Find the volume of the portion of the paraboloid formed by rotating the parabola \(y^2=4ax\) about the axis of \(x\), contained between the planes \(x=h\) and \(x=k\).
Solve the equation \(\frac{dy}{dx}-2y=x+\cos x\).
Solution: \begin{align*} && e^{-2x}y'-2e^{-2x}y &= e^{-2x}(x+\cos x) \\ \Rightarrow && e^{-2x}y &= \int e^{-2x}(x+\cos x) \d x \\ \Rightarrow && e^{-2x} y &= -e^{-2x} \frac{1}{20} \left (5 + 10 x + 8 \cos x - 4 \sin x \right) + C \\ \Rightarrow && y &= -\frac{1}{20} \left (5 + 10 x + 8 \cos x - 4 \sin x \right) +Ce^{2x} \end{align*}
Prove that if a rational integral function \(f(x)\) is divided by \(x-a\) the remainder is \(f(a)\). Prove that if \(f(x)\) is divided by \((x-a)(x-b)\) the remainder is \[ [\{f(a)-f(b)\}x+af(b)-bf(a)]/(a-b). \]
Solve the equations \[ x+y=3, \quad x^5+y^5=17. \] Prove that if \(\epsilon\) is small the equation \[ x^3-3x+2=\epsilon x^3 \] has a root approximately equal to \(1-\epsilon+4\epsilon^2\), and find approximations to the other two roots.
If \(\frac{A}{PQ}\) be a rational proper fraction whose denominator contains two integral factors \(P, Q\) having no common factor, then \(\frac{A}{PQ}\) can be expressed as the sum of two proper fractions \(\frac{P'}{P}+\frac{Q'}{Q}\). Find the coefficient of \(x^{2n}\) in the expansion in ascending powers of \(x\) of \[ \frac{1+x+x^2}{1-x-x^2+x^3}. \]
Prove the rule for the formation of successive convergents to a continued fraction \[ \frac{a_1}{b_1+} \frac{a_2}{b_2+} \frac{a_3}{b_3+} \dots. \] Prove that the \(n\)th convergent to the continued fraction \(\frac{1}{1-} \frac{3}{4-} \frac{3}{4-} \frac{3}{4-} \dots\) is \(\frac{1}{2}(3^n-1)\).
Explain the method of mathematical induction and use it to prove that if \[ {}^nS_r = 1^r+2^r+\dots+n^r \] then \[ {}^nS_1 + {}^nS_3 = 2({}^nS_1)^2. \]
Explain the meanings of \(\frac{\partial v}{\partial r}\) and \(\frac{\partial v}{\partial x}\), where \(x,y\) are the rectangular coordinates of a point, \(r, \theta\) its polar coordinates, and illustrate them geometrically. Prove that \[ r\left\{\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}\right\} = \left(\frac{\partial v}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial y}\right)^2. \]
Prove that if \(y^3+3ax^2+x^3=0\), then \[ \frac{d^2y}{dx^2} + \frac{2a^2x^2}{y^5} = 0. \] Shew that the curve given by the above equation is everywhere concave to the axis of \(x\), and that there is a point of inflexion where \(x=-3a\).
Shew that the locus of the intersections of pairs of tangents to the curve \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta) \] which are at right angles to one another is the curve \[ x=a\{\theta+\frac{1}{2}\pi(1+\cos\theta)\}, \quad y=\frac{1}{2}\pi a \sin\theta. \] Draw both curves.