(i)     Show that \(\frac{{\rm{d}}}{{{\rm{d}}\theta }}(\sec \theta \tan \theta  + \ln (\sec \theta  + \tan \theta )) = 2{\sec ^3}\theta \) .

(ii)     Hence write down \(\int {{{\sec }^3}\theta {\rm{d}}\theta } \) .

Consider the differential equation \((1 + {x^2})\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + xy = 1 + {x^2}\) given that \(y = 1\) when \(x = 0\) .

  (i)     Use Euler’s method with a step length of \(0.1\) to find an approximate value for y when \(x = 0.3\) .

  (ii)     Find an integrating factor for determining the exact solution of the differential equation.

  (iii)     Find the solution of the equation in the form \(y = f(x)\) .

  (iv)     To how many significant figures does the approximation found in part (i) agree with the exact value of \(y\) when \(x = 0.3\) ?

(i)     Show that the improper integral \(\int_0^\infty  {\frac{1}{{{x^2} + 1}}} {\rm{d}}x\) is convergent.

(ii)     Use the integral test to deduce that the series \(\sum\limits_{n = 0}^\infty  {\frac{1}{{{n^2} + 1}}} \) is convergent, giving reasons why this test can be applied.

(i)     Show that the series \(\sum\limits_{n = 0}^\infty  {\frac{{{{( - 1)}^n}}}{{{n^2} + 1}}} \) is convergent.

(ii)     If the sum of the above series is \(S\), show that \(\frac{3}{5} < S < \frac{2}{3}\) .

For the series \(\sum\limits_{n = 0}^\infty  {\frac{{{x^n}}}{{{n^2} + 1}}} \)

  (i)     determine the radius of convergence;

  (ii)     determine the interval of convergence using your answers to (b) and (c).

By considering integration as the reverse of differentiation, show that for

\(0 \leqslant x < \frac{\pi }{2}\)

\[\int {\sec x{\text{d}}x = \ln (\sec x + \tan x) + C.} \]

Hence, using integration by parts, show that

\[\int {{{\sec }^3}x{\text{d}}x = \frac{1}{2}\left( {\sec x\tan x + \ln (\sec x + \tan x)} \right) + C.} \]

Find an integrating factor and hence solve the differential equation, giving your answer in the form \(y = f(x)\).

Starting with the differential equation, show that

\[\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} + y = 2{\sec ^2}x\tan x.\]

Hence, by using your calculator to draw two appropriate graphs or otherwise, find the \(x\)-coordinate of the point of inflection on the graph of \(y = f(x)\).

Using Euler’s method with increments of \(0.2\), find an approximate value for \(y\) when \(x = 1\).

Explain how Euler’s method could be improved to provide a better approximation.

Solve the differential equation to find an exact value for \(y\) when \(x = 1\).

(i)     Find the first three non-zero terms of the Maclaurin series for \(y\).

(ii)     Hence find an approximate value for \(y\) when \(x = 1\).

Write down the general term.

Find the interval of convergence.

Solve the differential equation \((u + 3{v^3})\frac{{{\rm{d}}v}}{{{\rm{d}}u}} = 2v\) , giving your answer in the form \(u = f(v)\) .

Show that \(f''(x) = \frac{{ - 1}}{{1 + \sin x}}\) .

Determine the Maclaurin series for \(f(x)\) as far as the term in \({x^4}\) .

Deduce the Maclaurin series for \(\ln (1 - \sin x)\) as far as the term in \({x^4}\) .

By combining your two series, show that \(\ln \sec x = \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{12}} +  \ldots \) .

Hence, or otherwise, find \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln \sec x}}{{x\sqrt x }}\) .

He makes 5 independent measurements of the concentration of a particular solution and correctly calculates the following confidence interval for \(\mu \) .

[\(22.7\) , \(26.1\)]

Determine the confidence level of this interval.

He is now given a different solution and is asked to determine a \(95\%\) confidence interval for its concentration. The confidence interval is required to have a width less than \(2\). Find the minimum number of independent measurements required.

Show that, for \(n \ge 2\) , \({S_{2n}} > {S_n} + \frac{1}{2}\) .

Deduce that \({S_{2m + 1}} > {S_2} + \frac{m}{2}\) .

Hence show that the sequence \(\left\{ {{S_n}} \right\}\) is divergent.

Find the general solution of the differential equation, expressing your answer in the form \(f(x,{\text{ }}y) = c\), where \(c\) is a constant.

(i)     Hence find the particular solution passing through the points \((1,{\rm{  \pm }}\sqrt 2 )\).

(ii)     Sketch the graph of your solution and name the type of curve represented.

(i)     Write down the particular solution passing through the points \((1,{\text{ }} \pm 1)\).

(ii)     Give a geometrical interpretation of this solution in relation to part (b).

