Implicit differentiation

The equation of a curve is

The curve has two stationary points.

Use implicit differentiation to show that at the stationary points y =

Hence w that the x-coordinates of the stationary points can be written in the form ±√n where n is an integer to be found.

A curve has equation

y 3 e 2 x + 2 y – 16 x = k

where k is a constant.

The curve has a stationary point on the y-axis.

Determine the value of k

The curve with equation

has a single stationary point at P as shown in the diagram below.

Show that the y-coordinate of P satisfies the equation

Hence, find the coordinates of P

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 equation of a curve is

The curve intersects the positive x-axis at the point P.

Show that the gradient of the curve at P is

Find the equation of the normal to the curve at P, giving your answer in the form

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.

Find the coordinates of the stationary point of the curve with equation