Extended Questions (Section B HL Only) (DP IB Analysis & Approaches (AA): HL): Exam Questions

The function $f$ has a derivative given by $f\text{'}\left(x\right)=\frac{1}{3x\left(k-x\right)},x\in \mathrm{ℝ},x\ne 0,$ where $k$ is a positive constant.

The expression for $f\text{'}\left(x\right)$ can be written in the form $\frac{a}{3x}+\frac{b}{k-x}$  where $p,q\in \mathrm{ℝ}.$ Find $a$  and $b$ in terms of k.

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

R is the population of rabbits on an island. The rate of change of the population can be modelled by the differential equation $\frac{dR}{dt}=\frac{3R\left(k-R\right)}{4k}$,  where t is the time measured in years, $t\ge 0$,  and $k$ is the maximum population that the island can support. 

The initial population of the rabbits is 20. 

By solving the differential equation, show that  $R=\frac{20k{e}^{\frac{3}{4}t}}{k-20+20e^{\frac{3}{4}t}}$

After two years, the population of rabbits has risen to 70.

Find k.

Find the value of t at which the population of rabbits is growing at its fastest rate.

A particle is moving in a vertical line and its acceleration, in ${\mathrm{ms}}^{-2}$ , at time t seconds, $t\ge 0$ is given by $a=\frac{1-v}{2},$ where $v$ is the velocity in meters per second and $v<1.$

The particle starts at a fixed origin O with initial velocity $v_o{\mathrm{ms}}^{-1}$.

By solving a suitable differential equation, show that the particle’s velocity at time t is given by $v\left(t\right)=1-e^{-\frac{t}{2}}\left(1-v_o\right).$

The particle moves down in the negative direction, until its displacement relative to the origin reaches a minimum. Then the particle changes direction and starts moving up, in a positive direction. 

(i) If the initial velocity of the particle is $-3{\mathrm{ms}}^{-1}$, find the time at which the minimum displacement of the particle from the origin occurs, giving your answer in exact form.

(ii) If T is the time in seconds when the displacement reaches its smallest value, show that $T=2\ln \left(1-v_o\right)$.

(i) Find a general expression for the displacement, in terms of $t$ and $v_o$.

(ii) Combine this general expression with the result from part (b)(ii) to find an expression for the minimum displacement of the particle in terms of $v_o$.

Let $v\left(T-k\right)$ represent the particle’s velocity k seconds before the minimum displacement and $v\left(T+k\right)$ the particle’s velocity k seconds after the minimum displacement. 

(i) Show that $v\left(T-k\right)=1-e^{\frac{k}{2}}.$

(ii) Given that $v\left(T+k\right)=1-e^{-\frac{k}{2}},$ show that $v\left(T-k\right)+v\left(T+k\right)\ge 0.$

The diagram below shows the graph of $f\left(x\right)=\arctan \left(x\right),x\in \mathrm{ℝ}.$ The graph has rotational symmetry of order 2 about the origin.

A different function, $g$, is described by $g\left(x\right)=-\arctan \left(x-1\right),x\in \mathrm{ℝ}.$

(i) Describe the sequence of transformations that transforms $f\left(x\right)$to g$\left(x\right)$.

(ii) Sketch the graph of  g$\left(x\right)$ on the axes above.

(iii) Using your answers to parts (i) and (ii) to help you, describe the relationship between ${\int }_0^1\arctan \left(x\right)dx$and ${\int }_0^1-\arctan \left(x-1\right)dx.$ .

(i) Prove that  $\arctan p-\arctan q=\arctan \left(\frac{p-q}{1+pq}\right).$

(ii) Show that $\arctan \left(\frac{1}{x^2-x+1}\right)$ can be written as $\arctan \left(x\right)-\arctan \left(x-1\right).$

Using the results from parts (a) and (b), evaluate ${\int }_0^1\arctan \left(\frac{1}{x^2-x+1}\right)dx,$leaving your answer in exact form.

Paola is modelling a small vase from her house for her maths project. To model the edge of the vase in cross-section, she decides to use a function $f$ of the form

$f\left(x\right)=\frac{q{\mathrm{e}}^{\frac{x}{2}}}{2+{\mathrm{e}}^x}$ 

where $x\in \mathrm{ℝ},x\ge 0$ and $q\in {\mathrm{ℝ}}^{+}$. 

The function and the vase are represented in the diagrams below.

The vertical height of the vase, OB, is measured along the $x$-axis. The radius of the vase’s opening is OA, and its base radius is BC. 

To model the vase, she will rotate by $2\pi$ radians about the $x$-axis the region enclosed by the graph of $y=f\left(x\right)$ ,  the $x$-axis, the $y$-axis, and the line $x=\ln 43$ . 

Show that the volume of the solid of revolution thus formed is $\frac{14q^2\mathrm{\pi }}{45}{\mathrm{units}}^3$.

The volume of the actual vase is 100 ${\mathrm{cm}}^3.$

Use this information to find the value of $q$.

Find the cross-sectional radius of the vase

(i)     at its base,

(ii)    at its widest point.

Paola wants to investigate how the cross-sectional radius of the vase changes.

Sketch a graph of the derivative of $f$, and use it to find the value of $x$ at which the cross-sectional radius of the vase is decreasing most rapidly.