Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^3} - {x^3}\) for which \(y = 1\) when \(x = 0\). Use Euler’s method with a step length of \(0.1\) to find an approximation for the value of \(y\) when \(x = 0.4\).

Use the integral test to determine whether or not \(\sum\limits_{n = 2}^\infty  {\frac{1}{{n{{\left( {{\text{ln}}\,n} \right)}^2}}}} \) converges.

Using l’Hôpital’s rule, show that

\(\mathop {\lim }\limits_{x \to \infty } {x^n}{{\text{e}}^{ - x}} = 0\) where \(n \in \mathbb{N}\).

Show that, for \(n \in {\mathbb{Z}^ + }\),

\[{I_n} = \alpha {{\text{e}}^{ - 1}} + \beta n{I_{n - 1}}\]

where \(\alpha \), \(\beta \) are constants to be determined.

Determine the value of \({I_3}\), giving your answer as a multiple of \({{\text{e}}^{ - 1}}\).

Show that \(\frac{{{\text{d}}z}}{{{\text{d}}x}} = \frac{2}{{{{\left( {y + x} \right)}^2}}}\left( {x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y} \right)\).

Show that the differential equation \(f\left( x \right)\left( {x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y} \right) = {y^2} - {x^2}\) can be written as \(f\left( x \right)\frac{{{\text{d}}z}}{{{\text{d}}x}} = 2z\).

Hence show that the solution to the differential equation \(\left( {x - 3} \right)\left( {x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y} \right) = {y^2} - {x^2}\) given that \(x = 4\) when \(y = 5\) is \(\frac{{y - x}}{{y + x}} = {\left( {\frac{{x - 3}}{3}} \right)^2}\).

Find the interval of convergence of the series \(\sum\limits_{k = 1}^\infty  {\frac{{{{(x - 3)}^k}}}{{{k^2}}}} \).

continuous at \(x = 0\);

differentiable at \(x = 0\).

Show that the series is conditionally convergent but not absolutely convergent.

Show that \(S > 0.4\) .

Show that \({f^{(4)}}x = f(x)\);

By considering derivatives of \(f\), determine the first three non-zero terms of the Maclaurin series for \(f(x)\).

Write down a series in terms of \(\mu \) for the probability \(p = {\text{P}}[X \equiv 0(\bmod 4)]\).

Show that \(p = {{\text{e}}^{ - \mu }}f(\mu )\).

Determine the numerical value of \(p\) when \(\mu  = 3\).

Find the general solution of the differential equation \((1 - {x^2})\frac{{{\rm{d}}y}}{{{\rm{d}}x}} = 1 + xy\) , for \(\left| x \right| < 1\) .

(i)     Show that the solution \(y = f(x)\) that satisfies the condition \(f(0) = \frac{\pi }{2}\) is \(f(x) = \frac{{\arcsin x + \frac{\pi }{2}}}{{\sqrt {1 - {x^2}} }}\) .

(ii)     Find \(\mathop {\lim }\limits_{x \to  - 1} f(x)\) .

Calculate the following limit

\(\mathop {\lim }\limits_{x \to 0} \frac{{{2^x} - 1}}{x}\) .

Calculate the following limit

\(\mathop {\lim }\limits_{x \to 0} \frac{{{{(1 + {x^2})}^{\frac{3}{2}}} - 1}}{{\ln (1 + x) - x}}\) .

By evaluating successive derivatives at \(x = 0\) , find the Maclaurin series for \(\ln \cos x\) up to and including the term in \({x^4}\) .

Consider \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln \cos x}}{{{x^n}}}\) , where \(n \in \mathbb{R}\) .

Using your result from (a), determine the set of values of \(n\) for which

  (i)     the limit does not exist;

  (ii)     the limit is zero;

  (iii)     the limit is finite and non-zero, giving its value in this case.

Given that the series \(\sum\limits_{n = 1}^\infty  {{u_n}} \) is convergent, where \({u_n} > 0\), show that the series \(\sum\limits_{n = 1}^\infty  {u_n^2} \) is also convergent.

State the converse proposition.

By giving a suitable example, show that it is false.

(i)     Sum the series \(\sum\limits_{r = 0}^\infty  {{x^r}} \) .

(ii)     Hence, using sigma notation, deduce a series for

  (a)     \(\frac{1}{{1 + {x^2}}}\) ;

  (b)     \(\arctan x\) ;

  (c)     \(\frac{\pi }{6}\) .

Show that \(\sum\limits_{n = 1}^{100} {n! \equiv 3(\bmod 15)} \) .

(a)     Assuming the Maclaurin series for \({{\text{e}}^x}\), determine the first three non-zero terms in the Maclaurin expansion of \(\frac{{{{\text{e}}^x} - {{\text{e}}^{ - x}}}}{2}\).

(b)     The random variable \(X\) has a Poisson distribution with mean \(\mu \). Show that \({\text{P}}\left( {X \equiv 1(\bmod 2)} \right) = a + b{{\text{e}}^{c\mu }}\) where \(a\), \(b\) and \(c\) are constants whose values are to be found.

Use l’Hôpital’s rule to find \(\mathop {\lim }\limits_{x \to 0} (\csc x - \cot x)\).

QUESTION 1

QUESTION 1

Using l’Hôpital’s Rule, determine the value of\[\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{1 - \cos x}} .\]

(i)      Find the range of values of \(n\) for which \(\int_1^\infty  {{x^n}{\rm{d}}x} \) exists.

(ii)     Write down the value of \(\int_1^\infty  {{x^n}{\rm{d}}x} \) in terms of \(n\) , when it does exist.

Find the solution to the differential equation

\((\cos x - \sin x)\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + (\cos x + \sin x)y = \cos x + \sin x\) ,

given that \(y = - 1\) when \(x = \frac{\pi }{2}\) .

Show that \(f''(x) = - 2{{\rm{e}}^x}\sin x\) .

Determine the Maclaurin series for \(f(x)\) up to and including the term in \({x^4}\) .

By differentiating your series, determine the Maclaurin series for \({{\rm{e}}^x}\sin x\) up to the term in \({x^3}\) .

Differentiate the expression \({x^2}\tan y\) with respect to \(x\), where \(y\) is a function of \(x\).

Hence solve the differential equation \({x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} + x\sin 2y = {x^3}{\cos ^2}y\) given that \(y = 0\) when \(x = 1\). Give your answer in the form \(y = f(x)\).

Solve the differential equation \(x\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + 2y = \sqrt {1 + {x^2}} \)

given that \(y = 1\) when \(x = \sqrt 3 \) .

Consider the infinite series \(S = \sum\limits_{n = 1}^\infty  {\frac{{{x^n}}}{{{2^{2n}}\left( {2{n^2} - 1} \right)}}} \).

(a)     Determine the radius of convergence.

(b)     Determine the interval of convergence.

Solve the following differential equation\[(x + 1)(x + 2)\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + y = x + 1\]giving your answer in the form \(y = f(x)\) .

Given that \(\frac{{{\rm{d}}x}}{{{\rm{d}}y}} + 2y\tan x = \sin x\) , and \(y = 0\) when \(x = \frac{\pi }{3}\) , find the maximum value of y.