Prove by mathematical induction that $\frac{d^n}{d{x}^n}\left(x^2{\mathrm{e}}^x\right)=\left[x^2+2nx+n\left(n-1\right)\right]{\mathrm{e}}^x$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Hence or otherwise, determine the Maclaurin series of $f\left(x\right)=x^2{\mathrm{e}}^x$ in ascending powers of $x$, up to and including the term in $x^4$.

Hence or otherwise, determine the value of $\underset{x\to 0}{\lim }\left[\frac{{\left(x^2{\mathrm{e}}^x-x^2\right)}^3}{x^9}\right]$.

Determine whether $f_n$ is an odd or even function, justifying your answer.

By using mathematical induction, prove that

$f_n(x)=\frac{\sin 2^{n+1}x}{2^n\sin 2x},x\ne \frac{m\pi }{2}$ where $m\in \mathbb{Z}$.

Hence or otherwise, find an expression for the derivative of $f_n(x)$ with respect to $x$.

Show that, for $n>1$, the equation of the tangent to the curve $y=f_n(x)$ at $x=\frac{\pi }{4}$ is $4x-2y-\pi =0$.

Use integration by parts to show that $\int {\text{e}}^x\text{cos}2x\text{d}x=\frac{2{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{5}\text{cos}2x+c$,  $c\in \mathbb{R}$.

Hence, show that $\int {\text{e}}^x\text{cos}{}^{2}x\text{d}x=\frac{{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{10}\text{cos}2x+\frac{{\text{e}}^x}{2}+c$,  $c\in \mathbb{R}$.

Find the $x$-coordinates of A and of C , giving your answers in the form $a+\text{arctan}b$, where $a$, $b\in \mathbb{R}$.

Find the area enclosed by the curve and the $x$-axis between B and D, as shaded on the diagram.

Show that ${\left(\text{sin}x+\text{cos}x\right)}^2=1+\text{sin}2x$.

Show that $\text{sec}2x+\text{tan}2x=\frac{\text{cos}x+\text{sin}x}{\text{cos}x-\text{sin}x}$.

Hence or otherwise find ${\int }_0^{\frac{\pi }{6}}\left(\text{sec}2x+\text{tan}2x\right)\text{d}x$ in the form $\text{ln}\left(a+\sqrt{b}\right)$ where $a$, $b\in \mathbb{Z}$.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=f(x)$;

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=\frac{1}{f(x)}$;

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=\left|\frac{1}{f(x)}\right|$.

Find $\int f(x)\cos x\text{d}x$.

By finding $g^{\prime }(x)$ explain why $g$ is an increasing function.

Find the $x$-coordinates of the points of intersection of the two graphs.

Find the exact area of the shaded region, giving your answer in the form $p\pi +q\sqrt{3}$, where $p$, $q\in \mathbb{Q}$.

At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$-coordinate of A on the graph of $g$.

A continuous random variable $X$ has the probability density function

$f\left(x\right)=\left\{\begin{array}{cc}\frac{2}{\left(b-a\right)\left(c-a\right)}\left(x-a\right),& a\le x\le c\\ \frac{2}{\left(b-a\right)\left(b-c\right)}\left(b-x\right),& c<x\le b\\ 0,& \text{otherwise}\end{array}\right.$.

The following diagram shows the graph of $y=f\left(x\right)$ for $a\le x\le b$.

Given that $c\ge \frac{a+b}{2}$, find an expression for the median of $X$ in terms of $a, b$ and $c$.

Show that the function $f$ has a local maximum value when $x=\frac{3\pi }{4}$.

Find the $x$-coordinate of the point of inflexion of the graph of $f$.

Sketch the graph of $f$, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.

Find the area of the region enclosed by the graph of $f$ and the $x$-axis.

The curvature at any point $(x,y)$ on a graph is defined as $\kappa =\frac{\left|\frac{{\text{d}}^{2}y}{\text{d}{x}^2}\right|}{{\left(1+{\left(\frac{\text{d}y}{\text{d}x}\right)}^2\right)}^{\frac{3}{2}}}$.

Find the value of the curvature of the graph of $f$ at the local maximum point.

