MISCELLANEOUS EXAMPLES ON CHAPTER VII

1. Verify the terms given of the following Taylor’s Series: $$\begin{array}{rlrl}& (1)& \tan x& =x+\frac{1}{3}{x}^3+\frac{2}{15}{x}^5+\dots ,\\ & (2)& \sec x& =1+\frac{1}{2}{x}^2+\frac{5}{24}{x}^4+\dots ,\\ & (3)& x\csc x& =1+\frac{1}{6}{x}^2+\frac{7}{360}{x}^4+\dots ,\\ & (4)& x\cot x& =1–\frac{1}{3}{x}^2–\frac{1}{45}{x}^4–\dots .\end{array}$$

 

2. Show that if $f(x)$ and its first $n+2$ derivatives are continuous, and $f^{(n+1)}(0)\ne 0$, and ${\theta }_n$ is the value of $\theta$ which occurs in Lagrange’s form of the remainder after $n$ terms of Taylor’s Series, then $${\theta }_n=\frac{1}{n+1}+\frac{n}{2(n+1{)}^2(n+2)}\{\frac{f^{(n+2)}(0)}{f^{(n+1)}(0)}+{ϵ}_x\}x,$$ where ${ϵ}_x\to 0$ as $x\to 0$. [Follow the method of Ex. LV. 12.]

 

3. Verify the last result when $f(x)=1/(1+x)$. [Here $(1+{\theta }_{n}x{)}^{n+1}=1+x$.]

 

4. Show that if $f(x)$ has derivatives of the first three orders then $$f(b)=f(a)+\frac{1}{2}(b–a)\{f^{\prime }(a)+f^{\prime }(b)\}–\frac{1}{12}(b–a{)}^{3}f{”}^{\prime }(\alpha ),$$ where $a<\alpha <b$. [Apply to the function $$\begin{array}{c}f(x)–f(a)–\frac{1}{2}(x–a)\{f^{\prime }(a)+f^{\prime }(x)\}\\ –{\left(\frac{x–a}{b–a}\right)}^3[f(b)–f(a)–\frac{1}{2}(b–a)\{f^{\prime }(a)+f^{\prime }(b)\}]\end{array}$$ arguments similar to those of § 147.]

 

5. Show that under the same conditions $$f(b)=f(a)+(b–a)f^{\prime }\{\frac{1}{2}(a+b)\}+\frac{1}{24}(b–a{)}^{3}f{”}^{\prime }(\alpha ).$$

 

6. Show that if $f(x)$ has derivatives of the first five orders then $$f(b)=f(a)+\frac{1}{6}(b–a)[f^{\prime }(a)+f^{\prime }(b)+4f^{\prime }\{\frac{1}{2}(a+b)\}]–\frac{1}{2880}(b–a{)}^5{f}^{(5)}(\alpha ).$$

 

7. Show that under the same conditions $$f(b)=f(a)+\frac{1}{2}(b–a)\{f^{\prime }(a)+f^{\prime }(b)\}–\frac{1}{12}(b–a{)}^2\{f”(b)–f”(a)\}+\frac{1}{720}(b–a{)}^5{f}^{(5)}(\alpha ).$$

 

8. Establish the formulae

$$(i)\left|\begin{array}{cc}f(a)& f(b)\\ g(a)& g(b)\end{array}\right|=(b–a)\left|\begin{array}{cc}f(a)& f^{\prime }(\beta )\\ g(a)& g^{\prime }(\beta )\end{array}\right|$$

where $\beta$ lies between $a$ and $b$, and

$$(ii)\left|\begin{array}{ccc}f(a)& f(b)& f(c)\\ g(a)& g(b)& g(c)\\ h(a)& h(b)& h(c)\end{array}\right|=\frac{1}{2}(b–c)(c–a)(a–b)\left|\begin{array}{ccc}f(a)& f^{\prime }(\beta )& f”(\gamma )\\ g(a)& g^{\prime }(\beta )& g”(\gamma )\\ h(a)& h^{\prime }(\beta )& h”(\gamma )\end{array}\right|$$

