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

2. Show that if $$f(x)$$ and its first $$n + 2$$ derivatives are continuous, and $$f^{(n+1)}(0) \neq 0$$, 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 $\theta_{n} = \frac{1}{n + 1} + \frac{n}{2(n + 1)^{2}(n + 2)} \left\{\frac{f^{(n+2)}(0)}{f^{(n+1)}(0)} + \epsilon_{x}\right\}x,$ where $$\epsilon_{x} \to 0$$ 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 $f(b) = f(a) + \tfrac{1}{2}(b – a) \{f'(a) + f'(b)\} – \tfrac{1}{12}(b – a)^{3} f”'(\alpha),$ where $$a < \alpha < b$$. [Apply to the function $\begin{gathered} f(x) – f(a) – \tfrac{1}{2}(x – a) \{f'(a) + f'(x)\}\\ – \left(\frac{x – a}{b – a}\right)^{3} [f(b) – f(a) – \tfrac{1}{2}(b – a) \{f'(a) + f'(b)\}]\end{gathered}$ arguments similar to those of § 147.]

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

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

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

8. Establish the formulae

$(i) \quad \begin{vmatrix} f(a) & f(b)\\ g(a) & g(b) \end{vmatrix} = (b – a) \begin{vmatrix} f(a) & f'(\beta)\\ g(a) & g'(\beta) \end{vmatrix}$

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

$(ii) \quad\begin{vmatrix} f(a) & f(b) & f(c)\\ g(a) & g(b) & g(c)\\ h(a) & h(b) & h(c) \end{vmatrix} = \tfrac{1}{2} (b – c)(c – a)(a – b) \begin{vmatrix} f(a) & f'(\beta) & f”(\gamma)\\ g(a) & g'(\beta) & g”(\gamma)\\ h(a) & h'(\beta) & h”(\gamma) \end{vmatrix}$

where $$\beta$$ and $$\gamma$$ lie between the least and greatest of $$a$$, $$b$$, $$c$$. [To prove (ii) consider the function $\phi(x) = \begin{vmatrix} f(a) & f(b) & f(x)\\ g(a) & g(b) & g(x)\\ h(a) & h(b) & h(x) \end{vmatrix} – \frac{(x – a)(x – b)}{(c – a)(c – b)} \begin{vmatrix} f(a) & f(b) & f(c)\\ g(a) & g(b) & g(c)\\ h(a) & h(b) & h(c) \end{vmatrix},$ 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 $\begin{vmatrix} f(a) & f(b) & f(c)\\ g(a) & g(b) & g(c)\\ h(a) & h(b) & h(c) \end{vmatrix} = \tfrac{1}{2}(c – a)(c – b) \begin{vmatrix} f(a) & f(b) & f”(\gamma)\\ g(a) & g(b) & g”(\gamma)\\ h(a) & h(b) & h”(\gamma) \end{vmatrix}.$ 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 $$A \leq F^{(n)}(x) \leq B$$ when $$0 \leq x \leq h$$, then $$A(x^{n}/n!) \leq F(x) \leq B(x^{n}/n!)$$ when $$0 \leq x \leq h$$.

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

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

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

12. Suppose that $$y = \phi(x)$$ is a function of $$x$$ with continuous derivatives of at least the first four orders, and that $$\phi(0) = 0$$, $$\phi'(0) = 1$$, so that $y = \phi(x) = x + a_{2}x^{2} + a_{3}x^{3} + (a_{4} + \epsilon_{x})x^{4},$ where $$\epsilon_{x} \to 0$$ as $$x \to 0$$. Establish the formula $x = \psi(y) = y – a_{2}y^{2} + (2a_{2}^{2} – a_{3})y^{3} – (5a_{2}^{3} – 5a_{2}a_{3} + a_{4} + \epsilon_{y})y^{4},$ where $$\epsilon_{y} \to 0$$ as $$y \to 0$$, for that value of $$x$$ which vanishes with $$y$$; and prove that $\frac{\phi(x)\psi(x) – x^{2}}{x^{4}} \to a_{2}^{2}$ 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 $-(\xi – x)/y’ = (\eta – y)/x’ = (x’^{2} + y’^{2})/(x’y” – x”y’);$ and the radius of curvature of the curve is $(x’^{2} + y’^{2})^{3/2}/(x’y” – x”y’),$ dashes denoting differentiations with respect to $$t$$.

