Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y=f\left(x\right)$.

sketch the graph of $y=f\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

sketch the graph of $y=f^{-1}\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

Hence, or otherwise, solve the inequality $f\left(x\right)>f^{-1}\left(x\right)$.

Find the largest possible domain $D$ for $f$ to be a function.

Sketch the graph of $y=f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

Explain why $f$ is an even function.

Explain why the inverse function $f^{-1}$ does not exist.

Find the inverse function $g^{-1}$ and state its domain.

Find $g^{\prime }(x)$.

Hence, show that there are no solutions to $g^{\prime }(x)=0$;

Hence, show that there are no solutions to $(g^{-1}{)}^{\prime }(x)=0$.

Plant $B$.

Plant $A$ correct to three significant figures.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

For $t>6$, prove that Plant $A$ was always taller than Plant $B$.

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

Write down the maximum and minimum value of $v$.

Write down two transformations that will transform the graph of $y=v\left(t\right)$ onto the graph of $y=i\left(t\right)$.

Sketch the graph of $y=p\left(t\right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left(t\right)$ ≥ 3.

Find $p_{av}$(0.007).

With reference to your graph of $y=p\left(t\right)$ explain why $p_{av}\left(T\right)$ > 0 for all $T$ > 0.

Given that $p\left(t\right)$ can be written as $p\left(t\right)=a\text{sin}\left(b\left(t-c\right)\right)+d$ where $a$, $b$, $c$, $d$ > 0, use your graph to find the values of $a$, $b$, $c$ and $d$.

$x$-axis.

$y$-axis.

Write down the equation of the vertical asymptote of the graph of $f$.

The oblique asymptote of the graph of $f$ can be written as $y=ax+b$ where $a, b\in \mathrm{\mathbb{Q}}$.

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

Sketch the graph of $f$ for $-30\le x\le 30$, clearly indicating the points of intersection with each axis and any asymptotes.

Express $\frac{1}{f\left(x\right)}$ in partial fractions.

Hence find the exact value of $\underset{0}{\overset{3}{\int }}\frac{1}{f\left(x\right)}dx$, expressing your answer as a single logarithm.

Write down the range of $f$.

Find $f^{-1}\left(x\right)$, stating its domain.

In the context of the population model, interpret the meaning of $\frac{dP}{dt}$.

Show that $\frac{d^{2}P}{d{t}^2}=k^{2}P\left(1-\frac{P}{N}\right)\left(1-\frac{2P}{N}\right)$.

Hence show that the population of marsupials will increase at its maximum rate when $P=\frac{N}{2}$. Justify your answer.

Hence determine the maximum value of $\frac{dP}{dt}$ in terms of $k$ and $N$.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt=\ln \frac{P}{P_0}\left(\frac{N-P_0}{N-P}\right)$.

After $10$ years, the population of marsupials is $3P_0$. It is known that $N=4P_0$.

Find the value of $k$ for this population model.

Determine an expression for $f^{\prime }(x)$ in terms of $x$.

Sketch a graph of $y=f^{\prime }(x)$ for .

Find the $x$-coordinate(s) of the point(s) of inflexion of the graph of $y=f(x)$, labelling these clearly on the graph of $y=f^{\prime }(x)$.

Express $\sin x$ in terms of $u$.

Express $\sin 2x$ in terms of $u$.

Hence show that $f(x)=0$ can be expressed as $u^3-7u^2+15u-9=0$.

Solve the equation $f(x)=0$, giving your answers in the form $\arctan k$ where $k\in \mathbb{Z}$.

Sketch the graph of $y=x^4-6x^3-2x^2+58x-51$, stating clearly the coordinates of any maximum and minimum points and intersections with axes.

Hence, or otherwise, state the condition on $k\in \mathbb{R}$ such that all roots of the equation $P\left(z\right)=k$ are real.

Show that $\frac{\text{d}y}{\text{d}x}=-\left(\frac{1+y\text{sin}\left(xy\right)}{1+x\text{sin}\left(xy\right)}\right)$.

Find the coordinates of P and Q.

Given that the gradients of the tangents to C at P and Q are m₁ and m₂ respectively, show that m₁ × m₂ = 1.

Find the coordinates of the three points on C, nearest the origin, where the tangent is parallel to the line $y=-x$.

Show that $t$ + 1 is an integrating factor for this differential equation.

Hence, by solving this differential equation, show that $x\left(t\right)=\frac{200-40{\text{e}}^{-\frac{t}{4}}\left(t+5\right)}{t+1}$.

