# §1.4 Calculus of One Variable

## §1.4(i) Monotonicity

If $f(x_{1})\leq f(x_{2})$ for every pair $x_{1}$, $x_{2}$ in an interval $I$ such that $x_{1}, then $f(x)$ is nondecreasing on $I$. If the $\leq$ 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 $f(c+)\equiv\lim_{x\to c+}f(x)=f(c),$ Referenced by: §1.4(ii) Permalink: http://dlmf.nist.gov/1.4.E1 Encodings: TeX, pMML, png See also: Annotations for 1.4(ii)

that is, for every arbitrarily small positive constant $\epsilon$ there exists $\delta$ ($>0$) such that

 1.4.2 $|f(c+\alpha)-f(c)|<\epsilon,$ Permalink: http://dlmf.nist.gov/1.4.E2 Encodings: TeX, pMML, png See also: Annotations for 1.4(ii)

for all $\alpha$ such that $0\leq\alpha<\delta$. Similarly, it is continuous on the left (or from below) at $x=c$ if

 1.4.3 $f(c-)\equiv\lim_{x\to c-}f(x)=f(c).$ Referenced by: §1.4(ii) Permalink: http://dlmf.nist.gov/1.4.E3 Encodings: TeX, pMML, png See also: Annotations for 1.4(ii)

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 $c\in(a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in\mathop{C\/}\nolimits(a,b)$. 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 $f(x)\in\mathop{C\/}\nolimits[a,b]$.

A removable singularity of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. For example, $f(x)=(\mathop{\sin\/}\nolimits x)/x$ with $c=0$.

A simple discontinuity of $f(x)$ at $x=c$ occurs when $f(c+)$ and $f(c-)$ exist, but $f(c+)\not=f(c-)$. 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

## §1.4(iii) Derivatives

The derivative $f^{\prime}(x)$ of $f(x)$ is defined by

 1.4.4 $f^{\prime}(x)=\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{% h}.$ Defines: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ Referenced by: §1.8(ii) Permalink: http://dlmf.nist.gov/1.4.E4 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)

When this limit exists $f$ is differentiable at $x$.

 1.4.5 $(f+g)^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x),$ A&S Ref: 3.3.2 Permalink: http://dlmf.nist.gov/1.4.E5 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)
 1.4.6 $(fg)^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x),$ A&S Ref: 3.3.3 Permalink: http://dlmf.nist.gov/1.4.E6 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)
 1.4.7 $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)% }{(g(x))^{2}}.$ A&S Ref: 3.3.4 Permalink: http://dlmf.nist.gov/1.4.E7 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)

### Higher Derivatives

 1.4.8 $f^{(2)}(x)=\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}x}^{2}}=\frac{\mathrm{d}}{% \mathrm{d}x}\left(\frac{\mathrm{d}f}{\mathrm{d}x}\right),$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ Permalink: http://dlmf.nist.gov/1.4.E8 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)
 1.4.9 $f^{(n)}=f^{(n)}(x)=\frac{\mathrm{d}}{\mathrm{d}x}f^{(n-1)}(x).$

If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in\mathop{C^{n}\/}\nolimits(I)$. When $n\geq 1$, $f$ is continuously differentiable on $I$. When $n$ is unbounded, $f$ is infinitely differentiable on $I$ and we write $f\in\mathop{C^{\infty}\/}\nolimits(I)$.

### Chain Rule

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

 1.4.10 $h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x).$ A&S Ref: 3.3.5 Permalink: http://dlmf.nist.gov/1.4.E10 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)

### Maxima and Minima

A necessary condition that a differentiable function $f(x)$ has a local maximum (minimum) at $x=c$, that is, $f(x)\leq f(c)$, ($f(x)\geq f(c)$) in a neighborhood $c-\delta\leq x\leq c+\delta$ ($\delta>0$) of $c$, is $f^{\prime}(c)=0$.

