A column of water 30 feet long is moving behind a plug piston in a pipe of uniform diameter, with a velocity of 15 feet per second. Prove that the time average of the pressure of the water on the piston, caused by its stoppage in one-tenth of a second, is 610 lbs. per square inch.
Express the area \(S\) of a triangle in terms of the lengths of the sides. \par Prove that \[ \frac{\partial S}{\partial a} = R \cos A, \] where \(R\) is the radius of the circumcircle.
Prove that, if \(y\) is equal to \(e^x\), or if \(y\) is equal to the sum of the first \(n+1\) terms of the expansion of \(e^x\) in ascending powers of \(x\), \[ x \frac{d^2y}{dx^2} - (n+x)\frac{dy}{dx} + ny = 0. \]
Water issues vertically from the nozzle of a fire hose, the sectional area of which is one square inch, with a velocity of 130 feet per second. Find the discharge in cubic feet per second, and the horse-power of the pump engine, assuming the efficiency to be \(70\%\), and that the nozzle is 50 feet above the pump.
Prove that, if \[ y = A \cos(\log x) + B\sin(\log x), \] then \[ x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0. \]
Find the maxima and minima of the function \[ y = \sin x + \tfrac{1}{2}\sin 2x + \tfrac{1}{3}\sin 3x. \] Draw a graph of the function.
Integrate the functions \[ \frac{1}{x(x^2+a^2)}, \quad x^2\sin^2x, \quad e^x\cos 2x. \] Prove that \[ \int_0^1 x^2 \log x dx = -\tfrac{1}{9}. \]
Evaluate \[ \int \frac{x dx}{(x^2-a^2)^2+b^2x^2}, \quad a>0, b>0, \] distinguishing between the cases in which \(b<2a\) and \(b>2a\).
Sketch the curve defined by the equations \[ x=a\cos^3\theta, \quad y=a\sin^3\theta, \] and show that its total length is \(6a\).
Prove that, if \(\alpha\) is a constant, the function \[ y = A \cos\alpha x + B \sin\alpha x + \frac{1}{\alpha}\int_0^x f(\xi)\sin\alpha(x-\xi)d\xi \] satisfies the equation \[ \frac{d^2y}{dx^2} + \alpha^2y = f(x). \]