5.8 Advanced Differentiation

Let.

Solve the equation in the interval  .

Show that.

Find the derivative of each of the following functions: 

For the curve defined by , show that

For the curve defined by , show that 

Consider the function defined by, .

The following diagram shows the graph of the curve :

The points marked  and  are the turning points of the graph.

Find.

Hence find the coordinates of points  and .

Find the equation of the normal to the graph at the point where the -coordinate is equal to .

For each of the following, find  by differentiating implicitly with respect to . 

A curve is described by the equation

Use implicit differentiation with respect to to show that 

Use your result from part (a) to find the equation of the

to the curve at the point .

Verify that your answer to part (c)(ii) and the result from part (b)(i) both give the same value for the gradient of the tangent to the curve at the point .

An international mission has landed a rover on the planet Mars. After landing, the rover deploys a small drone on the surface of the planet, then rolls away to a distance of 6 metres in order to observe the drone as it lifts off into the air. Once the rover has finished moving away, the drone ascends vertically into the air at a constant speed of 2 metres per second.

Let  be the distance, in metres, between the rover and the drone at time  seconds. 

Let  be the height, in metres, of the drone above the ground at time seconds. The entire area where the rover and drone are situated may be assumed to be perfectly horizontal.

Show that 

Find

(ii)

the height of the drone above the ground at the moment when the distance between the rover and the drone is increasing at a rate of  .

In the diagram below, is the outline of a type of informational signboard that a county council plans to use in one of its parks. The shape is formed by a rectangle , to one side of which an equilateral triangle has been appended.

The signboards will be produced in various different sizes.  However because of the cost of the edging that must go around the perimeter of the signboards, the council is eager to design the signboards so that the area of a signboard is the maximum possible for a given perimeter.

Let and let .

Write down an expression in terms of and  for the perimeter of the signboard, .

Hence use implicit differentiation to find

Explain why, for a given perimeter, it must be true that , and use this fact to show that .

Show that the area , of the signboard is given by  .

Hence use implicit differentiation to find the ratio of to  that gives the maximum area.

Consider the function  defined by

By first calculating   and ,  show that .

Write down the value of .

Find the derivative of the function  .

Given that ,  find .  Simplify your answer as far as possible.

Let be the function defined by . Show that

 

where .

Use differentiation to show that  is a solution to the equation

Consider the curve defined by  ,  for values of  satisfying .

Show that

Given that the curve has exactly one point of inflection, show that that point of inflection occurs when ,  where  is the so-called ‘golden ratio’.

Consider the function  defined by . 

The following diagram shows the graph of the curve :

The point marked A is the inflection point of the graph. 

Determine the exact coordinates of the point where the normal to the graph at point A  intersects the -axis.

For each of the following, find  by differentiating implicitly with respect to .

A curve is described by the equation

where  is a constant.

Use implicit differentiation to show that

For a particular value of , the curve goes through the point. 

Find the value of .

Find the equation of the

to the curve at the point .

Two observers, Pamela and Quinlan, are standing at points P and Q respectively watching a hot air balloon take off.  The balloon takes off from point O, which is in between points P and Q and is such that points P, O and Q all lie on a straight horizontal line.

Let  be the distance OP, and let be the distance between point P and the balloon at any time .  Similarly let be the distance OQ, and let  be the distance between point Q and the balloon at any time .  Let  be the height of the balloon above the ground at any time .  The balloon ascends vertically upwards, but its velocity during the ascent is not necessarily constant.  All distances are measured in metres, and all times in seconds.

Show that an expression for  can be written solely in terms of ,  and .

Quinlan is standing a distance of 50 metres from where the balloon takes off.  At a certain moment in time, the balloon is at a distance of 112 metres from point Q and the distance between the balloon and point Q is increasing at a rate of 1.79 .  At the same moment in time the distance between point P and the balloon is increasing at a rate of 1.05 .

Use the above information and the results of part (a) to determine the distance that Pamela is standing from the point where the balloon takes off.

A third observer, Rhydderch, is standing at point R.  Point R is on the same side of point O as point P is, and it lies on the same horizontal line as points O, P and Q.  At the same moment described above, the distance between the balloon and point R is increasing at a rate of less than 0.8 metres per second. 

Find an inequality to express the minimum distance PR between the point where Rhydderch is standing and the point where Pamela is standing.

In the diagram below,  is a pentagon made up of a rectangle , to one side of which an isosceles triangle  has been appended.  In addition sides and  of the rectangle are the same length as the equal sides  and of the triangle.

The pentagon is intended to represent the cross-section of a new building, and the architect would like the area of the pentagon, A , to be the maximum possible for any given perimeter, P. 

Let units and let .

By first finding the derivative  in terms of  and , work out the value of the derivative .

By considering the derivative , show that when the area is maximal for a given perimeter the following equation must hold:

Hence determine (i) the ratio of to  (in the form  for some  to be determined) that gives the maximum area for a given perimeter, and (ii) the maximum possible area for a pentagon of the above form with a perimeter of 100 metres.