Stationary points

The point A lies on the curve with equation

Given that A has coordinates (– 5, 37), show that

5 p – q = 30

Given that A is a stationary point, show that

10 p – q = 75

Hence find the value of p and the value of q

The curve with equation

has a second stationary point B

Find the coordinates of B

Fully justify your answer.

A curve C has equation

where k is a positive integer.

Show that

where A is a constant to be found.

Hence show that the y-coordinate of the stationary point of C can be written as Fully justify your answer.

Given that the stationary point of C has coordinates – state the value of k

A curve has equation

The point P with coordinates (4, 10) lies on the curve.

Find an equation of the tangent to the curve at the point P

Show that the curve has no stationary points.

A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below.

The points P, O and Q lie on a straight line and the angle ROS is radians.

A large copy of the logo, with PQ 5 metres, is to be put on a wall.

Show that the area of the logo, A square metres, is given by

Show that the maximum value of A occurs when

Fully justify your answer.

Find the exact maximum value of A

Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if PQ were increased to 10 metres.

The graph of

where

is shown in Figure 1.

Figure 1

Tick () one box.

Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor.

The entrance to the cave joins the floor at the points O and P.

Garry models the shape of the cross section of the entrance to the cave using the

where a is a constant, and x and y are the horizontal and vertical distances respectively, in metres, measured from O.

The distance OP is 16 metres.

Find the value of a that Garry should use in the model.

Show that the maximum height of the cave above OP is approximately 10.5 metres.

Suggest one limitation of the model Garry has used.

The curve y 15 x² and the isosceles triangle OPQ are shown on the diagram

Vertices P and Q lie on the curve such that Q lies vertically above some point (q, 0)

The line PQ is parallel to the x-axis.

Show that the area, A, of the triangle OPQ is given by

where c is a constant to be found.

Find the exact maximum area of triangle OPQ.

Fully justify your answer.

The function f is defined by

where f has its maximum possible domain.

The curve y f (x) intersects the line y x at the points P and Q as shown below.

State the value of x which is not in the domain of f.

Explain how you know that the function f is many-to-one.

Show that the x-coordinates of P and Q satisfy the equation

Hence, find the exact x-coordinate of P and the exact x-coordinate of Q.

Show that P and Q are stationary points of the curve.

Fully justify your answer.

Using set notation, state the range of f.

Section B

all questions in the spaces provided.

A curve has equation y f (x)

The curve has a point of inflection at x 7

Identify which one of the statements below must be true.

Circle your answer.

The function h is defined by

where h has its maximum possible domain.

Find the domain of h.

Give your answer using set notation.

Alice correctly calculates

She then argues that since there is a change of sign there must be a value of x in the interval 1 x 4 that gives h(x) 0

Explain the error in Alice’s argument.

A gardener is creating flowerbeds in the shape of sectors of circles.

The gardener uses an edging strip around the perimeter of each of the flowerbeds.

The cost of the edging strip is £1.80 per metre and can be purchased for any length.

One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below.

Find the area of this flowerbed.

Find the cost of the edging strip required for this flowerbed.

A flowerbed is to be made with an area of 20 m²

Show that the cost, £ C, of the edging strip required for this flowerbed is given by

where r is the radius measured in metres.

Hence, show that the minimum cost of the edging strip for this flowerbed occurs

Fully justify your answer.

A function f is defined for all real values of x as

The function has exactly two stationary points when x 0 and x

f x 9 (a) (i) Find ()

Determine the nature of the stationary points.

Fully justify your answer.

State the range of values of x for which

is an increasing function.

A second function g is defined for all real values of x as

State the single transformation which maps f onto g.

A cylinder is to be cut out of the circular face of a solid hemisphere.

The cylinder and the hemisphere have the same axis of symmetry.

The cylinder has height h and the hemisphere has a radius of R.

Show that the volume, V, of the cylinder is given by

A curve, C, has equation

Show that C has exactly one stationary point.

Fully justify your answer.

Sketch the graph of any cubic function that has both three distinct real roots and

The function f (x) is defined by

Show that there is a turning point where the curve crosses the y-axis.

The equation f (x) 0 has three distinct real roots.

By considering the positions of the turning points find, in terms of p, the range of possible values of q.

A curve has equation

2 2 4

Prove that the curve does not intersect the coordinate axes.

Prove that the curve has no stationary points.

In the case when x 0, find the equation of the tangent to the curve when y 1

Section B

all questions in the spaces provided.

A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible.

The company models the logo on an x - y plane as shown in the diagram.

Use calculus to find the maximum area of the rectangle.

Fully justify your answer.

A curve has equation y x³ 48x

The point A on the curve has x coordinate 4

The point B on the curve has x coordinate 4 h

Show that the gradient of the line AB is h² 12h

Explain how the result of part (a) can be used to show that A is a stationary point on the curve.

A function f is defined by f (x)

State the maximum possible domain of f.

Show that the graph of y f (x) has exactly one point of inflection.

Write down the values of x for which the graph of y f (x) is convex.