Sketch the graph of $x$ versus $t$ for 0 ≤ $t$ ≤ 60 and hence find the maximum amount of salt in the tank and the value of $t$ at which this occurs.

Find the value of $t$ at which the amount of salt in the tank is decreasing most rapidly.

The rate of change of the amount of salt leaving the tank is equal to $\frac{x}{t+1}$.

Find the amount of salt that left the tank during the first 60 minutes.

Use Euler’s method, with a step length of $0.1$, to find an approximate value of $y$ when $x=1.5$.

Use the substitution $y=vx$ to show that $x\frac{dv}{dx}=v^2-v-2$.

By solving the differential equation, show that $y=\frac{8x+x^4}{4-x^3}$.

Find the actual value of $y$ when $x=1.5$.

Using the graph of $y=\frac{8x+x^4}{4-x^3}$, suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of $y$ at $x=1.5$.

Sketch the curve $y=f\left(x\right)$, clearly indicating the coordinates of the endpoints.

Show that the inverse function of $f$ is given by $f^{-1}\left(x\right)=\sqrt{x^2+1}$.

State the domain and range of $f^{-1}$.

Show that the volume, $V {\text{m}}^3$, of water in the container when it is filled to a height of $h$ metres is given by $V=\pi \left(\frac{1}{3}{h}^3+h\right)$.

Hence, determine the maximum volume of the container.

Find the time it takes to fill the container to its maximum volume.

Find the rate of change of the height of the water when the container is filled to half its maximum volume.

The following diagram shows the curve $\frac{x^2}{36}+\frac{{\left(y-4\right)}^2}{16}=1$, where $h\le y\le 4$.

The curve from point $\text{Q}$ to point $\text{B}$ is rotated $360°$ about the $y$-axis to form the interior surface of a bowl. The rectangle $\text{OPQR}$, of height $h \text{cm}$, is rotated $360°$ about the $y$-axis to form a solid base.

The bowl is assumed to have negligible thickness.

Given that the interior volume of the bowl is to be $285 {\text{cm}}^3$, determine the height of the base.

Use integration by parts to find $\int {\left(\text{ln}x\right)}^2\text{d}x$.

Find the three-figure bearing on which airplane $B$ is travelling.

Show that airplane $A$ travels at a greater speed than airplane $B$.

Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

Find the coordinates of $\text{P}$.

Determine the length of time between the first airplane arriving at $\text{P}$ and the second airplane arriving at $\text{P}$.

Let $D\left(t\right)$ represent the distance between airplane $A$ and airplane $B$ for $0\le t\le 2.5$.

Find the minimum value of $D\left(t\right)$.

Find his velocity when $t=15$.

Calculate the vertical distance Xavier travelled in the first 10 seconds.

Determine the value of $h$.

Show that a finite limit only exists for $k=\frac{\mathrm{\pi }}{4}$.

Using l’Hôpital’s rule, show algebraically that the value of the limit is $-\frac{1}{4}$.

Find the times when $P$ comes to instantaneous rest.

Find an expression for $s$ in terms of $t$.

Find the maximum displacement of $P$, in metres, from its initial position.

Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.

Show that, at these times, $\tan 6t=2$.

Hence show that $\frac{v_2}{v_1}=\frac{v_3}{v_2}=-{\text{e}}^{-\frac{\mathrm{\pi }}{2}}$.

Using implicit differentiation, find an expression for $\frac{\text{d}y}{\text{d}x}$.

Find the equation of the tangent to the curve at the point $\left(\frac{1}{4}\text{, }\frac{5\pi }{6}\right)$.

Find the area of $R$.

The region $R$ is now rotated about the $y$-axis, through $2\pi$ radians, to form a solid.

By writing $\text{si}{\text{n}}^{3}y$ as $\left(1-\text{co}{\text{s}}^{2}y\right)\text{sin}y$, show that the volume of the solid formed is $\frac{2\pi }{3}$.

Determine the values of $p$, $q$ and $r$.

Hence find the area of the shaded region.

A function $f$ satisfies the conditions $f\left(0\right)=-4$, $f\left(1\right)=0$ and its second derivative is $f^{″}\left(x\right)=15\sqrt{x}+\frac{1}{{\left(x+1\right)}^2}$, $x$ ≥ 0.

Find $f\left(x\right)$.

Determine the first time t₁ at which P has zero velocity.

Find an expression for the acceleration of P at time t.

Find the value of the acceleration of P at time t₁.

Let $l$ be the tangent to the curve $y=x{\text{e}}^{2x}$ at the point (1, ${\text{e}}^2$).

