Differentiation (DP IB Maths: AI HL)

The equation of a curve is   for  .

Find .

The gradient of the tangent to the curve at point  is .

Find the coordinates of point .

Find the equation of the normal to the curve at point Give your answer in the form   .

The volume of a sphere of radius  is given by the formula   .

Find .

Find the rate of change of the volume with respect to the radius when .

Give your answer in terms of .

Show that  is an increasing function for all relevant values of .

A curve has the equation

Points  and  are the two points on the curve where the gradient is equal to 1, and the   -coordinate of  is less than zero.

Find the coordinates of points and .

Find the equations of

the tangent to the curve at point

the normal to the curve at point .

Point  is the point of intersection of the two lines found in part (b).

Find the coordinates of point .

The gradient of the tangent to the curve with equation    at the point  is 14.

Find the values of  and .

Patroclus, a would-be Olympic javelin thrower, throws a javelin during a training session.  The height of the javelin’s point can be modelled by the equation

where  is the time, in seconds, that has passed since the javelin was released, and  is the height of the javelin above the ground, in metres.

Find .

Find the greatest vertical distance that the javelin’s point travels above the height from which it was released.

Check, Mate! is a company that produces luxury chess sets for discerning chess set connoisseurs.  The company’s profits , in thousands of UK pounds (£1000), can be modelled by the function

where  is the number of chess sets (in hundreds) sold per year.  Because of manufacturing constraints, the maximum number of chess sets that the company can sell in a year is 2500.

State why there is no need to consider values of  greater than 25.

Sketch a graph of  for  .

Calculate

the average rate of change of  between    and 

the instantaneous rate of change of  at  .

In each case include the units, and explain the meaning of the value you find.

State the values of for which the instantaneous rate of change of is negative.  Explain the meaning of this result.

The diagram below shows a part of the graph of the function  ,  where

Calculate the instantaneous rate of change of  when .

Calculate the average rate of change of  between and 

Explain what would happen if you continued to calculate the average rates of change in part (b), moving the second value closer and closer to 2 each time.

A manufacturing company is producing tins that must have a capacity of 470 cm³.  The tins are in the shape of a cylinder with a height of  cm and a base radius of  cm.

Show that the surface area of the cylinder in cm², including the two circular ends, may be written as

Sketch the graph of .

The company would like to minimise the amount of metal used to make the tins.

Find the stationary point on the graph of   ,  and justify that it is a minimum point.

A commercially available tin of chopped tomatoes on sale in the UK has a capacity of 470  and a base radius of 3.7 cm.

Determine the percentage difference between the surface area of that tin of chopped tomatoes and the minimum possible surface area for a tin with the same capacity.

Two numbers,  and , are such that    and the difference between the two numbers is .

Find the minimum possible value of the product , and the values of  and  that correspond to that minimum value.

A curve is given by the equation

Determine the coordinates of the points on the curve where the gradient is 2. You must show all your working, and give your answers as exact fractions.

Find the range of values for  for which the curve is increasing.

An engineer is designing a right cone that is to be produced on a 3D printer. The cone has a base radius of  cm and a height of  cm, and while the radius may vary freely the height must always be 7 cm more than the radius.

Write down, in terms of  only, the formula for the volume of the cone.

Find the exact value of the radius at the point where the instantaneous rate of change of the volume with respect to the radius is . 

A curve has the equation

Points  and  are the two points on the curve where the gradient is equal to 3, and the   -coordinate of  is less than zero.

Find the coordinates of points and .

Find the equations of

the tangent to the curve at point .

the normal to the curve at point .

Point  is the point of intersection of the two lines found in part (b).

Find the coordinates of point . Give your answers as exact fractions.

A curve has equation  .

The gradient of the tangent to the curve at the point  is 25.

The gradient of the tangent to the curve at the point  is .

Find the values of , ,  and .

A newly-commissioned attack submarine is performing a series of manoeuvres to test its propulsion and steering systems.  The vertical position of the submarine relative to sea level (where sea level is represented by ) is given by the equation

where is the time, in minutes, that has passed since the submarine began its manoeuvres, and  is the vertical position of the submarine in metres.

Find the stationary points for .

For each of the stationary points found in part (a), determine whether the point is a maximum point or a minimum point. Justify your answer in each case.

Explain why, in order to find the maximum and minimum depths reached by the submarine in the interval , it is not sufficient merely to consider the stationary points found in part (a).

Find the greatest vertical distances that the submarine travels in the interval    above and below the depth from which it started its manoeuvres.

Muggins! is a company that produces luxury cribbage boards for discerning collectors of pub game paraphernalia.  For sales of between 0 and 100 cribbage boards in a month, the company’s profits , in thousands of UK pounds (£1000), can be modelled by the function

where  is the number of cribbage boards (in hundreds) sold during the month.  For sales of between 100 and 1000 cribbage boards in a month, the corresponding formula is 

Because of manufacturing constraints, the maximum number of cribbage boards that the company can sell in a month is 1000.

State the ranges of  values for which each formula is valid.

On the same set of axes, sketch the two profit functions. Each function should only be sketched over the interval of  values for which it is valid.

Show that the combined profit function sketched in part (b) is an increasing function for all valid values greater than zero.

Considering only values of for which ,  find the value of  for which the instantaneous rate of change of  is a minimum.  Give the value of the corresponding instantaneous rate of change, and explain the meaning of that value in context.

The diagram below shows a part of the graph of the function  ,  where

Calculate the average rate of change of  between and

Explain what would happen to the values of the average rates of change in part (b) if you continued to calculate them, moving the second  value closer and closer to 3 each time.

An artist is producing large pieces of sculpture for an art installation.  Each piece is in the form of a cylinder with base radius  metres, on top of which is a hemisphere with the same radius as the cylinder’s base radius.  The hemisphere is fitted exactly to the top of the cylinder, so that the circular bottom of the hemisphere lines up exactly with the circular top of the cylinder. 

Every side of each piece of sculpture must be painted, so the artist is eager to find a design for his sculptures such that, for any given volume of a piece of sculpture, the total surface area will be the minimum possible.

Show that for a piece of sculpture with volume , the minimum surface area occurs when

Find the minimum possible surface area for a piece of sculpture with volume .  Give your answer as an exact value.

Two numbers, and , are such that    and the difference between the two numbers is , where  is a positive constant.

Find the minimum possible value of the sum , and the values of  and  that correspond to that minimum value.  Your answers should be given in terms of .

Justify that your answer in part (a) is a minimum value.