Find the value $\kappa$ for $x=\frac{\pi }{2}$ and comment on its meaning with respect to the shape of the graph.

The graph of $y=f\left(x\right)$ has a local maximum at A. Find the coordinates of A.

Show that there is exactly one point of inflexion, B, on the graph of $y=f\left(x\right)$.

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Sketch the graph of $y=f\left(x\right)$ showing clearly the position of the points A and B.

By using the substitution $u=\text{sec} x$ or otherwise, find an expression for $\underset{0}{\overset{\frac{\pi }{3}}{\int }}{\text{sec}}^n x \tan x dx$ in terms of $n$, where $n$ is a non-zero real number.

Find the value of $k$.

Find $\text{E}\left(X\right)$.

Show that $f$ is an odd function.

The range of $f$ is $a\le y\le b$, where $a, b\in \mathrm{ℝ}$.

Find the value of $a$ and the value of $b$.

State the equation of the vertical asymptote on the graph of $y=f\left(x\right)$.

State the equation of the horizontal asymptote on the graph of $y=f\left(x\right)$.

Use an algebraic method to determine whether $f$ is a self-inverse function.

Sketch the graph of $y=f\left(x\right)$, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

The region bounded by the $x$-axis, the curve $y=f\left(x\right)$, and the lines $x=5$ and $x=7$ is rotated through $2\mathrm{\pi }$ about the $x$-axis. Find the volume of the solid generated, giving your answer in the form $\mathrm{\pi }(a+b \ln 2)$, where $a, b \in \mathrm{\mathbb{Z}}$.

At the point (1, 1) , show that $\frac{\text{d}y}{\text{d}x}=\frac{2+\pi }{2-\pi }$.

Hence find the equation of the normal to $C$ at the point (1, 1).

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

Factorize $x^2+3x+2$.

Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates of the $y$-intercept and the local maximum.

Hence find the value of $p$ if ${\int }_0^{1}f(x)\text{d}x=\ln (p)$.

Sketch the graph of $y=f\left(\left|x\right|\right)$.

Determine the area of the region enclosed between the graph of $y=f\left(\left|x\right|\right)$, the $x$-axis and the lines with equations $x=-1$ and $x=1$.

Find $\frac{\text{d}y}{\text{d}x}$.

Find ${\int }_0^1\text{arccos}\left(\frac{x}{2}\right)\text{d}x$.

Show that $f\text{'}\text{'}\left(x\right)=-\frac{1}{4\sqrt{{\left(1+x\right)}^3}}$.

Use mathematical induction to prove that $f^{\left(n\right)}\left(x\right)={\left(-\frac{1}{4}\right)}^{n-1}\frac{\left(2n-3\right)!}{\left(n-2\right)!}{\left(1+x\right)}^{\frac{1}{2}-n}$ for $n\in \mathrm{\mathbb{Z}}, n\ge 2$.

Let $g\left(x\right)={\text{e}}^{mx}, m\in \mathrm{\mathbb{Q}}$.

Consider the function $h$ defined by $h\left(x\right)=f\left(x\right)\times g\left(x\right)$ for $x>-1$.

It is given that the $x^2$ term in the Maclaurin series for $h\left(x\right)$ has a coefficient of $\frac{7}{4}$.

Find the possible values of $m$.

Find $f^{\prime }\left(x\right)$.

Show that, if $f^{\prime }\left(x\right)=0$, then $x=2-\sqrt{3}$.

find the coordinates of the $y$-intercept.

show that there are no $x$-intercepts.

sketch the graph, showing clearly any asymptotic behaviour.

Show that $\frac{3}{x+1}-\frac{1}{x-1}=\frac{2x-4}{x^2-1}$.

The area enclosed by the graph of $y=f\left(x\right)$ and the line $y=4$ can be expressed as $\text{ln}v$. Find the value of $v$.

${\int }_{-2}^0\left(f\left(x\right)\text{ + 2}\right)\text{d}x$.

${\int }_{-2}^{0}f\left(x\text{ + 2}\right)\text{d}x$.

Using the substitution $u=\text{sin}x$, find $\int \frac{\text{co}{\text{s}}^{3}x\text{d}x}{\sqrt{\text{sin}x}}$.

