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

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

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

Very briefly, explain why the value of this integral must be negative.

Show that the general solution of this differential equation is $P=A{\text{e}}^{kt}$, where $A\in \mathbb{R}$.

$\underset{t\to \mathrm{\infty }}{\text{lim}}P$

the solution of the differential equation, giving your answer in the form $P=f\left(t\right)$.

Given that the initial population is 1000, $L=10000$  and $m=0.003$, find the number of years it will take for the population to triple.

Show that $\frac{\text{d}P}{\text{d}t}=\frac{m}{L}P\left(L-P\right)$, where $m\in \mathbb{R}$.

the number of years it will take for the population to triple.

Use part (a) to show that $f(x)$ is always decreasing.

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

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

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

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.

Hence find the area of the shaded region.

the number of years it will take for the population to triple.

Give a reason why the series expansion found in part (b) is not valid for $x=\frac{3}{4}$.

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 $k$.

the population after 10 years

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

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 $\frac{2\pi }{3}$.

Find the sum to infinity of this sequence.

Express the function $f\left(x\right)=\frac{-1}{x+x^2}$ in partial fractions.

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

$y$-axis.

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 $b$.

Use parts (a) and (b) to show that .

Find the value of $A$ and the value of $B$.

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

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

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

Solve the differential equation $\frac{\text{d}P}{\text{d}t}=\frac{m}{L}P\left(L-P\right)$, giving your answer in the form $P=g\left(t\right)$.

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 $a, b \in \mathrm{\mathbb{Z}}$.

Find the area of $R$.

Let $f(x)=\frac{1}{1-x^2}$ for . Use partial fractions to find $\int f\left(x\right)dx$.

Use part (a) to find the exact value of $\underset{-1}{\overset{0}{\int }}f(x)dx$, giving the answer in the form $\text{ln}q$,   $q\in \mathbb{Q}$.

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 $v$.

Hence, expand $\frac{2+7x}{\left(1+2x\right)\left(1-x\right)}$ in ascending powers of $x$, up to and including the term in $x^2$.

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.

$x$-axis.

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