where $\beta$ and $\gamma$ lie between the least and greatest of $a$, $b$, $c$. [To prove (ii) consider the function $$\varphi (x)=\left|\begin{array}{ccc}f(a)& f(b)& f(x)\\ g(a)& g(b)& g(x)\\ h(a)& h(b)& h(x)\end{array}\right|–\frac{(x–a)(x–b)}{(c–a)(c–b)}\left|\begin{array}{ccc}f(a)& f(b)& f(c)\\ g(a)& g(b)& g(c)\\ h(a)& h(b)& h(c)\end{array}\right|,$$ which vanishes when $x=a$, $x=b$, and $x=c$. Its first derivative, by Theorem B of § 121, must vanish for two distinct values of $x$ lying between the least and greatest of $a$, $b$, $c$; and its second derivative must therefore vanish for a value $\gamma$ of $x$ satisfying the same condition. We thus obtain the formula $$\left|\begin{array}{ccc}f(a)& f(b)& f(c)\\ g(a)& g(b)& g(c)\\ h(a)& h(b)& h(c)\end{array}\right|=\frac{1}{2}(c–a)(c–b)\left|\begin{array}{ccc}f(a)& f(b)& f”(\gamma )\\ g(a)& g(b)& g”(\gamma )\\ h(a)& h(b)& h”(\gamma )\end{array}\right|.$$ The reader will now complete the proof without difficulty.]

 

9. If $F(x)$ is a function which has continuous derivatives of the first $n$ orders, of which the first $n–1$ vanish when $x=0$, and $A\le F^{(n)}(x)\le B$ when $0\le x\le h$, then $A(x^n/n!)\le F(x)\le B(x^n/n!)$ when $0\le x\le h$.

Apply this result to $$f(x)–f(0)–x{f}^{\prime }(0)–\cdots –\frac{x^{n-1}}{(n–1)!}{f}^{(n-1)}(0),$$ and deduce Taylor’s Theorem.

 

10. If ${\mathrm{\Delta }}_h\varphi (x)=\varphi (x)–\varphi (x+h)$, ${\mathrm{\Delta }}_h^2\varphi (x)={\mathrm{\Delta }}_h\{{\mathrm{\Delta }}_h\varphi (x)\}$, and so on, and $\varphi (x)$ has derivatives of the first $n$ orders, then $${\mathrm{\Delta }}_h^n\varphi (x)=\sum _{r=0}^n(-1{)}^r(\genfrac{}{}{0ex}{}{n}{r})\varphi (x+rh)=(-h{)}^n{\varphi }^{(n)}(\xi ),$$ where $\xi$ lies between $x$ and $x+nh$. Deduce that if ${\varphi }^{(n)}(x)$ is continuous then $\{{\mathrm{\Delta }}_h^n\varphi (x)\}/h^n\to (-1{)}^n{\varphi }^{(n)}(x)$ as $h\to 0$. [This result has been stated already when $n=2$, in Ex. LV. 13.]

 

11. Deduce from Ex. 10 that $x^{n-m}{\mathrm{\Delta }}_h^n{x}^m\to m(m–1)\dots (m–n+1)h^n$ as $x\to \mathrm{\infty }$, $m$ being any rational number and $n$ any positive integer. In particular prove that $$x\sqrt{x}\{\sqrt{x}–2\sqrt{x+1}+\sqrt{x+2}\}\to -\frac{1}{4}.$$

 

12. Suppose that $y=\varphi (x)$ is a function of $x$ with continuous derivatives of at least the first four orders, and that $\varphi (0)=0$, ${\varphi }^{\prime }(0)=1$, so that $$y=\varphi (x)=x+a_2{x}^2+a_3{x}^3+(a_4+{ϵ}_x)x^4,$$ where ${ϵ}_x\to 0$ as $x\to 0$. Establish the formula $$x=\psi (y)=y–a_2{y}^2+(2a_2^2–a_3)y^3–(5a_2^3–5a_2{a}_3+a_4+{ϵ}_y)y^4,$$ where ${ϵ}_y\to 0$ as $y\to 0$, for that value of $x$ which vanishes with $y$; and prove that $$\frac{\varphi (x)\psi (x)–x^2}{x^4}\to a_2^2$$ as $x\to 0$.

