DP Mathematics: Analysis and Approaches Questionbank

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

Hence, given that $f^{-1}$ does not exist, show that $b^2-3ac>0$.

Show that $g^{-1}$ exists.

$g\left(x\right)$ can be written in the form $p(x-2{)}^3+q$ , where $p, q \in \mathrm{ℝ}$.

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

Hence find $g^{-1}\left(x\right)$.

State each of the transformations in the order in which they are applied.

Sketch the graphs of $y=g\left(x\right)$ and $y=g^{-1}\left(x\right)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.

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.

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 $x\in \mathrm{ℝ}$.

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

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.

$x$-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.

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.

Solve the inequality $f\left(x\right)\ge g\left(x\right)$.

Consider the graphs of $y=\frac{x^2}{x-3}$ and $y=m\left(x+3\right)$, $m\in \mathbb{R}$.

Find the set of values for $m$ such that the two graphs have no intersection points.

Show that $f\left(x\right)$ has no vertical asymptotes.

With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.

Write down the equation of the vertical asymptote.

Write down the equation of the vertical asymptote.

Find the equation of the horizontal asymptote. 

Find the co-ordinates of all stationary points.

As $x\to \pm \mathrm{\infty }$ the graph of $f\left(x\right)$ approaches an oblique straight line asymptote.

Divide $2x^2-5x-12$ by $x+2$ to find the equation of this asymptote.

Find all the intercepts of the graph of $f\left(x\right)$ with both the $x$ and $y$ axes.

Find the exact value of $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$, giving the answer in the form $\text{ln}q\text{,}q\in \mathbb{Q}$.

find the coordinates of the $y$-intercept.

show that there are no $x$-intercepts.

sketch the graph, showing clearly any asymptotic behaviour.

Find the expected number of weeks in the year in which Lucca eats no bananas.

By sketching the graph of $p$ as a function of $q$, determine the range of values of $p$ for which there are possible values of $q$.

Find an expression for ${\theta }_p$ in terms of $p$.

Sketch the graph of $y=\frac{1-3x}{x-2}$, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

Find the probability that Lucca eats at least one banana in a particular day.

Show that $p=\frac{q^2+q-1}{2q+1}$.