Differentiation (DP IB Maths: AA 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

(i)     the tangent to the curve at point 

(ii)    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 .

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

Calculate the instantaneous rate of change of when 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.

The equation of a curve is 

Show that the curve has exactly two stationary points. Determine the coordinates and nature of each point.

Show that the curve has exactly one point of inflection and determine its coordinates.

Give an example of a curve with equation , where  are real numbers and  and  are real numbers and  for which there is a point of inflection that is also a stationary point. Be sure to justify your answer.

A function is defined by

for all real numbers .

Show that the -coordinates of the stationary points on the curve must satisfy the equation

Determine the ranges of values of  for which is

Sketch the curve of , showing the coordinates of any minimum and maximum points, as well as the point where the curve crosses the -axis.

A function is defined for all for The derivative of is given by

Find

The graph of is concave up when  where  is the least possible number that makes that inequality true.

Find the value of .

Show that the curve has only one point of inflection, and find the gradient of the curve at that point of inflection.

Let be a function defined by for all in the interval. The following diagram shows the graph of:

The curve intercepts the -axis at points , and . There is a point of inflection at point  and a local maximum at point

Find the values of and .

Find and hence determine the -coordinate of the local maximum at point . You should give your answer as an exact value.

Find the equation of the tangent at point . Give your answer in the form where and  are integers.

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

(i)     the tangent to the curve at point .

(ii)    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 .

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

Calculate the average rate of change of  between and

(i)     

(ii)   

(iii)   

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.

Let be a function defined by for all

The curve intercepts the -axis at points and  where 

Find the values of  and

The curve has a local minimum at point .

Show that the -coordinate of point  is equal to Be sure to justify that the point corresponding to that -coordinate is indeed a local minimum, and that it is the only local minimum.

The curve  has a point of inflection at point

Find the gradient of the normal to the curve at point

A function  is defined by

for all real numbers .

Show that the curve has a stationary point when  and determine the corresponding -coordinate.

Find the -coordinates of any other stationary points on the curve.

Determine the ranges of values of  for which  is

Given that the value of the function when is greater than the value of the function at any other stationary point, sketch the curve of  Be sure to show clearly the -coordinates of any minimum and maximum points, as well as the coordinates of the point where the curve crosses the -axis.

A function  is defined for all for The derivative of  is given by

The graph of is concave up when ,  where  is the least possible number that makes that inequality true.

Find the value of .

Show that the curve  has only one point of inflection, and find the gradient of the normal line to the curve at that point.

A curve has equation  where  and

Show that the curve will only have stationary points if and satisfy the inequality

In the case where the inequality in part (a)(i) is satisfied, determine the -coordinate(s) of the stationary point(s), giving your answer as simply as possible in terms of and

Show that the curve will always have exactly one point of inflection, and determine its -coordinate in terms of  and b.

In the case where the point of inflection is also a stationary point, show that the curve will have no other stationary points.

In the case where the curve has two distinct stationary points, show that the -coordinate of the point of inflection will lie halfway between the -coordinates of the two stationary points.