial derivatives).
8 Derivative of the inverse function x = x(y) (for details see footnote*):
x
y = 1
y
x.
9 Derivative of a composite exponential function:
[u(x) v(x)] = u v ln u⋅v + vu v–1u .
10 Derivative of a composite function of two arguments:
[f (u(x), v(x))] = f u (u, v)u + f v (u, v)v (f u and f vare partial derivatives)
Example 1 Let us calculate the derivative of the function x
2
2x+ 1. Using the rule of differentiating the ratio of two functions, we obtain
x2
2x+ 1
=(x
2 )(2x+ 1) – x 2 (2x+ 1)
(2x+ 1) 2 = 2x(2x+ 1) – 2x2
(2x+ 1) 2 = 2x2+ 2x
(2x+ 1) 2
Example 2 Let us calculate the derivative of the function ln cos x.
Using the rule of differentiating composite functions and the formula for the logarithmic derivative from Paragraph 6.2.2-1, we get
(ln cos x) = 1
cos x (cos x)
= – tan x.
Example 3 Let us calculate the derivative of the function x x Using the rule of differentiating the
composite exponential function with u(x) = v(x) = x, we have
(x x) = x x ln x + xx x–1= x x (ln x +1).
* Let y = f (x) be a differentiable monotone function on the interval (a, b) and f (x0 ) ≠ 0, where x 0 (a, b) Then the inverse function x = g(y) is differentiable at the point y0= f (x0) and g (y0 ) = 1
f (x0 ).
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6.2.3 Theorems about Differentiable Functions L’Hospital Rule
6.2.3-1 Main theorems about differentiable functions
ROLLE THEOREM If the function y = f (x) is continuous on the segment [a, b], differ-entiable on the interval (a, b), and f (a) = f (b), then there is a point c (a, b) such that
f (c) =0
LAGRANGE THEOREM If the function y = f (x) is continuous on the segment [a, b] and differentiable on the interval (a, b), then there is a point c(a, b)such that
f (b) – f (a) = f (c)(b – a).
This relation is called the formula of finite increments.
CAUCHY THEOREM Let f (x) and g(x) be two functions that are continuous on the segment [a, b], differentiable on the interval (a, b), and g (x) ≠ 0for all x (a, b) Then there is a point c(a, b)such that
f (b) – f (a)
g (b) – g(a) =
f (c)
g (c).
6.2.3-2 L’Hospital’s rules on indeterminate expressions of the form0/0 and ∞/∞.
THEOREM1 Let f (x) and g(x) be two functions defined in a neighborhood of a point
a , vanishing at this point, f (a) = g(a) =0, and having the derivatives f (a) and g (a), with
g (a) ≠ 0 Then
lim
x→a
f (x)
g (x) =
f (a)
g (a).
Example 1 Let us calculate the limit lim
x→0
sin x
1– e– 2x.
Here, both the numerator and the denominator vanish for x =0 Let us calculate the derivatives
f (x) = (sin x) = cos x =⇒ f (0) = 1,
g (x) = (1 – e–2x)= 2e–2x =⇒ g (0) = 2 ≠ 0.
By the L’Hospital rule, we find that
lim
x→0
sin x
1– e– 2x = f
(0)
g (0) =
1
2. THEOREM2 Let f (x) and g(x) be two functions defined in a neighborhood of a point
a , vanishing at a, together with their derivatives up to the order n –1inclusively Suppose
also that the derivatives f(n) (a) and g(n) (a) exist and are finite, g(n) (a)≠ 0 Then
lim
x→a
f (x)
g (x) =
f(n) (a)
g(n) (a).
THEOREM3 Let f (x) and g(x) be differentiable functions and g (x)≠ 0in a
neighbor-hood of a point a (x≠a ) If f (x) and g(x) are infinitely small or infinitely large functions for x → a, i.e., the ratio f (x)
g (x) at the point a is an indeterminate expression of the form 0
0
or ∞
∞, then
lim
x→a
f (x)
g (x) = limx→a
f (x)
g (x)
(provided that there exists a finite or infinite limit of the ratio of the derivatives)
Remark. The L’Hospital rule 3 is applicable also in the case of a being one of the symbols ∞, +∞, –∞.