A window consists of a rectangular frame surmounted by a semicircle. If the perimeter of the window is given, prove that the area is a maximum when the diameter of the semicircle (the width of the window) is twice the height of the rectangle. Find the maximum area when the given perimeter is 24 feet.
Prove that at a point of inflexion on a curve, \(\frac{d^2y}{dx^2}=0\); and that if \(x,y\) are functions of a parameter \(t\), \(x'y''-y'x''=0\), where dashes denote differentiation with respect to \(t\).
Find the points of inflexion on the curve \(x=2a\cos t+b\cos 2t, y=2a\sin t+b\sin 2t\), where \(a,b\) are positive and show that they are real if \(b
For a curve defined by \(p=f(\psi)\), prove that the projection of the radius vector on the tangent is \(\frac{dp}{d\psi}\) and that \(\rho = p+\frac{d^2p}{d\psi^2}\). For the curve \(p=a\sin 2\psi\), prove that \(r^2=a^2-3p^2\) and that the \(p,r\) equation of the locus of centres of curvature is \(r^2=16a^2-3p^2\).
The line \(AB\) is equal in length to \(A'B'\) and in the same plane: shew that \(AB\) can always be moved to coincidence with \(A'B'\) by a rotation about a point in the plane except in the case when, \(AA'\) being equal, parallel and in the same sense as \(BB'\), translation is alone necessary. Shew also that a line can be chosen in the plane so that the image of \(AB\) by reflexion in the line can be moved by translation parallel to the line to coincide with \(A'B'\). Shew that two successive reflexions of a plane figure in lines in the plane are equivalent to a rotation and that any odd number of reflexions is equivalent to a single reflexion and translation.
Shew that, if two points at distance \(a\) apart be inverted with regard to an origin distant \(e\) and \(f\) from them respectively, the distance between the inverse points is \(\frac{a}{ef}\) if the radius of inversion be of unit length. Shew that the problem of inverting three points \(A, B, C\) with regard to a point in their plane so that the inverse points are the vertices of an equilateral triangle admits of two real solutions and that the two centres of inversion are inverse points with regard to the circle \(ABC\).
The coefficients \(a, b, c, a', b', c'\) are real in the quadratic expressions \[ f(x) = ax^2+bx+c, \quad \phi(x) = a'x^2+b'x+c' \] and a value of \(\lambda\) is taken so that the roots \(x_1, x_2\) of \(f(x)-\lambda\phi(x)=0\) are real. Prove (1) that, if \(f(x_1), f(x_2)\) have opposite signs, the like result holds whatever other such value \(\lambda\) has and moreover the roots are real for all real values of \(\lambda\): (2) that, if \(f(x_1), f(x_2)\) have the same sign, the like result holds for all such pairs of roots but the values of \(\lambda\) giving real roots are restricted.
Prove by means of the expansions or otherwise that, when \(n\) is a positive integer and \(x\) is positive and less than \(n\), \[ \left(1+\frac{x}{n}\right)^n < e^x < \left(1-\frac{x}{n}\right)^{-n}. \] Deduce that the inequalities are also true when \(x\) is negative and numerically less than \(n\). Prove also that, when \(x\) is positive, \(\left(1+\frac{x}{n}\right)^n\) is nearer to \(e^x\) than \(\left(1-\frac{x}{n}\right)^{-n}\) is.
Shew that for the special value \(\lambda = -\frac{2a^2b^2}{a^2+b^2}\) the conics \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda}=1 \] cut orthogonally whatever be the angle between the oblique axes of reference. Shew that the four common points are then given by \(\frac{x^2}{a^2} = \frac{y^2}{b^2} = \frac{1}{a^2+b^2}\). Shew also that these results mean geometrically that, if any rectangle be described about an ellipse, there is a concentric hyperbola through the points of contact having the sides of the rectangle as normals: shew that of the family of hyperbolas obtained by varying the rectangle one is confocal to the ellipse, the diagonals of the rectangle being then along the axes of the ellipse.
Shew that, if \(c^2
Shew from the differential coefficients that the functions \[ x - \log(1+x), \quad \frac{2x}{2+x} - \log(1+x) \] are respectively positive and negative when \(x\) is positive. Shew also that, when \(a\) and \(h\) are positive, \(\log(a+\theta h) - \log a - \theta\{\log(a+h)-\log a\}\) considered as a function of \(\theta\), has a maximum for a value of \(\theta\) between 0 and \(\frac{1}{2}\).