If, for all \(x\) such that \(0 \leq x \leq h\) (\(h > 0\)), $$|c_0 + c_1x + c_2x^2 + \ldots + c_nx^n| \leq Ax^{n+1}$$ where \(A\) is a constant and \(n\) a given positive integer), show that the constants \(c_0, c_1, \ldots, c_n\) are all zero. A function \(f(x)\) is differentiable as many times as we wish, and in the interval \(0 \leq x \leq h\) its \((n-1)\)th derivative \(f^{(n-1)}(x)\) lies between \(\pm K_n\), for each \(n\). Prove by successive integration that, for \(0 \leq x \leq h\), $$f(x) - f(0) - xf'(0) - \ldots - \frac{x^n}{n!}f^{(n)}(0)$$ is between \(\pm K_n x^{n+1}/(n+1)!\). Deduce that if \(g(x) \equiv f(x^2)\) then \(g^{(2p+1)}(0) = 0\), \(g^{(2p)}(0) = (2p)!f^{(p)}(0)/p!\) for every positive integer \(p\). [It must not be assumed that the infinite Taylor series of \(f(x)\) converges to sum \(f(x)\).] Without assuming the binomial theorem for fractional indices, find the sixth derivative at \(x = 0\) of \((1-x^2)^{1/2}\).
Sketch the family of curves $$(x-a)^2 - y^2 + y^3 = 0,$$ where \(a\) is a parameter. Show that the usual method of finding an envelope, by eliminating \(a\) between the equations \(F(x, y, a) = 0\) and \(\partial F/\partial a = 0\), yields in this case not only the envelope but also an extraneous locus. How do you account for this? Prove that for a family of straight lines of the form \(y = ax + P(a)\), where \(P\) is a polynomial of degree at least two, the standard method always yields a true envelope (that is, a curve whose slope at any point is the same as that of some straight line of the family which passes through that point). Find, in the form \(\phi(x, y) = 0\), the envelope of the family of lines \(y = ax + 3x - a^3\), and sketch it.
Show that, if \(P(x)\) is a polynomial of degree \(n\) such that the \(n\) repeated factors, then between any two consecutive real roots of the equation \(P(x) = 0\), the polynomial function must possess a maximum or minimum between any two consecutive real zeros of the function.) Show that, if the equation \(P(x) = 0\) has \(n\) real roots (counting multiple roots according to their multiplicity), then \(P'(x) = 0\) has \(n-1\) real roots. If \(P(x) = 0\) has \(m-1\) real roots, all distinct, where \(m < n\), show that \(P(x) = 0\) has \(m\) or \(m-2k\) real roots, where \(k\) is some positive integer. Show that this remains true if one, but only one, of the real roots of \(P'(x) = 0\) is repeated. A sequence of functions \(Q_n(x)\) is defined by the relations $$Q_0(x) = 1 \quad (\text{all } x),$$ $$Q_n(x) = xQ_{n-1}(x) - Q'_{n-1}(x) \quad (n \geq 1).$$ Prove that \(Q_n(x)\) is a polynomial of leading term \(x^n\) and with \(n\) distinct real zeros.
Obtain a reduction formula for $$I_n = \int \frac{x^n dx}{(ax^2 + c)^{1/2}}$$ where \(a\), \(c\) are real non-zero constants. Show that by the use of this formula (if necessary, with \(n\) negative), \(I_m\) can, for any positive or negative integer \(m\), be expressed in terms of known functions and \(I_0\) or \(I_{-1}\) according to the sign of \(a\) and \(c\). Obtain explicitly $$\int \frac{x^4 + 1}{(x^2 + 4)^{1/2}} dx \quad \text{and} \quad \int \frac{dx}{x^3(1-4x^2)^{1/2}}.$$
Show how to reduce the equation (in homogeneous coordinates) of any non-degenerate conic, \(ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0\), to the form \(\eta^2 = \xi\zeta\) where \(\xi\), \(\eta\), \(\zeta\) are certain linear expressions in \(x\), \(y\) and \(z\). In what circumstances can the equations of two given conics be simultaneously reduced to the forms \(\eta^2 = \xi\zeta\) and \(\eta^2 = k\xi\zeta\), where \(k\) is a constant? Two parabolas \(S_1\), \(S_2\) touch at a point \(P_0\) and have parallel axes. From any point \(P\) of \(S_1\) two tangents are drawn to \(S_2\), cutting \(S_1\) again at \(Q\), \(R\) respectively. Prove that, as \(P\) varies, the line \(QR\) envelopes a third parabola touching \(S_1\) and \(S_2\) at \(P_0\) and with axis parallel to those of \(S_1\) and \(S_2\).
