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§1.4 Calculus of One Variable

Contents

§1.4(i) Monotonicity

If f(x1)f(x2) for every pair x1, x2 in an interval I such that x1<x2, then f(x) is nondecreasing on I. If the sign is replaced by <, then f(x) is increasing (also called strictly increasing) on I. Similarly for nonincreasing and decreasing (strictly decreasing) functions. Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases.

§1.4(ii) Continuity

A function f(x) is continuous on the right (or from above) at x=c if

1.4.1 f(c+)limxc+f(x)=f(c),

that is, for every arbitrarily small positive constant ϵ there exists δ (>0) such that

1.4.2 |f(c+α)-f(c)|<ϵ,

for all α such that 0α<δ. Similarly, it is continuous on the left (or from below) at x=c if

1.4.3 f(c-)limxc-f(x)=f(c).

And f(x) is continuous at c when both (1.4.1) and (1.4.3) apply.

If f(x) is continuous at each point c(a,b), then f(x) is continuous on the interval (a,b) and we write fC(a,b). If also f(x) is continuous on the right at x=a, and continuous on the left at x=b, then f(x) is continuous on the interval [a,b], and we write f(x)C[a,b].

A removable singularity of f(x) at x=c occurs when f(c+)=f(c-) but f(c) is undefined. For example, f(x)=(sinx)/x with c=0.

A simple discontinuity of f(x) at x=c occurs when f(c+) and f(c-) exist, but f(c+)f(c-). If f(x) is continuous on an interval I save for a finite number of simple discontinuities, then f(x) is piecewise (or sectionally) continuous on I. For an example, see Figure 1.4.1

See accompanying text
Figure 1.4.1: Piecewise continuous function on [a,b). Magnify

§1.4(iii) Derivatives

The derivative f(x) of f(x) is defined by

1.4.4 f(x)=dfdx=limh0f(x+h)-f(x)h.

When this limit exists f is differentiable at x.

1.4.5 (f+g)(x)=f(x)+g(x),
1.4.6 (fg)(x)=f(x)g(x)+f(x)g(x),
1.4.7 (fg)(x)=f(x)g(x)-f(x)g(x)(g(x))2.

Higher Derivatives

1.4.8 f(2)(x)=d2fdx2=ddx(dfdx),
1.4.9 f(n)=f(n)(x)=ddxf(n-1)(x).

If f(n) exists and is continuous on an interval I, then we write fCn(I). When n1, f is continuously differentiable on I. When n is unbounded, f is infinitely differentiable on I and we write fC(I).

Chain Rule

For h(x)=f(g(x)),

1.4.10 h(x)=f(g(x))g(x).

Maxima and Minima

A necessary condition that a differentiable function f(x) has a local maximum (minimum) at x=c, that is, f(x)f(c), (f(x)f(c)) in a neighborhood c-δxc+δ (δ>0) of c, is f(c)=0.

Mean Value Theorem

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists a point c(a,b) such that

1.4.11 f(b)-f(a)=(b-a)f(c).

If f(x)0 (0) (=0) for all x(a,b), then f is nondecreasing (nonincreasing) (constant) on (a,b).

Leibniz’s Formula

1.4.12 (fg)(n)=f(n)g+(n1)f(n-1)g++(nk)f(n-k)g(k)++fg(n).

Faà Di Bruno’s Formula

1.4.13 dndxnf(g(x))=(n!m1!m2!mn!)f(k)(g(x))(g(x)1!)m1(g′′(x)2!)m2(g(n)(x)n!)mn,

where the sum is over all nonnegative integers m1,m2,,mn that satisfy m1+2m2++nmn=n, and k=m1+m2++mn.

L’Hôpital’s Rule

If

1.4.14 limxaf(x)=limxag(x)=0(or ),

then

1.4.15 limxaf(x)g(x)=limxaf(x)g(x),

when the last limit exists.

§1.4(iv) Indefinite Integrals

If F(x)=f(x), then fdx=F(x)+C, where C is a constant.

Integration by Parts

1.4.16 fgdx=(fdx)g-(fdx)dgdxdx.
1.4.17 xndx={xn+1n+1+C,n-1,ln|x|+C,n=-1.

For the function ln see §4.2(i).

See §§4.10, 4.26(ii), 4.26(iv), 4.40(ii), and 4.40(iv) for indefinite integrals involving the elementary functions.

For extensive tables of integrals, see Apelblat (1983), Bierens de Haan (1867), Gradshteyn and Ryzhik (2000), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

§1.4(v) Definite Integrals

Suppose f(x) is defined on [a,b]. Let a=x0<x1<<xn=b, and ξj denote any point in [xj,xj+1], j=0,1,,n-1. Then

1.4.18 abf(x)dx=limj=0n-1f(ξj)(xj+1-xj)

as max(xj+1-xj)0. Continuity, or piecewise continuity, of f(x) on [a,b] is sufficient for the limit to exist.

