Applications of Differentiation (AQA GCSE Further Maths)

Use differentiation to show that  is an increasing function for all values of .

[4]

 

Work out the value of     when  

[5]

This shape is made from two rectangles.
All dimensions are in centimetres.

The perimeter of the shape is 252 cm

Show that   

[2]

The area of the shape is 
Show that   

[2]

Use differentiation to work out the maximum value of as  varies.

[3]

) is the graph of a cubic function.

for

for

The function is

increasing for

decreasing for

increasing for

Draw a possible sketch of for values of from –2 to 6

[4]

is the point on the curve  where 

The gradient of the normal to the curve at is 

Work out the value of .

[4]

Show that has a minimum value when   

[5]

is a point on a curve.

The curve has gradient function 

The tangent to the curve at P is parallel to the line 3x – 2y = 9
Work out the -coordinate of .

[4]

Show that the curve has exactly two stationary points.

[4]

The curve has 
The curve has exactly one stationary point at where
Use the expression for  to show that is a point of inflection.

[3]

A curve has equation 

At a point P on the curve, the tangent is parallel to the line 

Work out the coordinates of P.
You must show your working.

[4]

The continuous curve  has exactly two stationary points.

Here is some information about the curve.

is positive	is zero	is negative	is zero	is negative

 and

State the coordinates and the nature of each of the stationary points.

[3]

stationary point (.............. , ..............) nature  ..........................

stationary point (.............. , ..............) nature  ..........................

The diagram shows a sketch of the cubic curve   

where is a constant.

The -axis is a tangent to the curve at its minimum point.

Work out the value of

[5]

The continuous curve g()  has exactly two stationary points.

The stationary points are

• a point of inflection at
•  a minimum point at where > 1 and < 0

On the axes below, sketch the curve.
Label points and on your sketch.

[3]

The curve intersects the -axis at .

The tangent to the curve at intersects the -axis at .

Work out the length .

[6]

......................units