 

13. The coordinates $(\xi ,\eta )$ of the centre of curvature of the curve $x=f(t)$, $y=F(t)$, at the point $(x,y)$, are given by $$-(\xi –x)/y^{\prime }=(\eta –y)/x^{\prime }=(x^{\prime 2}+y^{\prime 2})/(x^{\prime }y”–x”y^{\prime });$$ and the radius of curvature of the curve is $$(x^{\prime 2}+y^{\prime 2}{)}^{3/2}/(x^{\prime }y”–x”y^{\prime }),$$ dashes denoting differentiations with respect to $t$.

 

14. The coordinates $(\xi ,\eta )$ of the centre of curvature of the curve $27a{y}^2=4x^3$, at the point $(x,y)$, are given by $$3a(\xi +x)+2x^2=0,\eta =4y+(9ay)/x.$$

 

15. Prove that the circle of curvature at a point $(x,y)$ will have contact of the third order with the curve if $(1+y_1^2)y_3=3y_1{y}_2^2$ at that point. Prove also that the circle is the only curve which possesses this property at every point; and that the only points on a conic which possess the property are the extremities of the axes. [Cf. Ch. VI, Misc. Ex. 10 (iv).]

 

16. The conic of closest contact with the curve $y=a{x}^2+b{x}^3+c{x}^4+\cdots +k{x}^n$, at the origin, is $a^{3}y=a^4{x}^2+a^{2}bxy+(ac–b^2)y^2$. Deduce that the conic of closest contact at the point $(\xi ,\eta )$ of the curve $y=f(x)$ is $$18{\eta }_2^{3}T=9{\eta }_2^4(x–\xi {)}^2+6{\eta }_2^2{\eta }_3(x–\xi )T+(3{\eta }_2{\eta }_4–4{\eta }_3^2)T^2,$$ where $T=(y–\eta )–{\eta }_1(x–\xi )$.

 

17. Homogeneous functions.¹ If $u=x^{n}f(y/x,z/x,\dots )$ then $u$ is unaltered, save for a factor ${\lambda }^n$, when $x$, $y$, $z$, … are all increased in the ratio $\lambda :1$. In these circumstances $u$ is called a homogeneous function of degree $n$ in the variables $x$, $y$, $z$, …. Prove that if $u$ is homogeneous and of degree $n$ then $$x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}+\cdots =nu.$$ This result is known as Euler’s Theorem on homogeneous functions.

 

18. If $u$ is homogeneous and of degree $n$ then $\partial u/\partial x$, $\partial u/\partial y$, … are homogeneous and of degree $n–1$.

 

19. Let $f(x,y)=0$ be an equation in $x$ and $y$ (e.g. $x^n+y^n–x=0$), and let $F(x,y,z)=0$ be the form it assumes when made homogeneous by the introduction of a third variable $z$ in place of unity ( $x^n+y^n–x{z}^{n-1}=0$). Show that the equation of the tangent at the point $(\xi ,\eta )$ of the curve $f(x,y)=0$ is $$x{F}_{\xi }+y{F}_{\eta }+z{F}_{\zeta }=0,$$ where $F_{\xi }$, $F_{\eta }$, $F_{\zeta }$ denote the values of $F_x$, $F_y$, $F_z$ when $x=\xi$, $y=\eta$, $z=\zeta =1$.

 

20. Dependent and independent functions. Jacobians or functional determinants. Suppose that $u$ and $v$ are functions of $x$ and $y$ connected by an identical relation $$\begin{array}{}\text{(1)}& \varphi (u,v)=0.\end{array}$$

