MISCELLANEOUS EXAMPLES ON CHAPTER II

1. Show that if $y=f(x)=(ax+b)/(cx–a)$ then $x=f(y)$.

 

2. If $f(x)=f(-x)$ for all values of $x$, $f(x)$ is called an even function. If $f(x)=-f(-x)$, it is called an odd function. Show that any function of $x$, defined for all values of $x$, is the sum of an even and an odd function of $x$.

[Use the identity $f(x)=\frac{1}{2}\{f(x)+f(-x)\}+\frac{1}{2}\{f(x)–f(-x)\}$.]

 

3. Draw the graphs of the functions $$3\sin x+4\cos x,\sin (\frac{\pi }{\sqrt{2}}\sin x).$$

 

4. Draw the graphs of the functions $$\sin x(a{\cos }^{2}x+b{\sin }^{2}x),\frac{\sin x}{x}(a{\cos }^{2}x+b{\sin }^{2}x),{\left(\frac{\sin x}{x}\right)}^2.$$

 

5. Draw the graphs of the functions $x[1/x]$, $[x]/x$.

 

6. Draw the graphs of the functions $$\begin{array}{rl}(i)& \arccos (2x^2–1)–2\arccos x,\\ (ii)& \arctan \frac{a+x}{1–ax}–\arctan a–\arctan x,\end{array}$$ where the symbols $\arccos a$, $\arctan a$ denote, for any value of $a$, the least positive (or zero) angle, whose cosine or tangent is $a$.

 

7. Verify the following method of constructing the graph of $f\{\varphi (x)\}$ by means of the line $y=x$ and the graphs of $f(x)$ and $\varphi (x)$: take $OA=x$ along $OX$, draw $AB$ parallel to $OY$ to meet $y=\varphi (x)$ in $B$, $BC$ parallel to $OX$ to meet $y=x$ in $C$, $CD$ parallel to $OY$ to meet $y=f(x)$ in $D$, and $DP$ parallel to $OX$ to meet $AB$ in $P$; then $P$ is a point on the graph required.

 

8. Show that the roots of $x^3+px+q=0$ are the abscissae of the points of intersection (other than the origin) of the parabola $y=x^2$ and the circle $$x^2+y^2+(p–1)y+qx=0.$$

 

9. The roots of $x^4+n{x}^3+p{x}^2+qx+r=0$ are the abscissae of the points of intersection of the parabola $x^2=y–\frac{1}{2}nx$ and the circle $$x^2+y^2+(\frac{1}{8}{n}^2–\frac{1}{2}pn+\frac{1}{2}n+q)x+(p–1–\frac{1}{4}{n}^2)y+r=0.$$

 

10. Discuss the graphical solution of the equation $$x^m+a{x}^2+bx+c=0$$ by means of the curves $y=x^m$, $y=-a{x}^2–bx–c$. Draw up a table of the various possible numbers of roots.

 

11. Solve the equation $\sec \theta +\csc \theta =2\sqrt{2}$; and show that the equation $\sec \theta +\csc \theta =c$ has two roots between $0$ and $2\pi$ if $c^2<8$ and four if $c^2>8$.

 

12. Show that the equation $$2x=(2n+1)\pi (1–\cos x),$$ where $n$ is a positive integer, has $2n+3$ roots and no more, indicating their localities roughly.

 

13. Show that the equation $\frac{2}{3}x\sin x=1$ has four roots between $-\pi$ and $\pi$.

 

14. Discuss the number and values of the roots of the equations

$(1)\cot x+x–\frac{3}{2}\pi =0$,

$(2)x^2+{\sin }^{2}x=1$,

$(3)\tan x=2x/(1+x^2)$,

$(4)\sin x–x+\frac{1}{6}{x}^3=0$,

$(5)(1–\cos x)\tan \alpha –x+\sin x=0$.

 

15. The polynomial of the second degree which assumes, when $x=a$, $b$, $c$ the values $\alpha$, $\beta$, $\gamma$ is $$\alpha \frac{(x–b)(x–c)}{(a–b)(a–c)}+\beta \frac{(x–c)(x–a)}{(b–c)(b–a)}+\gamma \frac{(x–a)(x–b)}{(c–a)(c–b)}.$$ Give a similar formula for the polynomial of the $(n–1)$th degree which assumes, when $x=a_1$, $a_2$, … $a_n$, the values ${\alpha }_1$, ${\alpha }_2$, … ${\alpha }_n$.

 

16. Find a polynomial in $x$ of the second degree which for the values $0$, $1$, $2$ of $x$ takes the values $1/c$, $1/(c+1)$, $1/(c+2)$; and show that when $x=c+2$ its value is $1/(c+1)$.

 

17. Show that if $x$ is a rational function of $y$, and $y$ is a rational function of $x$, then $Axy+Bx+Cy+D=0$.

 

18. If $y$ is an algebraical function of $x$, then $x$ is an algebraical function of $y$.

 

19. Verify that the equation $$\cos \frac{1}{2}\pi x=1–\frac{x^2}{x+(x–1)\sqrt{{\displaystyle \frac{2–x}{3}}}}$$ is approximately true for all values of $x$ between $0$ and $1$. [Take $x=0$, $\frac{1}{6}$, $\frac{1}{3}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{5}{6}$, $1$, and use tables. For which of these values is the formula exact?]