### Mean Value Theorem

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

 1.4.11 $f(b)-f(a)=(b-a)f^{\prime}(c).$ Permalink: http://dlmf.nist.gov/1.4.E11 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)

If $f^{\prime}(x)\geq 0$ ($\leq 0$) ($=0$) for all $x\in(a,b)$, then $f$ is nondecreasing (nonincreasing) (constant) on $(a,b)$.

### Leibniz’s Formula

 1.4.12 $(fg)^{(n)}=f^{(n)}g+\binom{n}{1}f^{(n-1)}g^{\prime}+\dots+\binom{n}{k}f^{(n-k)% }g^{(k)}+\dots+fg^{(n)}.$ Symbols: $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient, $k$: integer and $n$: nonnegative integer A&S Ref: 3.3.8 Permalink: http://dlmf.nist.gov/1.4.E12 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)

### Faà Di Bruno’s Formula

 1.4.13 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}f(g(x))=\sum\left(\frac{n!}{m_{1}!m_% {2}!\cdots m_{n}!}\right)f^{(k)}(g(x))\*\left(\frac{g^{\prime}(x)}{1!}\right)^% {m_{1}}\left(\frac{g^{\prime\prime}(x)}{2!}\right)^{m_{2}}\dots\left(\frac{g^{% (n)}(x)}{n!}\right)^{m_{n}},$

where the sum is over all nonnegative integers $m_{1},m_{2},\dots,m_{n}$ that satisfy $m_{1}+2m_{2}+\dots+nm_{n}=n$, and $k=m_{1}+m_{2}+\dots+m_{n}$.

### L’Hôpital’s Rule

If

 1.4.14 $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}g(x)=0\;\;\mbox{(or \infty)},$ Permalink: http://dlmf.nist.gov/1.4.E14 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)

then

 1.4.15 $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)},$ A&S Ref: 3.4.1 Permalink: http://dlmf.nist.gov/1.4.E15 Encodings: TeX, pMML, png See also: Annotations for 1.4(iii)

when the last limit exists.

## §1.4(iv) Indefinite Integrals

If $F^{\prime}(x)=f(x)$, then $\int f\mathrm{d}x=F(x)+C$, where $C$ is a constant.

### Integration by Parts

 1.4.16 $\int fg\mathrm{d}x=\left(\int f\mathrm{d}x\right)g-\int\left(\int f\mathrm{d}x% \right)\frac{\mathrm{d}g}{\mathrm{d}x}\mathrm{d}x.$
 1.4.17 $\int x^{n}\mathrm{d}x=\begin{cases}\dfrac{x^{n+1}}{n+1}+C,&\quad n\not=-1,\\ \mathop{\ln\/}\nolimits\left|x\right|+C,&\quad n=-1.\end{cases}$

For the function $\mathop{\ln\/}\nolimits$ 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=x_{0}, and $\xi_{j}$ denote any point in $[x_{j},x_{j+1}]$, $j=0,1,\dots,n-1$. Then

 1.4.18 $\int^{b}_{a}f(x)\mathrm{d}x=\lim\sum^{n-1}_{j=0}f(\xi_{j})(x_{j+1}-x_{j})$

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

 1.4.19 $\int^{b}_{a}(cf(x)+dg(x))\mathrm{d}x=c\int^{b}_{a}f(x)\mathrm{d}x+d\int^{b}_{a% }g(x)\mathrm{d}x,$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E19 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

$c$ and $d$ constants.

 1.4.20 $\int^{b}_{a}f(x)\mathrm{d}x=-\int^{a}_{b}f(x)\mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E20 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)
 1.4.21 $\int^{b}_{a}f(x)\mathrm{d}x=\int^{c}_{a}f(x)\mathrm{d}x+\int^{b}_{c}f(x)% \mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E21 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

### Infinite Integrals

 1.4.22 $\int^{\infty}_{a}f(x)\mathrm{d}x=\lim_{b\to\infty}\int^{b}_{a}f(x)\mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §1.4(v), §1.5(v) Permalink: http://dlmf.nist.gov/1.4.E22 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

Similarly for $\int^{a}_{-\infty}$. Next, if $f(b)=\pm\infty$, then

 1.4.23 $\int^{b}_{a}f(x)\mathrm{d}x=\lim_{c\to b-}\int^{c}_{a}f(x)\mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §1.4(v), §1.5(v) Permalink: http://dlmf.nist.gov/1.4.E23 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

Similarly when $f(a)=\pm\infty$.

