Problems

The curve \(y=ax+bx^3\) passes through the points \((-0.2, 0.0167)\) and \((0.25, 0.026)\). Prove that the tangent at the origin makes an angle of approximately 34 seconds with the \(x\)-axis. Find the radius of curvature at the origin correct to six significant figures.

Prove that the function \[ \frac{1-\cos x}{\sin(x-a)} \quad (0 < a < \pi), \] has infinitely many maxima equal to 0 and minima equal to \(2\sin a\). Sketch the graph of the function.

Sketch the locus (the cycloid) given by \[ x=a(\theta+\sin\theta), \quad y=a(1+\cos\theta), \] for values of the parameter \(\theta\) between \(0\) and \(4\pi\). Prove that the normals to this curve all touch an equal cycloid, and draw this second curve in your diagram.

Discuss the convergence of the series \[ 1+z+z^2+\dots+z^n+\dots, \] where \(z\) may be real or complex.

The function \(f(x,y)\) has the property that, for all \(x, y, t\), \[ f(tx,ty) = t^k f(x,y), \] where \(k\) is a constant. Prove that \begin{align*} x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} &= kf, \\ x\frac{\partial^2 f}{\partial x^2} + y\frac{\partial^2 f}{\partial x \partial y} &= (k-1)\frac{\partial f}{\partial x}. \end{align*}

Prove that, if \(x_0\) is an approximate solution of the equation \[ x \log_e x - x = k, \] and \(k_0=x_0 \log_e x_0 - x_0\), then a better approximation to the root is given by \[ x_0 + \frac{k-k_0}{\log_e x_0}. \] Given that \(\log_e 10 = 2.3026\) (to four places), find as good an approximation as you can to the root of \[ x \log_e x - x = 13. \]

Criticize the following arguments:

1. The equation \(y=(4x^2+3)/(x^2+1)\) defines \(y\) for all values of \(x\). \(dy/dx=0\) if \(x=0\) or \(\pm 1/\sqrt{2}\); and \(x=0\) gives a minimum value 3 for \(y\), while \(x=\pm 1/\sqrt{2}\) give equal maxima \(y=4\). For all other values of \(x\), \(y\) lies between 3 and 4.
2. \(\displaystyle\int_2^3 \frac{dx}{1-x} = \left[-\log(1-x)\right]_2^3 = -\log 2\).
3. The substitution \(\tan\theta=t\) gives \[ \int_{\pi/4}^{3\pi/4} \frac{d\theta}{2\cos^2\theta+1} = \int_{\pi/4}^{3\pi/4} \frac{\sec^2\theta \, d\theta}{2+\sec^2\theta} = \int_{1}^{-1} \frac{dt}{3+t^2} = \frac{1}{\sqrt{3}}\left[\arctan\frac{t}{\sqrt{3}}\right]_1^{-1} = \frac{1}{\sqrt{3}}\left(\frac{5\pi}{6}-\frac{\pi}{6}\right) = \frac{2\pi}{3\sqrt{3}}. \]

Obtain a reduction formula connecting the integrals \[ \int \frac{x^m \, dx}{(1+x^2)^n} \quad \text{and} \quad \int \frac{x^{m-2} \, dx}{(1+x^2)^{n-1}}. \] Prove that the value of the integral \[ \int_0^\infty \frac{x^{2k} \, dx}{(1+x^2)^{k+1}} \] decreases as the positive integer \(k\) increases. Determine the smallest (integer) value of \(k\) which makes the value of the integral less than 0.4.

Prove that, according as \(n\) is an even or odd positive integer, \[ \int_0^\pi \frac{\sin n\theta}{\sin\theta} \, d\theta = 0 \text{ or } \pi. \] If \(n\) is a positive integer, evaluate \[ \int_0^\pi \frac{\sin^2 n\theta}{\sin^2\theta} \, d\theta. \]

Determine \(P, Q, R\) as functions of \(x\) such that the equation \[ \frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy=R \] may be satisfied by \(y=x\), \(y=1\) and \(y=1/x\) for all values of \(x\) (except \(x=0\) for \(y=1/x\)). With these values of \(P, Q, R\), state what condition must be satisfied by the numerical coefficients \(a,b,c\), if the equation is also satisfied, for all \(x\) except 0, by \[ y=ax+b+\frac{c}{x}. \]