6.2 Extended Questions (Section B, HL)

The function  has a derivative given by  where  is a positive constant.

The expression for can be written in the form   where  Find   and  in terms of k.

Hence find an expression for .

R is the population of rabbits on an island. The rate of change of the population can be modelled by the differential equation ,  where t is the time measured in years, ,  and  is the maximum population that the island can support. 

The initial population of the rabbits is 20. 

By solving the differential equation, show that 

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 , at time t seconds,  is given by  where is the velocity in meters per second and

The particle starts at a fixed origin O with initial velocity .

By solving a suitable differential equation, show that the particle’s velocity at time t is given by

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. 

If the initial velocity of the particle is , find the time at which the minimum displacement of the particle from the origin occurs, giving your answer in exact form.

If T is the time in seconds when the displacement reaches its smallest value, show that .

Let  represent the particle’s velocity k seconds before the minimum displacement and  the particle’s velocity k seconds after the minimum displacement. 

Show that

Given that show that

The diagram below shows the graph of  The graph has rotational symmetry of order 2 about the origin.

A different function, g, is described by g

(i)

Describe the sequence of transformations that transforms to g.

(ii)

Sketch the graph of  g on the axes above.

(iii)

Using your answers to parts (i) and (ii) to help you, describe the relationship between and  .

Prove that  

(ii)      Show that can be written as 

Using the results from parts (a) and (b), evaluate 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  of the form

where  and . 

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

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

To model the vase, she will rotate by  radians about the -axis the region enclosed by the graph of  ,  the -axis, the -axis, and the line  . 

Show that the volume of the solid of revolution thus formed is .

The volume of the actual vase is 100

Use this information to find the value of .

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 , and use it to find the value of  at which the cross-sectional radius of the vase is decreasing most rapidly.