Find the coordinates of the point where $l$ meets the $x$-axis.

Determine the coordinates of point $\text{A}$ and the coordinates of point $\text{B}$.

Hence, write down the maximum speed of the body.

the value of $t_1$.

the distance travelled between $t=0$ and $t=t_1$.

the acceleration when $t=t_1$.

Find the distance travelled in the first 30 seconds.

Find an expression for the volume of water $V({\text{m}}^3)$ in the trough in terms of $\theta$.

Calculate $\frac{\text{d}\theta }{\text{d}t}$ when $\theta =\frac{\pi }{3}$.

Find the value of $p$ and the value of $q$.

Find an expression for $f\text{'}\left(x\right)$.

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

Sketch the graph of $y=f\left(x\right)$ for $-3\le x\le 3$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.

Find the equations of all the asymptotes on the graph of $y=g\left(x\right)$.

By considering the graph of $y=g\left(x\right)-f\left(x\right)$, or otherwise, solve $f\left(x\right)<g\left(x\right)$ for $x\in \mathrm{ℝ}$.

Write down a definite integral to represent the area of $A$.

Calculate the area of $A$.

Show that there are exactly two points on the curve where the gradient is zero.

Find the equation of the normal to the curve at the point P.

The normal at P cuts the curve again at the point Q. Find the $x$-coordinate of Q.

The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.

The expression for $f\prime \left(x\right)$ can be written in the form $\frac{a}{x}+\frac{b}{k-x}$, where $a, b\in \mathrm{ℝ}$. Find $a$ and $b$ in terms of $k$.

Hence, find an expression for $f\left(x\right)$.

By solving the differential equation, show that $P=\frac{1200k}{\left(k-1200\right){\text{e}}^{-{\displaystyle \frac{t}{5}}}+1200}$.

Find the value of $k$, giving your answer correct to four significant figures.

Find the value of $t$ when the rate of change of the population is at its maximum.

Write down the first three terms of the binomial expansion of $(1+t{)}^{-1}$ in ascending powers of $t$.

By using the Maclaurin series for $\cos x$ and the result from part (a), show that the Maclaurin series for $\text{sec} x$ up to and including the term in $x^4$ is $1+\frac{x^2}{2}+\frac{5x^4}{24}$.

By using the Maclaurin series for $\text{arctan} x$ and the result from part (b), find $\underset{x\to 0}{\lim }\left(\frac{x\text{\hspace{0.05em}arctan} 2x}{\text{sec} x-1}\right)$.

Show that the volume of the solid formed is $\frac{15k^2\mathrm{\pi }}{34}$ cubic units.

Find the value of $k$ that satisfies the requirements of Pedro’s design.

Find $\text{OA}$.

Find $\text{BC}$.

By sketching the graph of a suitable derivative of $f$, find where the cross-sectional radius of the bowl is decreasing most rapidly.

State the cross-sectional radius of the bowl at this point.

The function $f$ is defined by $f\left(x\right)={\left(x-1\right)}^2$, $x$ ≥ 1 and the function $g$ is defined by $g\left(x\right)=x^2+1$, $x$ ≥ 0.

The region $R$ is bounded by the curves $y=f\left(x\right)$, $y=g\left(x\right)$ and the lines $y=0$, $x=0$ and $y=9$ as shown on the following diagram.

The shape of a clay vase can be modelled by rotating the region $R$ through 360˚ about the $y$-axis.

Find the volume of clay used to make the vase.

Show that the $x$-coordinate of the minimum point on the curve $y=f(x)$ satisfies the equation $\tan x=2x$.

Determine the values of $x$ for which $f(x)$ is a decreasing function.

Sketch the graph of $y=f(x)$ showing clearly the minimum point and any asymptotic behaviour.

Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y=-x$.

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

Use the substitution $y=vx$ to show that $\int \frac{dv}{f\left(v\right)-v}=\ln x+C$ where $C$ is an arbitrary constant.

By using the result from part (a) or otherwise, solve the differential equation and hence show that the curve has equation $y=x\left(\tan \left(\ln x\right)-1\right)$.

The curve has a point of inflexion at $\left(x_1, y_1\right)$ where ${\text{e}}^{-\frac{\mathrm{\pi }}{2}}<x_1<{\text{e}}^{\frac{\mathrm{\pi }}{2}}$. Determine the coordinates of this point of inflexion.