Find the Maclaurin series for $f\left(x\right)$ up to and including the $x^3$ term.

Hence, find an approximate value for ${\int }_0^1{\text{e}}^{x^2} \sin \left(x^2\right)dx$.

Show that $g\left(x\right)$ satisfies the equation $g\text{'}\text{'}\left(x\right)=2(g\text{'}(x)-g(x\left)\right)$.

Hence, deduce that $g^{\left(4\right)}\left(x\right)=2\left(g\text{'}\text{'}\text{'}\left(x\right)-g\text{'}\text{'}\left(x\right)\right)$.

Using the result from part (c), find the Maclaurin series for $g\left(x\right)$ up to and including the $x^4$ term.

Hence, or otherwise, determine the value of $\underset{x\to 0}{\lim }\frac{{\text{e}}^x \cos x-1-x}{x^3}$.

Write down the $x$-coordinate of the point of inflexion on the graph of $y=f\left(x\right)$.

find the value of $f\left(1\right)$.

find the value of $f\left(4\right)$.

Sketch the curve $y=f\left(x\right)$, 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

Show that $A=\frac{\sqrt{2}\mathrm{\pi }}{2}$.

Find the value of $k$.

Show that $m=-\frac{6x}{{\left(x^2+2\right)}^2}$.

Show that the maximum value of $m$ is $\frac{27}{32}\sqrt{\frac{2}{3}}$.

Show that $\text{lo}{\text{g}}_{r^2}x=\frac{1}{2}\text{lo}{\text{g}}_{r}x$ where $r,x\in {\mathbb{R}}^{+}$.

The folium of Descartes is a curve defined by the equation $x^3+y^3-3xy=0$, shown in the following diagram.

Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the $y$-axis.

Find $t_1$ and $t_2$.

Find the displacement of the particle when $t=t_1$

Use l’Hôpital’s rule to determine the value of

$\underset{x\to 0}{\lim } \frac{2 \sin x-\sin 2x}{x^3}$.

Write $2x-x^2$ in the form $a{\left(x-h\right)}^2+k$, where $a\text{, }h\text{, }k\in \mathbb{R}$.

Hence, find the value of ${\int }_{\frac{1}{2}}^{\frac{3}{2}}\frac{1}{\sqrt{2x-x^2}}\text{d}x$.

Find the value of $a$ and the value of $b$.

State the restriction which must be placed on $x$ for this expansion to be valid.

Using the substitution $x=\tan \theta$ show that $\underset{0}{\overset{1}{\int }}\frac{1}{{\left(x^2+1\right)}^2}\text{d}x=\underset{0}{\overset{\frac{\pi }{4}}{\int }}{\cos }^2\theta \text{d}\theta$.

Hence find the value of $\underset{0}{\overset{1}{\int }}\frac{1}{{\left(x^2+1\right)}^2}\text{d}x$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$.

Find the equations of the tangents to this curve at the points where the curve intersects the line $x=1$.

Find the value of ${\int }_1^9\left(\frac{3\sqrt{x}-5}{\sqrt{x}}\right)dx$.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

Show that $g^{-1}\left(x\right)=1+\frac{\sqrt{4x^2+x}}{x}$.

State the domain of $g^{-1}$.

Given that $\left(h\circ g\right)\left(a\right)=\frac{\mathrm{\pi }}{4}$, find the value of $a$.

Give your answer in the form $p+\frac{q}{2}\sqrt{r}$, where $p, q, r\in {\mathrm{\mathbb{Z}}}^{+}$.

Use the substitution $u=x^{\frac{1}{2}}$ to find $\int \frac{\text{d}x}{x^{\frac{3}{2}}+x^{\frac{1}{2}}}$.

Hence find the value of $\frac{1}{2}\underset{1}{\overset{9}{\int }}\frac{\text{d}x}{x^{\frac{3}{2}}+x^{\frac{1}{2}}}$, expressing your answer in the form arctan $q$, where $q\in \mathbb{Q}$.

A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at $t$ = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms⁻¹. Let the position of the car be X and let OĈX = θ.

