DP Mathematics: Analysis and Approaches Questionbank

Find $f\text{'}\left(p\right)$ in terms of $k$ and $p$.

Show that the equation of $L_1$ is $kx+p^{2}y-2pk=0$.

Find the area of triangle $\text{AOB}$ in terms of $k$.

The graph of $f$ is translated by $\left(\begin{array}{c}4\\ 3\end{array}\right)$ to give the graph of $g$.
In the following diagram:

• point $\text{Q}$ lies on the graph of $g$
• points $\text{C}$, $\text{D}$ and $\text{E}$ lie on the vertical asymptote of $g$
• points $\text{D}$ and $\text{F}$ lie on the horizontal asymptote of $g$
• point $\text{G}$ lies on the $x$-axis such that $\text{FG}$ is parallel to $\text{DC}$.

Line $L_2$ is the tangent to the graph of $g$ at $\text{Q}$, and passes through $\text{E}$ and $\text{F}$.

Given that triangle $\text{EDF}$ and rectangle $\text{CDFG}$ have equal areas, find the gradient of $L_2$ in terms of $p$.

The $x$-intercept of the graph of $f$ is $\left(5, 0\right)$.

On the following grid, sketch the graph of $f$.

Find the value of $a$.

Describe a sequence of transformations that transforms the graph of $y=\sqrt{x}$ for $x\ge 0$ to the graph of $y=-1-\sqrt{x+3}$ for $x\ge -3$.

Let $g\left(x\right)=\frac{1}{2}f\left(x\right)+1$ for $-4\le x\le 6$. On the axes above, sketch the graph of $g$.

Describe a sequence of transformations that transforms the graph of $y=\text{arctan\hspace{0.05em}}x$ to the graph of $y=\text{arctan}\left(2x+1\right)+\frac{\mathrm{\pi }}{4}$ for $x\in \mathrm{ℝ}$.

Find the values of $d$ for which $g$ is an increasing function.

Describe the transformation by which $f\left(x\right)$ is transformed to $g\left(x\right)$.

State the range of $g$.

Sketch the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ on the same axes, clearly stating the points of intersection with any axes.

Find the coordinates of P.

Consider the function $f\left(x\right)=x^4-6x^2-2x+4$, $x\in \mathbb{R}$.

The graph of $f$ is translated two units to the left to form the function $g\left(x\right)$.

Express $g\left(x\right)$ in the form $a{x}^4+b{x}^3+c{x}^2+dx+e$ where $a$, $b$, $c$, $d$, $e\in \mathbb{Z}$.

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 $d$.

Write down the value of $q$;

Write down the range of $f$.

Write down the domain of $g$.

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

Hence, find the area of the region enclosed by the graphs of $f$ and $g$.

Find the value of $b$.

The graph of $f\left(x\right)=x^2$ is transformed to obtain the graph of $g$.

Describe this transformation.