Prove, under conditions to be stated, that \[ f(b)-f(a)=(b-a)f'(x), \] where \(x\) is some value between \(a\) and \(b\). Illustrate this result geometrically. Extend the formula to obtain a measure of the departure of a curve from its tangent at a given point.
The graph in rectangular coordinates of a polynomial of the fourth degree in \(x\) is found to touch the \(x\)-axis at \((-a,0)\) and have a point of inflection at \((a,0)\). Show that the curve must also pass through the point \((2a,0)\) and possesses another point of inflection on the line \(x=-a/2\).
Solution: Let \(p(x) = c(x+a)^2(x-a)(x-p)\), since it touches \((-a,0)\) and also the point \((a,0)\). Since there is a point of inflection at \((a,0)\) the second derivative is \(0\) there, ie \begin{align*} && p(x) &= c(x^2-a^2)(x^2+(a-p)x-ap) \\ &&&=c(x^4+(a-p)x^3-(ap+a^2)x^2-a^2(a-p)x+a^3p)\\ && p''(x) &= c(12x^2+6(a-p)x-2(ap+a^2))\\ && 0 &= p''(a) \\ &&&= c(12a^2+6(a-p)a-2(ap+a^2)) \\ &&&= c(16a^2-8ap) \\ &&&= 8ac(2a-p) \\ \Rightarrow && p = 2a \\ \Rightarrow && p(2a) &=0 \quad \text{ie curve passes through }(2a,0) \\ && p''(x) &= c(12x^2-6ax-2(2a^2+a^2))\\ &&&= c(12x^2-6ax-6a^2) \\ &&&= 6c(2x^2-ax-a^2)\\ &&&= 6c(2x-a)(x-a) \\ \end{align*} Therefore there is a point of inflection when \(x = -a/2\) since it is clear \(p''\) changes sign at this point.
Find for what ranges of \(x\) the function \(\dfrac{\log x}{x}\) increases as \(x\) increases, and decreases as \(x\) increases. Hence show that if \(n\) is a given positive number and \(x\) is a positive real variable, the equation \(x=n^x\) has two roots, one root, or no root according to the value of \(n\), and state the critical values of \(n\) concerned.
A right circular cone has unit volume. Show that its total surface area, including the base, cannot be less than \(2(9\pi)^{\frac{1}{3}}\). If such a cone has unit total surface area, what would be its maximum volume?
Obtain an expression for the area of a closed oval curve of polar equation \(r=r(\theta)\) in the two cases when the pole (\(r=0\)) is inside the curve and when it is outside the curve. A circle of radius \(a\) has centre \(C\), and a point \(O\) is taken at distance \(c (< a)\) from its centre. The foot of the perpendicular from \(O\) to a tangent to the circle is \(P\). Show that the locus of \(P\) is a closed curve of area \(\pi(a^2+\frac{1}{2}c^2)\).
Two uniform thin rods \(AB, BC\), each of length \(2a\) and of weights \(W_1, W_2\) respectively, are held together rigidly by a bolt and nut at \(B\) so as to form a V, enclosing an angle \(2\beta\); the axis of the bolt is perpendicular to the plane of the V. The rods are hung from a smooth horizontal rail by light rings attached at \(A\) and \(C\). Find the force and the frictional couple exerted at \(B\) by the rod \(BC\) on the rod \(AB\). Show these in a clear diagram. Deduce that the frictional couple at \(B\) may be reduced to zero by the application of equal and opposite horizontal forces at \(A\) and \(C\) of magnitude \(\frac{1}{4}(W_1+W_2)\tan\beta\).
Show that the centre of gravity of a hemispherical bowl, of radius \(a\) and made of uniform thin sheet material, is distant \(\frac{1}{2}a\) from the plane of the rim. The bowl, whose weight is \(W\), can rest in equilibrium with its curved surface in contact with an inclined plane, rough enough to prevent slipping, and a particle of weight \(sW\) fastened to the lowest point of the rim. If \(\beta\) is the inclination of the plane to the horizontal and \(2s = \tan\gamma\), show that \((2\cos\gamma + \sin\gamma)\sin\beta \le 1\).
A uniform rigid square lamina \(ABCD\), of weight \(W\), rests, with the diagonal \(AC\) vertical and \(A\) uppermost, on two parallel horizontal rails which are perpendicular to the plane of the lamina and are in contact with the lamina at \(E, F\), the mid-points of \(DC, CB\) respectively. It may be supposed that the plane of the lamina is kept vertical by smooth constraints. A force \(P\), parallel to \(DB\), is applied at \(A\) and is increased from zero, and equilibrium is ultimately broken. If this occurs by the lamina beginning to rotate about \(F\), show that the coefficient of friction at \(F\) is at least \(\frac{1}{2}\). If, on the other hand, equilibrium is broken by slipping at \(E\) and \(F\), where the coefficients of friction are \(\mu_1, \mu_2\) respectively, show that, when slipping occurs, the ratio of the normal reaction at \(E\) to that at \(F\) is \((1-2\mu_2):(1+2\mu_1)\). Show also that in these circumstances the value of \(P\) is then \[ \frac{W(\mu_1+\mu_2)}{2+\mu_1-\mu_2+4\mu_1\mu_2}. \]
A circular cylinder of radius \(a\) is fixed with its axis horizontal. On the cylinder rests a thick plank with its length horizontal and perpendicular to the generators and its centre of gravity at a height \(b\) vertically above the highest generator. The surfaces are sufficiently rough to prevent slipping. Show that the plank is in stable or unstable equilibrium according as \(b< a\) or \(b \ge a\).
A gun, situated on level ground, is firing at a vehicle which is moving directly away from the gun with velocity \(V\). At the instant of firing the vehicle is distant \(l\) from the gun, and at the same level. Assuming that the vehicle is within range and that \(V\) is small compared with the speed \(U\) of projection, show that on account of the motion of the vehicle the elevation of the gun should be increased from \(\theta_0\) (corresponding to a stationary target) to \(\theta_0+(V/U)\sin\theta_0\sec 2\theta_0\), approximately, provided that \(\theta_0\) is not nearly equal to \(\frac{1}{4}\pi\). Comment on this restriction. Show also that the time of flight of the projectile is greater by \((V/g)\tan 2\theta_0\), approximately, than the value corresponding to a stationary target.