Find $\frac{\text{d}\theta }{\text{d}t}$, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .

Use l’Hôpital’s rule to determine the value of $\underset{x\to 0}{\lim }\left(\frac{2x \cos \left(x^2\right)}{5 \tan x}\right)$.

By solving an appropriate differential equation, show that the particle’s velocity at time $t$ is given by $v\left(t\right)=(1+v_0){\text{e}}^{-t}-1$.

Show that the time $T$ taken for the particle to reach $s_{\text{max}}$ satisfies the equation ${\text{e}}^T=1+v_0$.

By solving an appropriate differential equation and using the result from part (b) (i), find an expression for $s_{\text{max}}$ in terms of $v_0$.

By using the result to part (b) (i), show that $v\left(T-k\right)={\text{e}}^k-1$.

Deduce a similar expression for $v(T+k)$ in terms of $k$.

Hence, show that $v\left(T-k\right)+v\left(T+k\right)\ge 0$.

Use l’Hôpital’s rule to find $\underset{x\to 0}{\lim }\left(\frac{\text{arctan} 2x}{\tan 3x}\right)$.

Using implicit differentiation, or otherwise, find $\frac{\text{d}y}{\text{d}x}$ for each curve in terms of $x$ and $y$.

Let P($a$, $b$) be the unique point where the curves $C_1$ and $C_2$ intersect.

Show that the tangent to $C_1$ at P is perpendicular to the tangent to $C_2$ at P.

Solve the differential equation $\frac{dy}{dx}=\frac{\ln {\displaystyle }{\displaystyle 2}{\displaystyle x}}{x^2}-\frac{2y}{x}, x>0$, given that $y=4$ at $x=\frac{1}{2}$.

Give your answer in the form $y=f\left(x\right)$.

A particle moves in a straight line such that at time $t$ seconds $(t⩾0)$, its velocity $v$, in $\text{m}{\text{s}}^{-1}$, is given by $v=10t{\text{e}}^{-2t}$. Find the exact distance travelled by the particle in the first half-second.

Show that the graph of $y=q(x)$ is concave up for $x>1$.

Sketch the graph of $y=q(x)$ showing clearly any intercepts with the axes.

Show that $\frac{\text{d}y}{\text{d}x}=\frac{y \cos \left(xy\right)}{2y-x \cos \left(xy\right)}$.

Prove that, when $\frac{\text{d}y}{\text{d}x}=0 , y=\pm 1$.

Hence find the coordinates of all points on $C$, for $0<x<4\mathrm{\pi }$, where $\frac{\text{d}y}{\text{d}x}=0$.

Find the coordinates of the points on the curve $y^3+3x{y}^2-x^3=27$ at which $\frac{\text{d}y}{\text{d}x}=0$.

Find $\int \arcsin x\text{d}x$

Find the first two derivatives of $f\left(x\right)$ and hence find the Maclaurin series for $f\left(x\right)$ up to and including the $x^2$ term.

Show that the coefficient of $x^3$ in the Maclaurin series for $f\left(x\right)$ is zero.

Using the Maclaurin series for $\text{arctan}x$ and ${\text{e}}^{3x}-1$, find the Maclaurin series for $\text{arctan}\left({\text{e}}^{3x}-1\right)$ up to and including the $x^3$ term.

Hence, or otherwise, find $\underset{x\to 0}{\text{lim}}\frac{f\left(x\right)-1}{\text{arctan}\left({\text{e}}^{3x}-1\right)}$.

Find the equation of the tangent to the curve $y={\text{e}}^{2x}–3x$ at the point where $x=0$.

Given that ${\int }_0^{\text{ln}k}{\text{e}}^{2x}\text{d}x=12$, find the value of $k$.

By using the substitution $u=\sin x$, find $\int \frac{\sin x \cos x}{{\sin }^2 x-\sin x-2}dx$.

Show that the volume of the cone may be expressed by $V=\frac{\pi }{3}\left(2R{h}^2-h^3\right)$.

Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{32\pi R^3}{81}$.

Hence, or otherwise, find $\underset{0}{\overset{\pi }{\int }}{\text{e}}^{-x}\text{sin}x\text{d}x$.