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 $c\in(a,b)$ and assume that $\int_{a}^{c-\epsilon}f(x)\mathrm{d}x$ and $\int_{c+\epsilon}^{b}f(x)\mathrm{d}x$ exist when $0<\epsilon<\min(c-a,b-c)$, but not necessarily when $\epsilon=0$. Then we define

 1.4.24 $\pvint^{b}_{a}f(x)\mathrm{d}x=P\int^{b}_{a}f(x)\mathrm{d}x=\lim_{\epsilon\to 0% +}\left(\int^{c-\epsilon}_{a}f(x)\mathrm{d}x+\int^{b}_{c+\epsilon}f(x)\mathrm{% d}x\right),$ Defines: $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E24 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

when this limit exists.

Similarly, assume that $\int_{-b}^{b}f(x)\mathrm{d}x$ exists for all finite values of $b$ ($>0$), but not necessarily when $b=\infty$. Then we define

 1.4.25 $\pvint^{\infty}_{-\infty}f(x)\mathrm{d}x=P\int^{\infty}_{-\infty}f(x)\mathrm{d% }x=\lim_{b\to\infty}\int^{b}_{-b}f(x)\mathrm{d}x,$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Referenced by: §1.14(i) Permalink: http://dlmf.nist.gov/1.4.E25 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

when this limit exists.

### Fundamental Theorem of Calculus

For $F^{\prime}(x)=f(x)$ with $f(x)$ continuous,

 1.4.26 $\int^{b}_{a}f(x)\mathrm{d}x=F(b)-F(a),$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E26 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)
 1.4.27 $\frac{\mathrm{d}}{\mathrm{d}x}\int^{x}_{a}f(t)\mathrm{d}t=f(x).$

### Change of Variables

If $\phi^{\prime}(x)$ is continuous or piecewise continuous, then

 1.4.28 $\int^{b}_{a}f(\phi(x))\phi^{\prime}(x)\mathrm{d}x=\int^{\phi(b)}_{\phi(a)}f(t)% \mathrm{d}t.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\phi(x)$: function Permalink: http://dlmf.nist.gov/1.4.E28 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

### First Mean Value Theorem

For $f(x)$ continuous and $\phi(x)\geq 0$ and integrable on $[a,b]$, there exists $c\in[a,b]$, such that

 1.4.29 $\int^{b}_{a}f(x)\phi(x)\mathrm{d}x=f(c)\int^{b}_{a}\phi(x)\mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\phi(x)$: function Permalink: http://dlmf.nist.gov/1.4.E29 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

### Second Mean Value Theorem

For $f(x)$ monotonic and $\phi(x)$ integrable on $[a,b]$, there exists $c\in[a,b]$, such that

 1.4.30 $\int^{b}_{a}f(x)\phi(x)\mathrm{d}x=f(a)\int^{c}_{a}\phi(x)\mathrm{d}x+f(b)\int% ^{b}_{c}\phi(x)\mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\phi(x)$: function Permalink: http://dlmf.nist.gov/1.4.E30 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

### Repeated Integrals

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

 1.4.31 $\int_{a}^{b}\mathrm{d}x_{n}\int_{a}^{x_{n}}\mathrm{d}x_{n-1}\cdots\int_{a}^{x_% {2}}\mathrm{d}x_{1}\int_{a}^{x_{1}}f(x)\mathrm{d}x=\frac{1}{n!}\int_{a}^{b}(b-% x)^{n}f(x)\mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral and $n$: nonnegative integer Referenced by: §1.15(vi), §1.4(v) Permalink: http://dlmf.nist.gov/1.4.E31 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