1.4.19 ab(cf(x)+dg(x))dx=cabf(x)dx+dabg(x)dx,

c and d constants.

1.4.20 abf(x)dx=-baf(x)dx.
1.4.21 abf(x)dx=acf(x)dx+cbf(x)dx.

Infinite Integrals

1.4.22 af(x)dx=limbabf(x)dx.

Similarly for -a. Next, if f(b)=±, then

1.4.23 abf(x)dx=limcb-acf(x)dx.

Similarly when f(a)=±.

When the limits in (1.4.22) and (1.4.23) exist, the integrals are said to be convergent. If the limits exist with f(x) replaced by |f(x)|, then the integrals are absolutely convergent. Absolute convergence also implies convergence.

Cauchy Principal Values

Let c(a,b) and assume that ac-ϵf(x)dx and c+ϵbf(x)dx exist when 0<ϵ<min(c-a,b-c), but not necessarily when ϵ=0. Then we define

1.4.24 abf(x)dx=Pabf(x)dx=limϵ0+(ac-ϵf(x)dx+c+ϵbf(x)dx),

when this limit exists.

Similarly, assume that -bbf(x)dx exists for all finite values of b (>0), but not necessarily when b=. Then we define

1.4.25 -f(x)dx=P-f(x)dx=limb-bbf(x)dx,

when this limit exists.

Fundamental Theorem of Calculus

For F(x)=f(x) with f(x) continuous,

1.4.26 abf(x)dx=F(b)-F(a),
1.4.27 ddxaxf(t)dt=f(x).

Change of Variables

If ϕ(x) is continuous or piecewise continuous, then

1.4.28 abf(ϕ(x))ϕ(x)dx=ϕ(a)ϕ(b)f(t)dt.

First Mean Value Theorem

For f(x) continuous and ϕ(x)0 and integrable on [a,b], there exists c[a,b], such that

1.4.29 abf(x)ϕ(x)dx=f(c)abϕ(x)dx.

Second Mean Value Theorem

For f(x) monotonic and ϕ(x) integrable on [a,b], there exists c[a,b], such that

1.4.30 abf(x)ϕ(x)dx=f(a)acϕ(x)dx+f(b)cbϕ(x)dx.

Repeated Integrals

If f(x) is continuous or piecewise continuous on [a,b], then

1.4.31 abdxnaxndxn-1ax2dx1ax1f(x)dx=1n!ab(b-x)nf(x)dx.

Square-Integrable Functions

A function f(x) is square-integrable if

1.4.32 f22ab|f(x)|2dx<.

Functions of Bounded Variation

With a<b, the total variation of f(x) on a finite or infinite interval (a,b) is

1.4.33 𝒱a,b(f)=supj=1n|f(xj)-f(xj-1)|,

where the supremum is over all sets of points x0<x1<<xn in the closure of (a,b), that is, (a,b) with a,b added when they are finite. If 𝒱a,b(f)<, then f(x) is of bounded variation on (a,b). In this case, g(x)=𝒱a,x(f) and h(x)=𝒱a,x(f)-f(x) are nondecreasing bounded functions and f(x)=g(x)-h(x).

If f(x) is continuous on the closure of (a,b) and f(x) is continuous on (a,b), then

1.4.34 𝒱a,b(f)=ab|f(x)dx|,

whenever this integral exists.

Lastly, whether or not the real numbers a and b satisfy a<b, and whether or not they are finite, we define 𝒱a,b(f) by (1.4.34) whenever this integral exists. This definition also applies when f(x) is a complex function of the real variable x. For further information on total variation see Olver (1997b, pp. 27–29).

§1.4(vi) Taylor’s Theorem for Real Variables

If f(x)Cn+1[a,b], then

1.4.35 f(x)=k=0nf(k)(a)k!(x-a)k+Rn,
1.4.36 Rn=f(n+1)(c)(n+1)!(x-a)n+1,
a<c<x,

and

1.4.37 Rn=1n!ax(x-t)nf(n+1)(t)dt.

§1.4(vii) Maxima and Minima

If f(x) is twice-differentiable, and if also f(x0)=0 and f′′(x0)<0 (>0), then x=x0 is a local maximum (minimum) (§1.4(iii)) of f(x). The overall maximum (minimum) of f(x) on [a,b] will either be at a local maximum (minimum) or at one of the end points a or b.

§1.4(viii) Convex Functions

A function f(x) is convex on (a,b) if

1.4.38 f((1-t)c+td)(1-t)f(c)+tf(d)

for any c,d(a,b), and t[0,1]. See Figure 1.4.2. A similar definition applies to closed intervals [a,b].

If f(x) is twice differentiable, then f(x) is convex iff f′′(x)0 on (a,b). A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.

See accompanying text
Figure 1.4.2: Convex function f(x). g(t)=f((1-t)c+td), l(t)=(1-t)f(c)+tf(d), c,d(a,b), 0t1. Magnify