Advanced Differentiation (DP IB Maths: AA HL)

Let .

By differentiating from first principles, show that .

Let .

Find the positive solution to the equation   that is closest to zero.

Given that   find .

For the function g defined by g,  show that

Find the derivative of the function .

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 A and B are the turning points of the graph. 

Find the coordinates of points A and B.

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 .

Consider the curve defined by the equation

Use implicit differentiation to find  in terms of  and .

By first rewriting the equation of the curve in the form :

(i)

Determine the coordinates of the point on the curve where  .

(ii)

Explain why  is an increasing function on all intervals  for which the interval  does not include the -coordinate of the point identified in part (b)(i).

(iii)

Describe the asymptotic behaviour of the curve as  .

After setting up a firework rocket on a stretch of level ground, the firework engineer lights the fuse and steps back to a safe distance of 10 metres from the rocket.  The rocket then begins to ascend vertically into the air at a constant velocity of 64 metres per second. 

Let be the distance, in metres, between the rocket and the point on the ground where the engineer is standing at time  seconds after the rocket takes off.  Let  be the height, in metres, of the rocket above the ground at time  seconds.

Write an expression for in terms of only.

Use implicit differentiation to show that

Find

(ii)

the height of the rocket above the ground at the moment when the distance between the rocket and the point where the engineer is standing is increasing at a rate of  .

Quadrilateral  represents a corral for unicorns.  There are fences along the four sides of the corral, as well as a straight fence across the middle connecting points .  Because of the way unicorns are trained, it is essential that triangles  and  be identical isosceles triangles, with .  The length of side , however, can vary.

Gonzolph is a unicorn trainer who is concerned about the high cost of unicorn fencing.  He would therefore like the total length of fencing, , used in his corral to be the minimum possible for a given area, , to be enclosed. 

Let   and let  .

By first finding the derivative  in terms of and , show that for a given area the equation  must be satisfied.

By considering the derivative , show that when the length of fencing required to enclose a given area is the minimum possible then .

Hence find the size of angle in a corral that minimises the amount of fencing required to enclose a given area.

Let  , where  are constants with  . 

By differentiating from first principles, show that .

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.