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

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 $$y = ax^{2} + bx^{3} + cx^{4} + \dots + kx^{n}$$, 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 $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z} + \dots = nu.$ 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 ( $$x^{n} + y^{n} – xz^{n-1} = 0$$). Show that the equation of the tangent at the point $$(\xi, \eta)$$ of the curve $$f(x, y) = 0$$ is $xF_{\xi} + yF_{\eta} + zF_{\zeta} = 0,$ 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 $\begin{equation*} \phi(u, v) = 0. \tag{1}\end{equation*}$

Differentiating (1) with respect to $$x$$ and $$y$$, we obtain $\begin{equation*} \frac{\partial \phi}{\partial u}\, \frac{\partial u}{\partial x} + \frac{\partial \phi}{\partial v}\, \frac{\partial v}{\partial x} = 0,\quad \frac{\partial \phi}{\partial u}\, \frac{\partial u}{\partial y} + \frac{\partial \phi}{\partial v}\, \frac{\partial v}{\partial y} = 0, \tag{2} \end{equation*}$ and, eliminating the derivatives of $$\phi$$, $\begin{equation*} J = \begin{vmatrix} u_{x} & u_{y}\\ v_{x} & v_{y} \end{vmatrix} = u_{x}v_{y} – u_{y}v_{x} = 0, \tag{3} \end{equation*}$ 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 $$\phi(u, v, w) = 0$$ according as $J = \begin{vmatrix} u_{x} & u_{y} & u_{z}\\ v_{x} & v_{y} & v_{z}\\ w_{x} & w_{y} & w_{z} \end{vmatrix} = \frac{\partial(u, v, w)}{\partial(x, y, z)}$ does or does not vanish for all values of $$x$$, $$y$$, $$z$$.

21. Show that $$ax^{2} + 2hxy + by_{2}$$ and $$Ax^{2} + 2Hxy + By^{2}$$ are independent unless $$a/A = h/H = b/B$$.

22. Show that $$ax^{2} + by^{2} + cz^{2} + 2fyz + 2gzx + 2hxy$$ can be expressed as a product of two linear functions of $$x$$, $$y$$, and $$z$$ if and only if $abc + 2fgh – af^{2} – bg^{2} – ch^{2} = 0.$

[Write down the condition that $$px + qy + rz$$ and $$p’x + q’y + r’z$$ 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 $u = f(x) + f(y) + f(z),\quad v = f(y)f(z) + f(z)f(x) + f(x)f(y),\quad w = f(x)f(y)f(z),$ then $$f$$ must be a constant. [The condition for a functional relation will be found to be $f'(x)f'(y)f'(z) \{f(y) – f(z)\} \{f(z) – f(x)\} \{f(x) – f(y)\} = 0.]$

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 $\begin{vmatrix} A & A’ & A”\\ B & B’ & B”\\ C & C’ & C” \end{vmatrix}$ 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 $$\alpha = BC’ – B’C$$, …. Then $$\alpha’ = BC” – B”C$$, …, and it follows from the vanishing of the determinant that $$\beta\gamma’ – \beta’\gamma = 0$$, …; 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 $x_{y} = – z_{y}/z_{x},\quad x_{z} = 1/z_{x}.$

[We have $dz = z_{x}\, dx + z_{y}\, dy,\quad dx = x_{y}\, dy + x_{z}\, dz.$ 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 $y_{z}^{u}z_{x}^{u}x_{y}^{u} = -y_{z}^{x}z_{x}^{y}x_{y}^{z} = 1,\quad x_{z}^{u}z_{x}^{y} + y_{z}^{u}z_{y}^{x} = 1,$ 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 $\int_{x_{0}}^{x_{1}} \frac{dx}{ax^{2} + 2bx + c} = \frac{1}{\sqrt{ac – b^{2}}} \arctan\left\{ \frac{(x_{1} – x_{0}) \sqrt{ac – b^{2}}} {ax_{1}x_{0} + b(x_{1} + x_{0}) + c} \right\},$ 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 $$ax^{2} + 2bx + c > 0$$ when $$x_{0} \leq x \leq x_{1}$$, $$f(x) = \sqrt{ax^{2} + 2bx + c}$$, and $y = f(x),\quad y_{0} = f(x_{0}),\quad y_{1} = f(x_{1}),\quad X = (x_{1} – x_{0})/(y_{1} + y_{0}),$ then $\int_{x_{0}}^{x_{1}} \frac{dx}{y} = \frac{1}{\sqrt{a}} \log \frac{1 + X\sqrt{a}}{1 – X\sqrt{a}},\quad \frac{-2}{\sqrt{-a}} \arctan\{X\sqrt{-a}\},$ 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 $$t = \dfrac{x – x_{0}}{y + y_{0}}$$ reduces the integral to the form $$2\int_{0}^{X} \frac{dt}{1 – at^{2}}$$.]

