Applications of Differentiation (Edexcel A Level Maths: Pure): Exam Questions

A curve has equation

Find writing your answer in simplest form.

Hence find the range of values of for which is decreasing.

A curve has equation

Find

(i)

(ii)

Verify that has a stationary point when .

Determine the nature of this stationary point, giving a reason for your answer.

Figure 1 shows part of the curve with equation

The point lies on the curve.

Find the gradient of the tangent to the curve at .

The point with coordinate also lies on the curve.

Find the gradient of the line , giving your answer in terms of in simplest form.

Explain briefly the relationship between part (b) and the answer to part (a).

In this question your must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve has equation

The point lies on and has coordinate

The line is tangent to at .

Show that has equation

Find the values of x for which is an increasing function.

Show that the function is increasing for all .

The curve C has equation.

Show that the point P(2, 9) lies on C.

Show that the value of  at  P  is  16.

Find an equation of the tangent to C at P.

The curve C has equation .  The point P lies on C.

Find an expression for .

Show that an equation of the normal to C at point P is .

This normal cuts the x-axis at the point Q.

Find the length of PQ, giving your answer as an exact value.

Given that , find

A curve has the equation .

Find expressions forand .

Determine the coordinates of the local minimum of the curve.

The diagram below shows part of the curve with equation . The curve touches the x-axis at A and cuts the x-axis at C. The points A and B are stationary points on the curve.

Using calculus, and showing all your working, find the coordinates of A and B.

Show that (-1, 0) is a point on the curve and explain why those must be the coordinates of point C.

A company manufactures food tins in the shape of cylinders which must have a constant volume of 150π cm³. To lessen material costs the company would like to minimise the surface area of the tins.

By first expressing the height h of the tin in terms of its radius r, show that the surface area of the cylinder is given by .

Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.

The curve has equation where

and and are constants.

Given

• the point lies on

• the gradient of the curve at is

(i) show that the value of is

(ii) find the value of .

Hence show that has no stationary points.

Factorise completely

The curve has equation

Sketch showing the coordinates of the points at which the curve cuts the -axis.

The line has equation where is a constant.

Given that and intersect at distinct points, find the range of values for , writing your answer in set notation.

Solutions relying on calculator technology are not acceptable.

The curve has equation

Find

(i)

(ii)

(i) Verify that has a stationary point at

(ii) Show that this stationary point is a point of inflection, giving reasons for your answer.

A company makes drinks containers out of metal.

The containers are modelled as closed cylinders with base radius cm and height cm and the capacity of each container is cm³

The metal used

• for the circular base and the curved sides costs pence/cm²

• for the circular top costs pence/cm²

Both metals used are of negligible thickness.

Show that the total cost, pence, of the metal for one container is given by

Use calculus to find the value of for which is a minimum, giving your answer to 3 significant figures.

Using prove that the cost is minimised for the value of found in part (b).

Hence find the minimum value of , giving your answer to the nearest integer.

A company makes toys for children.

Figure 5 shows the design for a solid toy that looks like a piece of cheese.

The toy is modelled so that

• face is a sector of a circle with radius cm and centre

• angle radians

• faces and are congruent

• edges and are perpendicular to faces and

• edges and have length cm

Given that the volume of the toy is show that the surface area of the toy, , is given by

making your method clear.

Using algebraic differentiation, find the value of for which has a stationary point.

Prove, by further differentiation, that this value of gives the minimum surface area of the toy.

Find the values of x for which is a decreasing function.

Show that the function is decreasing for all

The curve C has equation .  The point P(2,  2) lies on C.

Find an equation of the tangent to C at P.

The curve C has equation   The point P lies on C.

The normal to C at P intersects the x-axis at the point Q.

Find the coordinates of Q.

Given that , find

A curve has the equation

The point P is the stationary point of the curve.

Find the coordinates of P and determine its nature.

The diagram below shows a part of the curve with equation , where

Point A is the maximum point of the curve.

Find .

Use your answer to part (a) to find the coordinates of point A.

A garden bed is to be divided by fencing into four identical isosceles triangles, arranged as shown in the diagram below:

The base of each triangle is 2x metres, and the equal sides are each y metres in length.

Although x and y can vary, the total amount of fencing to be used is fixed at P metres.

Explain why .

Show that

where A is the total area of the garden bed.

Using your answer to (b) find, in terms of P, the maximum possible area of the garden bed.

Describe the shape of the bed when the area has its maximum value.

[A sphere of radius has volume and surface area ]

A manufacturer produces a storage tank.

The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9.

The walls of the tank are assumed to have negligible thickness.

The cylinder has radius metres and height metres and the hemisphere has radius metres.

The volume of the tank is 6 m³.

Show that, according to the model, the surface area of the tank, in m², is given by

The manufacturer needs to minimise the surface area of the tank.

Use calculus to find the radius of the tank for which the surface area is a minimum.

Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

Find the values of x for which is a decreasing function, where .

Show that the function,  is increasing for all x in its domain.

A curve has equation .  

A is the point on the curve with x coordinate 0, and B is the point on the curve with x coordinate 6.  

C is the point of intersection of the tangents to the curve at A and B. 

Find the coordinates of point C.

Calculate the area of triangle ABC.

A curve is described by the equation , where

P is the point on the curve such that the normal to the curve at P also passes through the origin.

Find the coordinates of point P. Give your answer in the form , where a and b are rational numbers to be found.

Write down the equation of the normal to the curve at P.

Show that an equation of the tangent to the curve at P is

A curve is described by the equation , where

Find and .

P is the stationary point on the curve.

Find the coordinates of P and determine its nature.

The diagram below shows the part of the curve with equation for which . The marked point P lies on the curve. O is the origin.

Show that

Find the minimum distance from O to the curve, using calculus to prove that your answer is indeed a minimum.

The top of a patio table is to be made in the shape of a sector of a circle with radius r and central angle , where .

Although r and  may be varied, it is necessary that the table have a fixed area of  A m².

Explain why .  

Show that the perimeter, P, of the table top is given by the formula

Show that the minimum possible value for P is equal to the perimeter of a square with area A. Be sure to prove that your value is a minimum.