DP Mathematics: Analysis and Approaches Questionbank

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

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

Hence find the area of the shaded region.

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

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 $x^2$.

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 $k$.

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

$x$-axis.

$y$-axis.

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

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

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.

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

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

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

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

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

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

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

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

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

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

Express $f(x)$ in partial fractions.

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

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

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

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

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.

the population after 10 years

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

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

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 $P=g\left(t\right)$.

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 $v$.

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

Find the sum to infinity of this sequence.