7.1 Applications of Differentiation

Show that the function  is decreasing for all .

A curve has the equation . 

The point  is the stationary point of the curve. 

Find the coordinates of  and determine its nature.

The diagram below shows a part of the curve with equation , where

       ,   

Point  is the maximum point of the curve.

Find .

Use your answer to part [a] to find the coordinates of point .

A garden bed is to be divided by fencing into four identical isosceles triangles, arranged as shown in the diagram below:

The base of each triangle is metres, and the equal sides are each  metres in length. 

Although  and  can vary, the total amount of fencing to be used is fixed at  metres. 

Explain why .

Show that

      
where  is the total area of the garden bed.

Using your answer to [b] find, in terms of , the maximum possible area of the garden bed.

Describe the shape of the bed when the area has its maximum value.

Find the coordinates of the stationary points, and their nature, on the graph with equation .

Show that the function  is increasing for all .

A curve has the equation . 

Find expressions for  and .

Determine the coordinates of the local minimum of the curve.

The diagram below shows part of the curve with equation . The curve touches the -axis at  and cuts the -axis at . The points  and  are stationary points on the curve.

Using calculus, and showing all your working, find the coordinates of and .

Show that  is a point on the curve and explain why those must be the coordinates of point .

A company manufactures food tins in the shape of cylinders which must have a constant volume of .  To lessen material costs the company would like to minimise the surface area of the tins. 

By first expressing the height  of the tin in terms of its radius , show that the surface area of the cylinder is given by   .

Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.

Find the -coordinates of the stationary points on the graph with equation .

Find the nature of the stationary points found in part [a].

Show that the function , ,  is increasing for all  in its domain.

A curve is described by the equation , where .

Find and .

 is the stationary point on the curve. 

Find the coordinates of  and determine its nature.

The diagram below shows the part of the curve with equation  for which .  The marked point  lies on the curve.  is the origin.

Show that .

Find the minimum distance from  to the curve, using calculus to prove that your answer is indeed a minimum.

The top of a patio table is to be made in the shape of a sector of a circle with radius  and central angle , where .

Although  and may be varied, it is necessary that the table have a fixed area of  . 

Explain why .

Show that the perimeter, P, of the table top is given by the formula

Show that the minimum possible value for  is equal to the perimeter of a square with area . Be sure to prove that your value is a minimum.