36. Prove that $\int_{0}^{a} \frac{dx}{x + \sqrt{a^{2} – x^{2}}} = \tfrac{1}{4}\pi.$

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 $\left(\int_{a}^{b} \phi\psi\, dx\right)^{2} \leq \int_{a}^{b} \phi^{2}\, dx \int_{a}^{b} \psi^{2}\, dx.$

[Use the definitions of §§ 156 and 157, and the inequality $\left(\sum\phi_{\nu}\psi_{\nu}\, \delta_{\nu}\right)^{2} \leq \sum\phi_{\nu}^{2}\, \delta_{\nu} \sum\psi_{\nu}^{2}\, \delta_{\nu}$ (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 $$m \neq n$$, 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 $\int_{\alpha}^{\beta} Q_{n}(Q_{n} – \kappa P_{n})\, dx = 0,\quad \int_{\alpha}^{\beta} P_{n}(Q_{n} – \kappa P_{n})\, dx = 0,$ 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 $$\tfrac{1}{2}(b – a) \{\phi(a) + \phi(b)\}$$ as the value of the integral $$\int_{a}^{b} \phi(x)\, dx$$ is less than $$\tfrac{1}{12}M(b – a)^{3}$$, where $$M$$ is the maximum of $$|\phi”(x)|$$ in the interval $${[a, b]}$$; and that the error in taking $$(b – a)\phi\{\tfrac{1}{2}(a + b)\}$$ is less than $$\tfrac{1}{24}M(b – a)^{3}$$. [Write $$f'(x)= \phi(x)$$ in Exs. 4 and 5.] Show that the error in taking $\tfrac{1}{6}(b – a)[\phi(a) + \phi(b) + 4\phi\{\tfrac{1}{2}(a + b)\}]$ as the value is less than $$\tfrac{1}{2880}M(b – a)^{5}$$, where $$M$$ is the maximum of $$\phi^{(4)}(x)$$. [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 $$y = \phi(x)$$ whose abscissae are $$a$$, $$\tfrac{1}{2}(a + b)$$, $$b$$.

It should be observed that if $$\phi(x)$$ is any cubic polynomial then $$\phi^{(4)}(x) = 0$$, and Simpson’s Rule is exact. That is to say, given three points whose abscissae are $$a$$, $$\tfrac{1}{2}(a + b)$$, $$b$$, 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 $$\phi(x)$$ is a polynomial of the fifth degree, then $\int_{0}^{1} \phi(x)\, dx = \tfrac{1}{18}\{5\phi(\alpha) + 8\phi(\tfrac{1}{2}) + 5\phi(\beta)\},$ $$\alpha$$ 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 $$\tfrac{1}{4}\pi = \int_{0}^{1} \frac{dx}{1 + x^{2}}$$. [The result is $$.7833\dots$$. If we divide the integral into two, from $$0$$ to $$\tfrac{1}{2}$$ and $$\tfrac{1}{2}$$ to $$1$$, and apply Simpson’s Rule to the two integrals separately, we obtain $$.785\ 391\ 6\dots$$. The correct value is $$.785\ 398\ 1\dots$$.]

49. Show that $8.9 < \int_{3}^{5} \sqrt{4 + x^{2}}\, dx < 9.$

50. Calculate the integrals $\int_{0}^{1} \frac{dx}{1 + x},\quad \int_{0}^{1} \frac{dx}{\sqrt{1 + x^{4}}},\quad \int_{0}^{\pi} \sqrt{\sin x}\, dx,\quad \int_{0}^{\pi} \frac{\sin x}{x}\, dx,$ 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.]

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.↩︎