Differentiating (1) with respect to $x$ and $y$, we obtain $$\begin{array}{}\text{(2)}& \frac{\partial \varphi }{\partial u}\frac{\partial u}{\partial x}+\frac{\partial \varphi }{\partial v}\frac{\partial v}{\partial x}=0,\frac{\partial \varphi }{\partial u}\frac{\partial u}{\partial y}+\frac{\partial \varphi }{\partial v}\frac{\partial v}{\partial y}=0,\end{array}$$ and, eliminating the derivatives of $\varphi$, $$\begin{array}{}\text{(3)}& J=\left|\begin{array}{cc}{u}_x& u_y\\ v_x& v_y\end{array}\right|=u_x{v}_y–u_y{v}_x=0,\end{array}$$ where $u_x$, $u_y$, $v_x$, $v_y$ are the derivatives of $u$ and $v$ with respect to $x$ and $y$. This condition is therefore necessary for the existence of a relation such as (1). It can be proved that the condition is also sufficient; for this we must refer to Goursat’s Cours d’ Analyse, vol. i, pp. 125 et seq.

Two functions $u$ and $v$ are said to be dependent or independent according as they are or are not connected by such a relation as (1). It is usual to call $J$ the Jacobian or functional determinant of $u$ and $v$ with respect to $x$ and $y$, and to write $$J=\frac{\partial (u,v)}{\partial (x,y)}.$$

Similar results hold for functions of any number of variables. Thus three functions $u$, $v$, $w$ of three variables $x$, $y$, $z$ are or are not connected by a relation $\varphi (u,v,w)=0$ according as $$J=\left|\begin{array}{ccc}{u}_x& u_y& u_z\\ v_x& v_y& v_z\\ w_x& w_y& w_z\end{array}\right|=\frac{\partial (u,v,w)}{\partial (x,y,z)}$$ does or does not vanish for all values of $x$, $y$, $z$.

 

21. Show that $a{x}^2+2hxy+b{y}_2$ and $A{x}^2+2Hxy+B{y}^2$ are independent unless $a/A=h/H=b/B$.

 

22. Show that $a{x}^2+b{y}^2+c{z}^2+2fyz+2gzx+2hxy$ can be expressed as a product of two linear functions of $x$, $y$, and $z$ if and only if $$abc+2fgh–a{f}^2–b{g}^2–c{h}^2=0.$$

[Write down the condition that $px+qy+rz$ and $p^{\prime }x+q^{\prime }y+r^{\prime }z$ should be connected with the given function by a functional relation.]

 

23. If $u$ and $v$ are functions of $\xi$ and $\eta$, which are themselves functions of $x$ and $y$, then $$\frac{\partial (u,v)}{\partial (x,y)}=\frac{\partial (u,v)}{\partial (\xi ,\eta )}\frac{\partial (\xi ,\eta )}{\partial (x,y)}.$$ Extend the result to any number of variables.

 

24. Let $f(x)$ be a function of $x$ whose derivative is $1/x$ and which vanishes when $x=1$. Show that if $u=f(x)+f(y)$, $v=xy$, then $u_x{v}_y–u_y{v}_x=0$, and hence that $u$ and $v$ are connected by a functional relation. By putting $y=1$, show that this relation must be $f(x)+f(y)=f(xy)$. Prove in a similar manner that if the derivative of $f(x)$ is $1/(1+x^2)$, and $f(0)=0$, then $f(x)$ must satisfy the equation $$f(x)+f(y)=f\left(\frac{x+y}{1–xy}\right).$$

 

25. Prove that if $f(x)={\int }_0^x\frac{dt}{\sqrt{1–t^4}}$ then $$f(x)+f(y)=f\left\{\frac{x\sqrt{1–y^4}+y\sqrt{1–x^4}}{1+x^2{y}^2}\right\}.$$

 

26. Show that if a functional relation exists between $$u=f(x)+f(y)+f(z),v=f(y)f(z)+f(z)f(x)+f(x)f(y),w=f(x)f(y)f(z),$$ then $f$ must be a constant. [The condition for a functional relation will be found to be $$f^{\prime }(x)f^{\prime }(y)f^{\prime }(z)\{f(y)–f(z)\}\{f(z)–f(x)\}\{f(x)–f(y)\}=0.]$$

 

27. If $f(y,z)$, $f(z,x)$, and $f(x,y)$ are connected by a functional relation then $f(x,x)$ is independent of $x$.

If $u=0$, $v=0$, $w=0$ are the equations of three circles, rendered homogeneous as in Ex. 19, then the equation $$\frac{\partial (u,v,w)}{\partial (x,y,z)}=0$$ represents the circle which cuts them all orthogonally.