(i)     Find the general solution of the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{x}{y} + \frac{y}{x}\), where \(xy \ne 0\).

(ii)     Find the particular solution passing through the point \((1,{\text{ }}\sqrt 2 )\).

(iii)     Sketch the particular solution.

(iv)     The graph of the solution only contains points with \(\left| x \right| > a\).

Find the exact value of \(a,{\text{ }}a > 0\).

Using a Taylor series, find a quadratic approximation for \(f(x) = \sin x\) centred about \(x = \frac{{3\pi }}{4}\).

When using this approximation to find angles between \(130^\circ\) and \(140^\circ\), find the maximum value of the Lagrange form of the error term.

Hence find the largest number of decimal places to which \(\sin x\) can be estimated for angles between \(130^\circ\) and \(140^\circ\).

Explain briefly why the same maximum value of error term occurs for \(g(x) = \cos x\) centred around \(\frac{\pi }{4}\) when finding approximations for angles between \(40^\circ\) and \(50^\circ\).

Assuming the series for \({{\rm{e}}^x}\) , find the first five terms of the Maclaurin series for\[\frac{1}{{\sqrt {2\pi } }}{{\rm{e}}^{\frac{{ - {x^2}}}{2}}} {\rm{  .}}\]

(i)      Use your answer to (a) to find an approximate expression for the cumulative distributive function of \({\rm{N}}(0,1)\) .

(ii)     Hence find an approximate value for \({\rm{P}}( - 0.5 \le Z \le 0.5)\) , where \(Z \sim {\rm{N}}(0,1)\) .

State and justify an appropriate test procedure giving the null and alternate hypotheses.

What is the critical region for the sample mean if the probability of a Type I error is to be \(3.5\%\)?

If the mean weight of the bags is actually \(28\).1 kg, what would be the probability of a Type II error?

The diagram shows a sketch of the graph of \(y = {x^{ - 4}}\) for \(x > 0\) .

By considering this sketch, show that, for \(n \in {\mathbb{Z}^ + }\) ,\[\sum\limits_{r = n + 1}^\infty  {\frac{1}{{{r^4}}}}  < \int_n^\infty  {\frac{{{\rm{d}}x}}{{{x^4}}}}  < \sum\limits_{r = n}^\infty  {\frac{1}{{{r^4}}}} .\]

Let \(S = \sum\limits_{r = 1}^\infty  {\frac{1}{{{r^4}}}} \) .

Use the result in (a) to show that, for \(n \ge 2\) , the value of \(S\) lies between

\(\sum\limits_{r = 1}^{n - 1} {\frac{1}{{{r^4}}}}  + \frac{1}{{3{n^3}}}\) and \(\sum\limits_{r = 1}^n {\frac{1}{{{r^4}}}}  + \frac{1}{{3{n^3}}}\) .

(i)     Show that, by taking \(n = 8\) , the value of \(S\) can be deduced correct to three decimal places and state this value.

(ii)     The exact value of \(S\) is known to be \(\frac{{{\pi ^4}}}{N}\)where \(N \in {\mathbb{Z}^ + }\) . Determine the value of \(N\) .

Now let \(T = \sum\limits_{r = 1}^\infty  {\frac{{{{( - 1)}^{r + 1}}}}{{{r^4}}}} \) .

Find the value of \(T\) correct to three decimal places.

Show that

(i)     \(\frac{{{\text{d}}{f_n}(x)}}{{{\text{d}}x}} = n{g_n}(x)\);

(ii)     \(\frac{{{\text{d}}{g_n}(x)}}{{{\text{d}}x}} = (n + 1){f_{n + 2}}(x) - n{f_n}(x)\).

(i)     Use these results to show that the Maclaurin series for the function \({f_5}(x)\) up to and including the term in \({x^4}\) is \(1 + \frac{5}{2}{x^2} + \frac{{85}}{{24}}{x^4}\).

(ii)     By considering the general form of its higher derivatives explain briefly why all coefficients in the Maclaurin series for the function \({f_5}(x)\) are either positive or zero.

(iii)     Hence show that \({\sec ^5}(0.1) > 1.02535\).

Use Euler’s method with a step length of \(0.1\) to find an approximate value for \(y\) when \(x = 0.3\) .

(i)     By differentiating the above differential equation, obtain an expression involving \(\frac{{{{\rm{d}}^{\rm{2}}}y}}{{{\rm{d}}{x^2}}}\) .

(ii)     Hence determine the Maclaurin series for \(y\) up to the term in \({{x^2}}\) .

(iii)     Use the result in part (ii) to obtain an approximate value for \(y\) when \(x = 0.3\) .

(i)     Show that \(\sec x + \tan x\) is an integrating factor for solving this differential equation.