Use the differential equation $\frac{dy}{dx}=\frac{y^2+3xy+2x^2}{x^2}$ to show that the points of zero gradient on the curve lie on two straight lines of the form $y=mx$ where the values of $m$ are to be determined.

Find the volume of the solid formed when the region bounded by the curve, the $x$-axis for $x⩾0$ and the $y$-axis for $y⩾0$ is rotated through $2\pi$ about the $x$-axis.

Show that $f$ is an even function.

By considering limits, show that the graph of $y=f\left(x\right)$ has a horizontal asymptote and state its equation.

Show that $f\text{'}\left(x\right)=\frac{2x}{\sqrt{x^2}\left(x^2+1\right)}$ for $x\in \mathrm{ℝ}, x\ne 0$.

By using the expression for $f\text{'}\left(x\right)$ and the result $\sqrt{x^2}=\left|x\right|$, show that $f$ is decreasing for $x<0$.

Find an expression for $g^{-1}\left(x\right)$, justifying your answer.

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

Sketch the graph of $y=g^{-1}\left(x\right)$, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.

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

Determine the equation of the tangent to $C$ at the point $\left(\frac{2}{\text{e}},\text{ e}\right)$

Differentiate from first principles the function $f\left(x\right)=3x^3-x$.

By using the substitution $x^2=2\sec \theta$, show that $\int \frac{\text{d}x}{x\sqrt{x^4-4}}=\frac{1}{4}\arccos \left(\frac{2}{x^2}\right)+c$.

Show that $\sqrt{16+k}-\sqrt{k}=\sqrt{k}\sqrt{16+k}$.

Find the value of $k$.

Show that $\frac{dy}{dx}=\frac{3x^2}{2{\text{e}}^{2y}-1}$.

The tangent to $C$ at the point Ρ is parallel to the $y$-axis.

Find the $x$-coordinate of Ρ.

Given that $2x^3-3x+1$ can be expressed in the form $Ax\left(x^2+1\right)+Bx+C$, find the values of the constants $A$, $B$ and $C$.

Hence find $\int \frac{2x^3-3x+1}{x^2+1}\text{d}x$.

A particle moves along a horizontal line such that at time $t$ seconds, $t$ ≥ 0, its acceleration $a$ is given by $a$ = 2$t$ − 1. When $t$ = 6 , its displacement $s$ from a fixed origin O is 18.25 m. When $t$ = 15, its displacement from O is 922.75 m. Find an expression for $s$ in terms of $t$.

Show that $\frac{dy}{dx}+\left(x\frac{{\displaystyle dy}}{{\displaystyle dx}}+y\right)\left(1+\ln \left(xy\right)\right)=1$.

Hence find the equation of the tangent to $C$ at the point where $x=1$.

Solve the differential equation, giving your answer in the form $f\left(x, y\right)=0$.

The graph of $y$ against $x$ has a local maximum between $x=2$ and $x=3$. Determine the coordinates of this local maximum.

Show that there are no points of inflexion on the graph of $y$ against $x$.

Assuming the Maclaurin series for $\cos x$ and $\ln (1+x)$, show that the Maclaurin series for $\cos \left(\ln \right(1+x\left)\right)$ is

$1-\frac{1}{2}{x}^2+\frac{1}{2}{x}^3-\frac{5}{12}{x}^4+\dots$

By differentiating the series in part (a), show that the Maclaurin series for $\sin \left(\ln \right(1+x\left)\right)$ is $x-\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\dots$ .

Hence determine the Maclaurin series for $\tan \left(\ln \right(1+x\left)\right)$ as far as the term in $x^3$.

Show that $y=x+50 \text{cot} \theta$ .

At time $T$, the following conditions are true.

Boat $\text{B}$ has travelled $10$ metres further than boat $\text{A}$.
Boat $\text{B}$ is travelling at double the speed of boat $\text{A}$.
The rate of change of the angle $\theta$ is $-0.1$ radians per second.

Find the speed of boat $\text{A}$ at time $T$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

At the point on the curve where $x=4$, it is given that $\frac{\text{d}y}{\text{d}t}=-0.1 {\text{m\hspace{0.05em}s}}^{-1}$

Find the value of $\frac{\text{d}x}{\text{d}t}$ at this exact same instant.

An earth satellite moves in a path that can be described by the curve $72.5x^2+71.5y^2=1$ where $x=x(t)$ and $y=y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{\text{d}x}{\text{d}t}=7.75\times {10}^{-5}$ when $x=3.2\times {10}^{-3}$, find the possible values of $\frac{\text{d}y}{\text{d}t}$.

Give your answers in standard form.