 

29. If $A$, $B$, $C$ are three functions of $x$ such that $$\left|\begin{array}{ccc}A& A^{\prime }& A”\\ B& B^{\prime }& B”\\ C& C^{\prime }& C”\end{array}\right|$$ vanishes identically, then we can find constants $\lambda$, $\mu$, $\nu$ such that $\lambda A+\mu B+\nu C$ vanishes identically; and conversely. [The converse is almost obvious. To prove the direct theorem let $\alpha =B{C}^{\prime }–B^{\prime }C$, …. Then ${\alpha }^{\prime }=BC”–B”C$, …, and it follows from the vanishing of the determinant that $\beta {\gamma }^{\prime }–{\beta }^{\prime }\gamma =0$, …; and so that the ratios $\alpha :\beta :\gamma$ are constant. But $\alpha A+\beta B+\gamma C=0$.]

 

30. Suppose that three variables $x$, $y$, $z$ are connected by a relation in virtue of which (i) $z$ is a function of $x$ and $y$, with derivatives $z_x$, $z_y$, and (ii) $x$ is a function of $y$ and $z$, with derivatives $x_y$, $x_z$. Prove that $$x_y=–z_y/z_x,x_z=1/z_x.$$

[We have $$dz=z_{x}dx+z_{y}dy,dx=x_{y}dy+x_{z}dz.$$ The result of substituting for $dx$ in the first equation is $$dz=(z_x{x}_y+z_y)dy+z_x{x}_{z}dz,$$ which can be true only if $z_x{x}_y+z_y=0$, $z_x{x}_z=1$.]

 

31. Four variables $x$, $y$, $z$, $u$ are connected by two relations in virtue of which any two can be expressed as functions of the others. Show that $$y_z^u{z}_x^u{x}_y^u=-y_z^x{z}_x^y{x}_y^z=1,x_z^u{z}_x^y+y_z^u{z}_y^x=1,$$ where $y_z^u$ denotes the derivative of $y$, when expressed as a function of $z$ and $u$, with respect to $z$.

 

32. Find $A$, $B$, $C$, $\lambda$ so that the first four derivatives of $${\int }_a^{a+x}f(t)dt–x[Af(a)+Bf(a+\lambda x)+Cf(a+x)]$$ vanish when $x=0$; and $A$, $B$, $C$, $D$, $\lambda$, $\mu$ so that the first six derivatives of $${\int }_a^{a+x}f(t)dt–x[Af(a)+Bf(a+\lambda x)+Cf(a+\mu x)+Df(a+x)]$$ vanish when $x=0$.

 

33. If $a>0$, $ac–b^2>0$, and $x_1>x_0$, then $${\int }_{x_0}^{x_1}\frac{dx}{a{x}^2+2bx+c}=\frac{1}{\sqrt{ac–b^2}}\arctan \left\{\frac{(x_1–x_0)\sqrt{ac–b^2}}{a{x}_1{x}_0+b(x_1+x_0)+c}\right\},$$ the inverse tangent lying between $0$ and $\pi$.²

 

34. Evaluate the integral ${\int }_{-1}^1\frac{\sin \alpha dx}{1–2x\cos \alpha +x^2}$. For what values of $\alpha$ is the integral a discontinuous function of $\alpha$?

[The value of the integral is $\frac{1}{2}\pi$ if $2n\pi <\alpha <(2n+1)\pi$, and $-\frac{1}{2}\pi$ if $(2n–1)\pi <\alpha <2n\pi$, $n$ being any integer; and $0$ if $\alpha$ is a multiple of $\pi$.]