(a) A light inextensible string is pulled against a rough curve in a plane. Given that at a point \(P\) on the string, which lies at a distance \(s\) measured along the string from a fixed point of it, the radius of curvature is \(\rho\), the tension \(T\), normal and tangential reactions per unit length \(N\) and \(R\) respectively, set up the equations of equilibrium. (b) A weightless string of length \(2l\) is fixed at the endpoints \(A\) and \(B\) which are at distance \(2a\) apart. A wind perpendicular to \(AB\) blows the string into a curve, exerting on it a normal force \(k \cos \psi\) per unit length, where \(\psi\) is the angle between the normal and the direction of the wind and \(k\) is constant. Find the intrinsic equation of the curve. Show that the tension in the string must exceed \(2kl/\pi\).
Two boys, \(A\) and \(B\), each of mass \(m\), hang at rest at the ends of a light inextensible rope which runs without slipping over a smoothly mounted pulley of radius \(a\) and moment of inertia \(I\). At a signal the boys begin to race, \(A\) and \(B\) climbing with constant speeds \(u\) and \(v\) respectively \((u > v)\) relative to the rope. Show that, in a race through height \(h\), \(A\) can give \(B\) any start less than \(h\{(I + ma^2) u + ma^2 v\}\) and win.
A light inextensible string, carrying equal masses \(m\) at the two ends, hangs over two smooth pegs \(A\), \(B\) at the same level and at distance \(2a\) apart. A mass \(2m\) is attached at the point \(C\) of the string which lies midway between \(A\) and \(B\), and the system is then released from rest. In the subsequent motion the angle between \(AC\) and the vertical is \(\theta\). Find the velocity of the mass \(2m\) as a function of \(\theta\) as long as neither mass \(m\) has reached the corresponding peg. Find also the tension in the string when \(\theta = \frac{1}{4}\pi\).
A string \(ABCD\), whose elasticity can be neglected, is stretched at tension \(T\) the fixed points \(A\) and \(D\) on a smooth horizontal table. Equal masses \(m\) are attached along the string, with small velocity \(v\). Assuming that the tension in the string off at right angles, and \(C\), Hence find the subsequent displacements of \(B\) and \(C\) as functions of time.
The displacement \(x\) of the indicator in a seismograph is related to the displacement \(s\) of the ground by the equation \(\ddot{x} + 2\lambda\omega\dot{x} + \omega^2x = -M\ddot{s}\) where \(\lambda\), \(\omega\), \(M\) are (positive) constants of the instrument and \((\,)\) denotes differentiation with respect to the time \(t\). (i) Show that the free motion of the seismograph (given by the solution of (1) identically zero) takes different forms according as \(\lambda \gtrless 1\). (ii) Show that if \(\lambda < 1\) the ratio \(\epsilon\) of the amplitudes of two successive half-swings is given by \(\log \epsilon = \pi\lambda/\sqrt{1-\lambda^2}\). (iii) Given \(\lambda = 1\), find a formula for the response \((x)\) of the instrument to an earth movement in which \(s\) is a given function \(f(t)\). How should the constants of the instrument be adjusted so that \(x\) is approximately proportional to \((a)\) the displacement \(s\), or \((b)\) the acceleration \(\ddot{s}\)?