If \(\alpha\) stands for the fifth root of 2, and \(x = \alpha+\alpha^4\), prove that \[ x^5=10x^2+10x+6. \]
Solve the equations:
\(OC\) touches a circle at \(C\) and \(OAB\) is a chord. Prove that \[ AB:OC :: BC^2-AC^2 : BC.AC. \] Prove also that, if \(OC\) is equal to the radius, the triangle \(ABC\) is greatest when the angle \(COA\) is \(60^\circ\).
The lower part of a flagstaff, of height \(a\), and the upper part, of height \(b\), subtend equal angles \(\theta\) at the top of a pedestal, of height \(c\), and at distance \(x\). Prove that \[ (b-a)x^2=(a-c)\{a^2+a(b-c)+bc\}, \text{ and that } \frac{2ab\sin^2\theta}{b-a} = a+\frac{c(a+b)}{a+b-2c}. \]
Justify the formula for measuring the length of an arc of a circle. `From \(\frac{8}{3}\) of the chord of half the arc subtract \(\frac{1}{3}\) of the chord of the whole arc.' Estimate the error involved, and, by applying the formula to an arc of \(60^\circ\), find \(\sin 15^\circ\) correct to 4 decimal places.
Trace roughly the curves \(x^2-y=2\) and \((y-3)(x+1)+8=0\) between \(x=-4\) and \(x=4\). Use your figure to find their points of intersection.
Find the limit as \(x \to a\) of \((x^n-a^n)/(x-a)\) for commensurable values of \(n\), whether positive or negative, and apply the result to the differentiation of \(x^n\). Prove that \[ \frac{d^4}{dx^4}(x^a e^x) = a^x e^x. \]
Use Maclaurin's Theorem to expand \(e^{-\cos x}\) in ascending powers of \(x\).
Express in partial fractions, and integrate with respect to \(x\), the expression \[ \frac{x^4+4x^2+9x}{(x+1)^2(x^2+1)}. \]
Evaluate \(\int_0^\infty \frac{dx}{\sqrt{x(4-x)(x-3)}}\) and \(\int_0^\infty \frac{dx}{(2+x)\sqrt{x(1+x)}}\). Shew without integration that \(\int_0^{\frac{2\pi}{3}} \frac{64d\theta}{(5+3\cos\theta)^2}\) lies between 644 and 753; and, by integrating, that its value is about 68. (Take arc tan \(\frac{1}{2}=.322\) and arc tan \(\frac{1}{4}=.165\).)