 

35. If $a{x}^2+2bx+c>0$ when $x_0\le x\le x_1$, $f(x)=\sqrt{a{x}^2+2bx+c}$, and $$y=f(x),y_0=f(x_0),y_1=f(x_1),X=(x_1–x_0)/(y_1+y_0),$$ then $${\int }_{x_0}^{x_1}\frac{dx}{y}=\frac{1}{\sqrt{a}}\log \frac{1+X\sqrt{a}}{1–X\sqrt{a}},\frac{-2}{\sqrt{-a}}\arctan \{X\sqrt{-a}\},$$ according as $a$ is positive or negative. In the latter case the inverse tangent lies between $0$ and $\frac{1}{2}\pi$. [It will be found that the substitution $t={\displaystyle \frac{x–x_0}{y+y_0}}$ reduces the integral to the form $2{\int }_0^X\frac{dt}{1–a{t}^2}$.]

 

36. Prove that $${\int }_0^a\frac{dx}{x+\sqrt{a^2–x^2}}=\frac{1}{4}\pi .$$

 

37. If $a>1$ then $${\int }_{-1}^1\frac{\sqrt{1–x^2}}{a–x}dx=\pi \{a–\sqrt{a^2–1}\}.$$

 

38. If $p>1$, $0<q<1$, then $${\int }_0^1\frac{dx}{\sqrt{\{1+(p^2–1)x\}\{1–(1–q^2)x\}}}=\frac{2\omega }{(p+q)\sin \omega },$$ where $\omega$ is the positive acute angle whose cosine is $(1+pq)/(p+q)$.

 

39. If $a>b>0$, then $${\int }_0^{2\pi }\frac{{\sin }^2\theta d\theta }{a–b\cos \theta }=\frac{2\pi }{b^2}\{a–\sqrt{a^2–b^2}\}.$$

 

40. Prove that if $a>\sqrt{b^2+c^2}$ then $${\int }_0^{\pi }\frac{d\theta }{a+b\cos \theta +c\sin \theta }=\frac{2}{\sqrt{a^2–b^2–c^2}}\arctan \left\{\frac{\sqrt{a^2–b^2–c^2}}{c}\right\},$$ the inverse tangent lying between $0$ and $\pi$.

 

41. If $f(x)$ is continuous and never negative, and ${\int }_a^{b}f(x)dx=0$, then $f(x)=0$ for all values of $x$ between $a$ and $b$. [If $f(x)$ were equal to a positive number $k$ when $x=\xi$, say, then we could, in virtue of the continuity of $f(x)$, find an interval $[\xi –\delta ,\xi +\delta ]$ throughout which $f(x)>\frac{1}{2}k$; and then the value of the integral would be greater than $\delta k$.]

 

42. Schwarz’s inequality for integrals. Prove that $${\left({\int }_a^b\varphi \psi dx\right)}^2\le {\int }_a^b{\varphi }^{2}dx{\int }_a^b{\psi }^{2}dx.$$

[Use the definitions of §§ 156 and 157, and the inequality $${(\sum {\varphi }_{\nu }{\psi }_{\nu }{\delta }_{\nu })}^2\le \sum {\varphi }_{\nu }^2{\delta }_{\nu }\sum {\psi }_{\nu }^2{\delta }_{\nu }$$ (Ch. I, Misc. Ex. 10).]

 

43. If $$P_n(x)=\frac{1}{(\beta –\alpha {)}^{n}n!}{\left(\frac{d}{dx}\right)}^n\{(x–\alpha )(\beta –x){\}}^n,$$ then $P_n(x)$ is a polynomial of degree $n$, which possesses the property that $${\int }_{\alpha }^{\beta }{P}_n(x)\theta (x)dx=0$$ if $\theta (x)$ is any polynomial of degree less than $n$. [Integrate by parts $m+1$ times, where $m$ is the degree of $\theta (x)$, and observe that ${\theta }^{(m+1)}(x)=0$.]

 

44. Prove that $${\int }_{\alpha }^{\beta }{P}_m(x)P_n(x)dx=0$$ if $m\ne n$, but that if $m=n$ then the value of the integral is $(\beta –\alpha )/(2n+1)$.

 

45. If $Q_n(x)$ is a polynomial of degree $n$, which possesses the property that $${\int }_{\alpha }^{\beta }{Q}_n(x)\theta (x)dx=0$$ if $\theta (x)$ is any polynomial of degree less than $n$, then $Q_n(x)$ is a constant multiple of $P_n(x)$.

