Differentiating standard functions

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

Given that

Hence find

The area, A, bounded by the curve with equation y 2^x, the x-axis, the y-axis and the line x 4 is approximated using eight rectangles of equal width as shown in the diagram below.

Show that the exact area of the largest rectangle is

The areas of these rectangles form a geometric sequence with common ratio

Find the exact value of the total area of the eight rectangles. Give your answer in the form k(1 2) where k is a rational number.

More accurate approximations for A can be found by increasing the number, n, of rectangles used.

Find the exact value of the limit of the approximations for A as n

A curve has equation y 3e^2x

Find the gradient of the curve at the point where y 10

The diagram shows a sector of a circle OAB.

The point C lies on OB such that AC is perpendicular to OB.

Angle AOB is radians.

Given the area of the triangle OAC is half the area of the sector OAB, show that

Use a suitable change of sign to show that a solution to the equation

lies in the interval given by

The Newton-Raphson method is used to find an approximate solution to the equation

Using ₁ as a first approximation for apply the Newton-Raphson method twice

Give your answer to three decimal places.

Explain how a more accurate approximation for can be found using the Newton-Raphson method.

Explain why using ₁ as a first approximation in the Newton-Raphson method does not lead to a solution for .

Given that y ln (5x)

Circle your answer.

A particle’s displacement, r metres, with respect to time, t seconds, is defined by the equation

Circle your answer.

Given y e^kx, where k is a constant, find

Circle your answer.

A curve is defined by the parametric equations

Find the Cartesian equation of the curve in the form xy ax by c, where a, b and c are integers.