Differentiation

Use differentiation to find  for the following:

Use differentiation to find  for the following:

For the curve with equation :

find

Find the coordinates of the point on the curve where the gradient is 2.

A curve has equation

Find 

Find the gradient of the curve at the point where:

What can you say about the tangents to the curves at these two points?

A particle  passes the fixed point  whilst moving along a straight line.

The displacement of , from , at time  seconds is  metres where

Find expressions for the velocity, , and the acceleration, of the particle at time  seconds.

Find the time at which the acceleration is 3 .

The curve  has equation .

Find  .

    = ..............................................

There are two points on the curve  at which the gradient of the curve is .

Find the  coordinate of each of these two points.
Show clear algebraic working.

.

Find .

....................................

The curve with equation  has two stationary points.

Work out the coordinates of these two stationary points.

The curve  has equation .

Find  .

Find the range of values of  for which  has a negative gradient.

Calculate the gradient of  at .

Differentiate . 

Find the coordinates of the turning point of the graph of .

( ...................... , ...................... )

Part of the graph with equation is shown below.

The graph has three stationary points, indicated on the graph by points , and.
Find the area of the triangle .

The diagram shows a cuboid with a square cross-section.

The sides of the square face are cm and the length of the cuboid is cm.
The cuboid is to have a fixed surface area, , of 25 cm².
Show that the volume of the cuboid,  cm³ is given by

Show that the value of  that maximises the volume of the cuboid is

Find the maximum volume of the cuboid, correct to 3 significant figures.

A particle moves along a straight line that passes through the fixed point
The displacement, metres, offrom at time t seconds, where , is given by

The direction of motion of  reverses when is at the point on the line.
The acceleration of at the instant when is at is .
Find the value of  .

a = ..................................... 

Two particles, and , move along a straight line.
The fixed point lies on this line.

The displacement of from at time seconds is metres, where

The displacement of from at time seconds is metres, where

Find the range of values of where for which both particles are moving in the same direction along the straight line.

The point A is the only stationary point on the curve with equation   where is a constant.

Given that the coordinates of are 

find the value of .
Show your working clearly.

................................................. 

The curve has equation where and are constants.

The point with coordinates (2, –6) lies on .
The gradient of the curve at is 16.

Find the coordinate of the point on the curve whose coordinate is 3.
Show clear algebraic working.

A particle is moving along a straight line.
The fixed point lies on the line.

At time t seconds , the displacement of from is s metres where

Find the minimum speed of .

...................................................... m/s

is a five-sided shape.

is a rectangle.
is an equilateral triangle.



The perimeter of is 100 cm.
The area of is cm²

Show that 

Find the value of for which has its maximum value.

Give your answer in the form where and are integers.

....................................................... [2]

Explain why the maximum value of is given by this value of .

A particle moves along a straight line.
The fixed point  lies on this line.
The displacement of the particle from  at time  seconds , , is  metres where

At time  seconds the velocity of  is  where 

Find an expression for  in terms of .

Give your expression in the form  where  and  are integers to be found.