[We can choose $\kappa$ so that $Q_n–\kappa P_n$ is of degree $n–1$: then $${\int }_{\alpha }^{\beta }{Q}_n(Q_n–\kappa P_n)dx=0,{\int }_{\alpha }^{\beta }{P}_n(Q_n–\kappa P_n)dx=0,$$ and so $${\int }_{\alpha }^{\beta }(Q_n–\kappa P_n{)}^{2}dx=0.$$ Now apply Ex. 41.]

 

46. Approximate Values of definite integrals. Show that the error in taking $\frac{1}{2}(b–a)\{\varphi (a)+\varphi (b)\}$ as the value of the integral ${\int }_a^b\varphi (x)dx$ is less than $\frac{1}{12}M(b–a{)}^3$, where $M$ is the maximum of $|\varphi ”(x)|$ in the interval $[a,b]$; and that the error in taking $(b–a)\varphi \{\frac{1}{2}(a+b)\}$ is less than $\frac{1}{24}M(b–a{)}^3$. [Write $f^{\prime }(x)=\varphi (x)$ in Exs. 4 and 5.] Show that the error in taking $$\frac{1}{6}(b–a)[\varphi (a)+\varphi (b)+4\varphi \{\frac{1}{2}(a+b)\}]$$ as the value is less than $\frac{1}{2880}M(b–a{)}^5$, where $M$ is the maximum of ${\varphi }^{(4)}(x)$. [Use Ex. 6. This rule, which gives a very good approximation, is known as Simpson’s Rule. It amounts to taking one-third of the first approximation given above and two-thirds of the second.]

Show that the approximation assigned by Simpson’s Rule is the area bounded by the lines $x=a$, $x=b$, $y=0$, and a parabola with its axis parallel to $OY$ and passing through the three points on the curve $y=\varphi (x)$ whose abscissae are $a$, $\frac{1}{2}(a+b)$, $b$.

It should be observed that if $\varphi (x)$ is any cubic polynomial then ${\varphi }^{(4)}(x)=0$, and Simpson’s Rule is exact. That is to say, given three points whose abscissae are $a$, $\frac{1}{2}(a+b)$, $b$, we can draw through them an infinity of curves of the type $y=\alpha +\beta x+\gamma x^2+\delta x^3$; and all such curves give the same area. For one curve $\delta =0$, and this curve is a parabola.

 

47. If $\varphi (x)$ is a polynomial of the fifth degree, then $${\int }_0^1\varphi (x)dx=\frac{1}{18}\{5\varphi (\alpha )+8\varphi (\frac{1}{2})+5\varphi (\beta )\},$$ $\alpha$ and $\beta$ being the roots of the equation $x^2–x+\frac{1}{10}=0$.

 

48. Apply Simpson’s Rule to the calculation of $\pi$ from the formula $\frac{1}{4}\pi ={\int }_0^1\frac{dx}{1+x^2}$. [The result is $.7833\dots$. If we divide the integral into two, from $0$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $1$, and apply Simpson’s Rule to the two integrals separately, we obtain $.7853916\dots$. The correct value is $.7853981\dots$.]

 

49. Show that $$8.9<{\int }_3^5\sqrt{4+x^2}dx<9.$$

 

50. Calculate the integrals $${\int }_0^1\frac{dx}{1+x},{\int }_0^1\frac{dx}{\sqrt{1+x^4}},{\int }_0^{\pi }\sqrt{\sin x}dx,{\int }_0^{\pi }\frac{\sin x}{x}dx,$$ to two places of decimals. [In the last integral the subject of integration is not defined when $x=0$: but if we assign to it, when $x=0$, the value $1$, it becomes continuous throughout the range of integration.]

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1. In this and the following examples the reader is to assume the continuity of all the derivatives which occur.↩︎
2. In connection with Exs. 33–35, 38, and 40 see a paper by Dr Bromwich in vol. xxxv of the Messenger of Mathematics.↩︎

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$\leftarrow$ 164. Integrals of complex functions

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