### Square-Integrable Functions

A function $f(x)$ is square-integrable if

 1.4.32 $\|f\|^{2}_{2}\equiv\int^{b}_{a}|f(x)|^{2}\mathrm{d}x<\infty.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E32 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

### Functions of Bounded Variation

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

 1.4.33 $\mathop{\mathcal{V}_{a,b}\/}\nolimits\!\left(f\right)=\sup\sum^{n}_{j=1}|f(x_{% j})-f(x_{j-1})|,$ Defines: $\mathop{\mathcal{V}\/}\nolimits\!\left(\NVar{f}\right)$: total variation and $\mathop{\mathcal{V}_{\NVar{a,b}}\/}\nolimits\!\left(\NVar{f}\right)$: total variation Symbols: $\sup$: least upper bound (supremum), $j$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.4.E33 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

where the supremum is over all sets of points $x_{0} in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. If $\mathop{\mathcal{V}_{a,b}\/}\nolimits\!\left(f\right)<\infty$, then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)=\mathop{\mathcal{V}_{a,x}\/}\nolimits\!\left(f\right)$ and $h(x)=\mathop{\mathcal{V}_{a,x}\/}\nolimits\!\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$.

If $f(x)$ is continuous on the closure of $(a,b)$ and $f^{\prime}(x)$ is continuous on $(a,b)$, then

 1.4.34 $\mathop{\mathcal{V}_{a,b}\/}\nolimits\!\left(f\right)=\int^{b}_{a}|f^{\prime}(% x)\mathrm{d}x|,$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\mathop{\mathcal{V}_{\NVar{a,b}}\/}\nolimits\!\left(\NVar{f}\right)$: total variation Referenced by: §1.4(v), §1.4(v) Permalink: http://dlmf.nist.gov/1.4.E34 Encodings: TeX, pMML, png See also: Annotations for 1.4(v)

whenever this integral exists.

Lastly, whether or not the real numbers $a$ and $b$ satisfy $a, and whether or not they are finite, we define $\mathop{\mathcal{V}_{a,b}\/}\nolimits\!\left(f\right)$ 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 $f(x)\in\mathop{C^{n+1}\/}\nolimits[a,b]$, then

 1.4.35 $f(x)=\sum^{n}_{k=0}\frac{f^{(k)}(a)}{k!}(x-a)^{k}+R_{n},$ Symbols: $!$: factorial (as in $n!$), $k$: integer, $n$: nonnegative integer and $R_{n}$: remainder A&S Ref: 3.6.4 Permalink: http://dlmf.nist.gov/1.4.E35 Encodings: TeX, pMML, png See also: Annotations for 1.4(vi)
 1.4.36 $R_{n}=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1},$ $a, Symbols: $!$: factorial (as in $n!$), $n$: nonnegative integer and $R_{n}$: remainder A&S Ref: 3.6.5 Permalink: http://dlmf.nist.gov/1.4.E36 Encodings: TeX, pMML, png See also: Annotations for 1.4(vi)

and

 1.4.37 $R_{n}=\frac{1}{n!}\int^{x}_{a}(x-t)^{n}f^{(n+1)}(t)\mathrm{d}t.$

## §1.4(vii) Maxima and Minima

If $f(x)$ is twice-differentiable, and if also $f^{\prime}(x_{0})=0$ and $f^{\prime\prime}(x_{0})<0$ ($>0$), then $x=x_{0}$ 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 $f((1-t)c+td)\leq(1-t)f(c)+tf(d)$ Permalink: http://dlmf.nist.gov/1.4.E38 Encodings: TeX, pMML, png See also: Annotations for 1.4(viii)

for any $c,d\in(a,b)$, and $t\in[0,1]$. 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 $f^{\prime\prime}(x)\geq 0$ on $(a,b)$. A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.