Problems

Define exactly what is meant by the derivative \(dy/dx\) of a function \(y = f(x)\). Obtain from first principles the derivatives of

1. [(i)] \(y = \sqrt{x}\),
2. [(ii)] \(y = (1+x)^2\cos 2x\).

Are there any values of \(x\) for which these functions do not possess derivatives?

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 \((r, \theta)\).