(ii)     Solve the differential equation, giving your answer in the form \(y = f(x)\) .

(iii)     Hence determine which of the two approximate values for y when \(x = 0.3\) , obtained in parts (a) and (b), is closer to the true value.

The function \(f\) is defined by \(f(x) = \frac{{{{\rm{e}}^x} + {{\rm{e}}^{ - x}}}}{2}\) .

  (i)     Obtain an expression for \({f^{(n)}}(x)\) , the nth derivative of \(f(x)\) with respect to \(x\).

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

  (iii)     Use your result to find a rational approximation to \(f\left( {\frac{1}{2}} \right)\) .

  (iv)     Use the Lagrange error term to determine an upper bound to the error in this approximation.

Use the integral test to determine whether the series \(\sum\limits_{n = 1}^\infty  {\frac{{\ln n}}{{{n^2}}}} \) is convergent or divergent.

Use l’Hôpital’s rule to show that \(\mathop {\lim }\limits_{x \to \infty } \frac{{{x^3}}}{{{{\text{e}}^x}}} = 0\).

(i)     Find \({\text{E}}({X^2})\).

(ii)     Show that \({\text{Var}}(X) = 2\).

State the central limit theorem.

Find the probability that the sample mean is less than 2.3.

Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).

Show that \(\left( {5x + y} \right)\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 1 - {\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)^2}\).

Show that \(\left( {5x + y} \right)\frac{{{{\text{d}}^3}y}}{{{\text{d}}{x^3}}} =  - 5\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} - 3\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)\left( {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right)\).

Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).

Find the value of \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{x} - \cot x} \right)\) .

Find the interval of convergence of the infinite series\[\frac{{(x + 2)}}{{3 \times 1}} + \frac{{{{(x + 2)}^2}}}{{{3^2} \times 2}} + \frac{{{{(x + 2)}^3}}}{{{3^3} \times 3}} +  \ldots \]

(i)     Find the Maclaurin series for \(\ln (1 + \sin x)\) up to and including the term in \({x^3}\) .

(ii)     Hence find a series for \(\ln (1 - \sin x)\) up to and including the term in \({x^3}\) .

(iii)     Deduce, by considering the difference of the two series, that \(\ln 3 \simeq \frac{\pi }{3}\left( {1 + \frac{{{\pi ^2}}}{{216}}} \right)\) .

Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} + y\tan x = 2{\cos ^4}x\) given that \(y = 1\) when \(x = 0\).

(a)     Solve the differential equation, giving your answer in the form \(y = f(x)\).

(b)     (i)     By differentiating both sides of the differential equation, show that

\[\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} + y =  - 10\sin x{\cos ^3}x\]

(ii)     Hence find the first four terms of the Maclaurin series for \(y\).

(a)     (i)     Using l’Hôpital’s rule, show that

\[\mathop {\lim }\limits_{x \to \infty } \frac{{{x^n}}}{{{{\text{e}}^{\lambda x}}}} = 0;{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}\lambda  \in {\mathbb{R}^ + }\]

(ii)     Using mathematical induction on \(n\), prove that

\[\int_0^\infty  {{x^n}{{\text{e}}^{ - \lambda x}}{\text{d}}x = \frac{{n!}}{{{\lambda ^{n + 1}}}};{\text{ }}} n \in \mathbb{N},{\text{ }}\lambda  \in {\mathbb{R}^ + }\]

(b)     The random variable \(X\) has probability density function

\[f(x) = \left\{ \begin{array}{l}\frac{{{\lambda ^{n + 1}}{x^n}{{\rm{e}}^{ - \lambda x}}}}{{n!}}x \ge 0,n \in {\mathbb{Z}^ + },\lambda  \in {\mathbb{R}^ + }\\{\rm{otherwise}}\end{array} \right.\]

Giving your answers in terms of \(n\) and \(\lambda \), determine

(i)     \({\text{E}}(X)\);

(ii)     the mode of \(X\).

(c)     Customers arrive at a shop such that the number of arrivals in any interval of duration \(d\) hours follows a Poisson distribution with mean \(8d\). The third customer on a particular day arrives \(T\) hours after the shop opens.

(i)     Show that \({\text{P}}(T > t) = {{\text{e}}^{ - 8t}}\left( {1 + 8t + 32{t^2}} \right)\).

(ii)     Find an expression for the probability density function of \(T\).

(iii)     Deduce the mean and the mode of \(T\).

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( x \right)\).

The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\) for \( - 4 \leqslant x \leqslant 4,\,\, - 4 \leqslant y \leqslant 4\) is shown in the following diagram.

Explain why the slope field indicates that the only linear solution is \(y =  - x - 1\).

Given that all the isoclines from a slope field of a differential equation are straight lines through the origin, find two examples of the differential equation.