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.

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

Find the value of $k$.

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

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

Find the value of $k$.

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

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

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

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

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 $v$.

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

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 $a$, $b\in \mathbb{Z}$.

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

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

Write down the range of $f$.