10273 problems found
Prove Taylor's theorem for a function \(f(x)\), in the range \(a \le x \le b\), stating the necessary restrictions on the behaviour of \(f(x)\) and its derivatives. Find the first four terms in the expansion of \(\log_e(1+\sin^2 x)\) in increasing powers of \(x\), and also the first three terms of the expansion of \(x\) in increasing powers of this function.
Explain the application of the Calculus to the discussion of inequalities, giving simple illustrations. Hence or otherwise prove that if \(0<\theta<1\), \[ \theta + \frac{\theta^3}{3} < \tan\theta < \theta + \frac{2\theta^3}{3}. \]
Trace the curve \(r\cos\theta+a\cos 2\theta = 0\). Shew that the area of the loop is \(a^2(2-\frac{\pi}{2})\), and that the area enclosed between the curve and its asymptote is \(a^2(2+\frac{\pi}{2})\).
Prove that for odd values of \(n\), \[ \int_0^\pi \frac{\cos n\theta}{\cos\theta} d\theta = (-1)^{\frac{n-1}{2}}\pi. \] If, for odd values of \(n\), \(I_n = \int_0^\pi \frac{\cos^2 n\theta}{\cos^2\theta}d\theta\), shew that \begin{align*} I_n &= I_{n-2}+2\pi \\ &= n\pi. \end{align*}
Shew that the area of the surface of the prolate spheroid obtained by the rotation of an ellipse of eccentricity \(e\) about its major axis (\(2a\)) is \[ A = 2\pi a^2\left[ \sqrt{1-e^2} + \frac{\sin^{-1} e}{e} \right] \] and that the centroid of the half surface bounded by the central circular section is at a distance \(d\) from the plane of that section, where \[ Ad = \frac{4\pi a^3}{3} \frac{1}{e^2}\left[ \sqrt{1-e^2} - (1-e^2)^{\frac{3}{2}} \right]. \]
Find the maximum and minimum values for real values of \(x,y,z\), of the quantity \(x^2+y^2+z^2\), subject to the conditions that \begin{align*} lx+my+nz &= 0, \\ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} &= 1, \end{align*} where \(a,b,c\) are positive and \(l,m,n\) are real. Verify that the values determined are real and positive.
(a) Prove that the three lines joining the mid-points of opposite edges of a tetrahedron meet in a point. (b) \(ABCD\) is a tetrahedron in which the edge \(AD\) is at right angles to the edge \(BC\), and the edge \(BD\) to the edge \(CA\). Prove that the edge \(CD\) is at right angles to the edge \(AB\). Prove also that in this case the perpendiculars from the vertices on to the opposite faces meet in a point.
Shew that the locus of a point \(P\) whose rectangular Cartesian coordinates are given by \[ x:y:1 = at^2+2bt+c : a't^2+2b't+c' : a''t^2+2b''t+c'', \] where \(t\) is a variable parameter, is in general a conic. Examine the particular cases in which (i) the determinant \[ \Delta = \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} \] vanishes, (ii) all first minors of \(\Delta\) vanish. Find the equation of the tangent at the point \(t\), and prove that the coordinates \((\xi, \eta)\) of the pole of the line joining the points \(t_1\) and \(t_2\) are given by \[ \xi:\eta:1 = at_1t_2+b(t_1+t_2)+c : a't_1t_2+b'(t_1+t_2)+c' : a''t_1t_2+b''(t_1+t_2)+c''. \]
\(g(x), h(x)\) are given polynomials, of degrees \(m, n\) respectively (\(m \ge n\)). Prove that the degree of a polynomial (not vanishing identically) which can be written in the form \[ \text{(1)} \qquad G(x)g(x) + H(x)h(x), \] where \(G(x)\) and \(H(x)\) are polynomials, can be as small as, but not smaller than, a definite integer \(\nu (\ge 0)\). Prove also that the polynomial \(\chi(x)\), which is of the form (1) and of degree \(\nu\), and in which the coefficient of \(x^\nu\) is unity, is unique. Prove that \(\chi(x)\) is a common factor of \(g(x)\) and \(h(x)\). Prove also that in the expression of \(\chi(x)\) in the form (1), \(G(x)\) and \(H(x)\) can be found of degrees less than \(n\) and \(m\) respectively.
A function \(\psi_n(x)\) is defined by the equation \[ \psi_n(x) = \frac{d^n}{dx^n}\frac{\sqrt{x}}{1+x} = \frac{\sqrt{x}}{(x+1)^{n+1}}\Psi_n(x). \] Shew