 

20. What is the form of the graph of the functions $$z=[x]+[y],z=x+y–[x]–[y]?$$

21. What is the form of the graph of the functions $z=\sin x+\sin y$, $z=\sin x\sin y$, $z=\sin xy$, $z=\sin (x^2+y^2)$?

22. Geometrical constructions for irrational numbers. In Chapter I we indicated one or two simple geometrical constructions for a length equal to $\sqrt{2}$, starting from a given unit length. We also showed how to construct the roots of any quadratic equation $a{x}^2+2bx+c=0$, it being supposed that we can construct lines whose lengths are equal to any of the ratios of the coefficients $a$, $b$, $c$, as is certainly the case if $a$, $b$, $c$ are rational. All these constructions were what may be called Euclidean constructions; they depended on the ruler and compasses only.

It is fairly obvious that we can construct by these methods the length measured by any irrational number which is defined by any combination of square roots, however complicated. Thus $$\sqrt[4]{\sqrt{\frac{17+3\sqrt{11}}{17–3\sqrt{11}}}–\sqrt{\frac{17–3\sqrt{11}}{17+3\sqrt{11}}}}$$ is a case in point. This expression contains a fourth root, but this is of course the square root of a square root. We should begin by constructing $\sqrt{11}$, as the mean between $1$ and $11$: then $17+3\sqrt{11}$ and $17–3\sqrt{11}$, and so on. Or these two mixed surds might be constructed directly as the roots of $x^2–34x+190=0$.

Conversely, only irrationals of this kind can be constructed by Euclidean methods. Starting from a unit length we can construct any rational length. And hence we can construct the line $Ax+By+C=0$, provided that the ratios of $A$, $B$, $C$ are rational, and the circle $$(x–\alpha {)}^2+(y–\beta {)}^2={\rho }^2$$ (or $x^2+y^2+2gx+2fy+c=0$), provided that $\alpha$, $\beta$, $\rho$ are rational, a condition which implies that $g$, $f$, $c$ are rational.

Now in any Euclidean construction each new point introduced into the figure is determined as the intersection of two lines or circles, or a line and a circle. But if the coefficients are rational, such a pair of equations as $$Ax+By+C=0,x^2+y^2+2gx+2fy+c=0$$ give, on solution, values of $x$ and $y$ of the form $m+n\sqrt{p}$, where $m$, $n$, $p$ are rational: for if we substitute for $x$ in terms of $y$ in the second equation we obtain a quadratic in $y$ with rational coefficients. Hence the coordinates of all points obtained by means of lines and circles with rational coefficients are expressible by rational numbers and quadratic surds. And so the same is true of the distance $\sqrt{(x_1–x_2{)}^2+(y_1–y_2{)}^2}$ between any two points so obtained.

With the irrational distances thus constructed we may proceed to construct a number of lines and circles whose coefficients may now themselves involve quadratic surds. It is evident, however, that all the lengths which we can construct by the use of such lines and circles are still expressible by square roots only, though our surd expressions may now be of a more complicated form. And this remains true however often our constructions are repeated. Hence Euclidean methods will construct any surd expression involving square roots only, and no others.

One of the famous problems of antiquity was that of the duplication of the cube, that is to say of the construction by Euclidean methods of a length measured by $\sqrt[3]{2}$. It can be shown that $\sqrt[3]{2}$ cannot be expressed by means of any finite combination of rational numbers and square roots, and so that the problem is an impossible one. See Hobson, Squaring the Circle, pp. 47 et seq.; the first stage of the proof, viz. the proof that $\sqrt[3]{2}$ cannot be a root of a quadratic equation $a{x}^2+2bx+c=0$ with rational coefficients, was given in Ch. I (Misc. Exs. 24).

 

23. Approximate quadrature of the circle. Let $O$ be the centre of a circle of radius $R$. On the tangent at $A$ take $AP=\frac{11}{5}R$ and $AQ=\frac{13}{5}R$, in the same direction. On $AO$ take $AN=OP$ and draw $NM$ parallel to $OQ$ and cutting $AP$ in $M$. Show that $$AM/R=\frac{13}{25}\sqrt{146},$$ and that to take $AM$ as being equal to the circumference of the circle would lead to a value of $\pi$ correct to five places of decimals. If $R$ is the earth’s radius, the error in supposing $AM$ to be its circumference is less than $11$ yards.

 

24. Show that the only lengths which can be constructed with the ruler only, starting from a given unit length, are rational lengths.

 

25. Constructions for $\sqrt[3]{2}$.$O$ is the vertex and $S$ the focus of the parabola $y^2=4x$, and $P$ is one of its points of intersection with the parabola $x^2=2y$. Show that $OP$ meets the latus rectum of the first parabola in a point $Q$ such that $SQ=\sqrt[3]{2}$.

 

26. Take a circle of unit diameter, a diameter $OA$ and the tangent at $A$. Draw a chord $OBC$ cutting the circle at $B$ and the tangent at $C$. On this line take $OM=BC$. Taking $O$ as origin and $OA$ as axis of $x$, show that the locus of $M$ is the curve $$(x^2+y^2)x–y^2=0$$ (the Cissoid of Diocles). Sketch the curve. Take along the axis of $y$ a length $OD=2$. Let $AD$ cut the curve in $P$ and $OP$ cut the tangent to the circle at $A$ in $Q$. Show that $AQ=\sqrt[3]{2}$.

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$\leftarrow$ 33. Loci in space

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