Problems

Explain how turning values and points of inflexion of the function \(y = f(x)\) can be found by studying the successive derivatives of \(y\). Find the values of \(x\) for which the function $$y = \frac{x^3 - x^2 + 4}{x^3 + x^2 + 4}$$ has turning values and discuss their character. How many real roots has the equation $$x^3(a-1) + x^2(a+1) + 4(a-1) = 0$$ for different values of \(a\)?

State Maclaurin's theorem for the expansion of a function \(y = f(x)\) in powers of \(x\). Use the theorem to obtain expansions in powers of \(x\) (up to terms in \(x^3\)) for

1. [(i)] \(\sin x\),
2. [(ii)] \(\cos x\),
3. [(iii)] \(\log_e(1 + x)\).

Find the limit as \(x\) tends towards zero of $$\frac{6[x\sin x - \sin(x^2)] + x^4}{\log_e(1 + x^2) - x^2\cos x}.$$

Specify the loci in the complex plane given by

1. [(i)] \(|z - 9| + |z + 2| = 4\),
2. [(ii)] \(|z| + |z - 1 - i| = 2\),
3. [(iii)] \(|z| = |z - x + 2ia|\),
4. [(iv)] \(|z + 1 + i| - |z - 1 - i| = 2\),

where \(z = x + iy\).

By considering the graph of \(1/x\) or otherwise, show that $$\log_e n - \log_e(n-1) > \frac{1}{n} \quad \text{and} \quad \log_e n - \log_e(n-1) < \frac{1}{2}\left(\frac{1}{n} + \frac{1}{n-1}\right),$$ where \(n\) is an integer greater than 1. The function \(f(n)\) is defined by $$f(n) = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} - \log_e n.$$ Show that \(f(n)\) decreases as \(n\) increases and that \(f(n) - \frac{1}{(2n)}\) increases as \(n\) increases. Deduce that \(f(n)\) tends to a finite limit as \(n\) tends to infinity.

The indefinite integral \(I_n\) is defined by $$I_n = \int \frac{dx}{(a^2 + x^2)^{1/n}},$$ where \(n\) is an integer greater than or equal to one. Obtain a reduction formula relating \(I_{n+2}\) to \(I_n\). If $$J_n = \int_{-a}^{a} \frac{dx}{(a^2 + x^2)^{1/n}},$$ evaluate (i) \(J_0\), (ii) \(J_5\).

Find the indefinite integrals

1. [(i)] \(\int x^2 \tan^{-1} x \, dx\),
2. [(ii)] \(\int \frac{dx}{\sqrt{x} + \sqrt{1-x}}\).

Evaluate $$\int_a^b x^2\sqrt{(x-a)(b-x)} \, dx,$$ when \(0 < a < b\).

Find the general solution of the following equations for \(y\) as a function of \(x\):

1. [(i)] \(\sec x \frac{dy}{dx} + y = \sin x\),
2. [(ii)] \((x^2 + 1)\left(x\frac{dy}{dx} - y\right) = xy^4\).

The function \(L_n(x)\) is defined by $$L_n(x) = e^x \frac{d^n}{dx^n}(x^n e^{-x}),$$ where \(n\) is a positive integer or zero. Show that \(L_n(x)\) is a polynomial of degree \(n\), that the coefficient of \(x^n\) is \((-1)^n\) and that \(L_n(0) = n!\). By substituting for \(L_n(x)\), but not for \(L_m(x)\), and integrating by parts, or otherwise, show that $$\int_0^{\infty} L_m(x)L_n(x)e^{-x}dx = \begin{cases} 0 & (n > m \geq 0), \\ (n!)^2 & (m = n). \end{cases}$$

Sketch the curve \(r = a(1 + \cos\theta)\) and find its total length. Find also the perpendicular distance between the origin and the tangent to the curve at the point \((r, \theta)\).

Find all the values of \(x\), \(y\) and \(z\) which satisfy the equations \begin{align} -y + z &= u,\\ x - z &= v,\\ -x + y &= w, \end{align} where \begin{align} v - 2w &= a,\\ -u + 3w &= b,\\ 2u - 3v &= c. \end{align}