Handbook of mathematics for engineers and scienteists part 41 pot

248 LIMITS AND DERIVATIVES 3. Let f (x)beafunctiononafinite segment [a, b] satisfying the Lipschitz condition   f(x 1 )–f(x 2 )   ≤ L|x 1 – x 2 |, for any x 1 and x 2 in [a, b], where L is a constant. Then f(x) has bounded variation and b V a f(x) ≤ L(b – a). 4. Let f(x)beafunctiononafinite segment [a, b] with a bounded derivative |f  (x)| ≤ L, where L = const. Then, f(x) is of bounded variation and b V a f(x) ≤ L(b – a). 5. Let f(x)beafunctionon[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 V a f(x)=  b a |ϕ(x)| dx. C OROLLARY. 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)andg(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.Iff(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 V a f(x)= c V a f(x)+ b V c 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 V a 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→x 0 –0 f(x) and a right-hand limit lim x→x 0 +0 f(x) at any point x 0 [a, b]. 6.1. BASIC CONCEPTS OF MAT H E M AT I CAL ANALYSIS 249 6.1.7-4. 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 function Φ(x) such that for all x 1 , x 2 [a, b](x 1 < x 2 ), the following inequality holds: |f(x 2 )–f(x 1 )| ≤ Φ(x 2 )–Φ(x 1 ). 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)=g 2 (x)–g 1 (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 x 0 (a < x 0 < b), then the function F (x)= x V 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 = x 0 < x 1 < x 2 < ··· < 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,weget lim λ→0 v = b V a 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)asn →∞if for any fixed x X, 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)onX as n →∞if for any ε > 0 there is an integer N = N(ε) and such that for all n > N and all x X, 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)asn →∞,bydefinition, for any ε > 0 and any x X,thereis 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 > 0 such that for any N > 0 there is k N > N and x N X such that |f k N (x N )–f(x N )| ≥ δ), then one says that the sequence {f n (x)} converges nonuniformly to f(x)onthesetX. 250 LIMITS AND DERIVATIVES 6.1.8-2. Basic theorems. Let X be an interval on the real axis. T HEOREM. 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 n (x)} = {x n } converges to f (x) ≡ 0 as 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 X R uniformly converges to f(x) as n →∞ if and only if for any ε > 0 there 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 x X . 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)asn →∞. 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 ba yfx= () yfx= ()+ε yfx=-() ε yfx= () n Figure 6.3. Geometrical meaning of uniform convergence of a sequence of functions {f n (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 = lim Δx→0 f(x + Δx)–f (x) Δx , where Δy = f(x+Δx)–f(x) is the increment of the function corresponding to the increment of the argument Δx. The derivative y  is also denoted by y  x , ˙y, dy dx , f  (x), df (x) dx . 6.2. DIFFERENTIAL CALCULUS FOR FUNCTIONS OF A SINGLE VARIABLE 251 Example 1. Let us calculate the derivative of the function f(x)=x 2 . By definition, we have f  (x) = lim Δx→0 (x + Δx) 2 – x 2 Δx = lim Δx→0 (2x + Δx)=2x. The increment Δx 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 x D 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 x D 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,sothat dy = f  (x)dx, Δy = dy + o(dx). The approximate relation Δy ≈ 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 ◦ .Lety = 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 ◦ .Thetangent line or simply the tangent to the graph of the function y = f (x) at a point M(x 0 , y 0 ), where y 0 = f(x 0 ), is defined as the straight line determined by the limit position of the secant MN 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  (x 0 )=tanα is the slope ratio of the tangential line (Fig. 6.4). O x y y dy M N α α Δy 0 yfx= () x 0 x+ xΔ 0 Figure 6.4. The tangent to the graph of a function y = f(x) at a point (x 0 , y 0 ). Equation of the tangent line to the graph of a function y = f (x) at a point (x 0 , y 0 ): y – y 0 = f  (x 0 )(x – x 0 ). 252 LIMITS AND DERIVATIVES Equation of the normal to the graph of a function y = f(x) at a point (x 0 , y 0 ): y – y 0 =– 1 f  (x 0 ) (x – x 0 ). 6.2.1-3. One-sided derivatives. One-sided derivatives are defined as follows: f  + (x) = lim Δx→+0 Δy Δx = lim Δx→+0 f(x + Δx)–f (x) Δx right-hand derivative, f  – (x) = lim Δx→–0 Δy Δx = lim Δx→–0 f(x + Δx)–f(x) Δx left-hand derivative. Example 2. The function y = |x| at the point x = 0 has 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 = x 0 and has equal one-sided derivatives at that point, y  + (x 0 )=y  – (x 0 )=a. Then this function has a derivative at x = x 0 and y  (x 0 )=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 ,(log a x)  = 1 x ln a , (sin x)  =cosx,(cosx)  =–sinx, (tan x)  = 1 cos 2 x ,(cotx)  =– 1 sin 2 x , (arcsin x)  = 1 √ 1 – x 2 , (arccos x)  =– 1 √ 1 – x 2 , (arctan x)  = 1 1 + x 2 , (arccot x)  =– 1 1 + x 2 , (sinh x)  =coshx,(coshx)  =sinhx, (tanh x)  = 1 cosh 2 x ,(cothx)  =– 1 sinh 2 x , (arcsinh x)  = 1 √ 1 + x 2 , (arccosh x)  = 1 √ x 2 – 1 , (arctanh x)  = 1 1 – x 2 , (arccoth x)  = 1 x 2 – 1 . 6.2. DIFFERENTIAL CALCULUS FOR FUNCTIONS OF A SINGLE VARIABLE 253 6.2.2-2. 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) v 2 (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 x F y (F x and F y are 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–1 u  . 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 v are 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  x 2 2x + 1   = (x 2 )  (2x + 1)–x 2 (2x + 1)  (2x + 1) 2 = 2x(2x + 1)–2x 2 (2x + 1) 2 = 2x 2 + 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)  =–tanx. 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,wehave (x x )  = x x ln x + xx x–1 = x x (ln x + 1). *Lety = f(x) be a differentiable monotone function on the interval (a, b)andf  (x 0 ) ≠ 0,wherex 0 (a, b). Then the inverse function x = g(y) is differentiable at the point y 0 = f(x 0 )and g  (y 0 )= 1 f  (x 0 ) . 254 LIMITS AND DERIVATIVES 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. C AUCHY 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) ≠ 0 for 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 form 0/0 and ∞ /∞. THEOREM 1. 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)=(sinx)  =cosx =⇒ 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 . THEOREM 2. 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 – 1 inclusively. 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) . T HEOREM 3. Let f(x) and g(x) be differentiable functions and g  (x) ≠ 0 in 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) = lim x→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 ∞,+∞,–∞. . Geometrical meaning of uniform convergence of a sequence of functions {f n (x)} to a continuous function f (x). 6.2. Differential Calculus for Functions of a Single Variable 6.2.1. Derivative and Differential,. expressions of the form 0/0 and ∞ /∞. THEOREM 1. 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) ,. sequence of functions {f n (x)} defined on a set X R uniformly converges to f(x) as n →∞ if and only if for any ε > 0 there is an integer N = N(ε)>0 such that for all n > N and m >