Applications of Differentiation (Edexcel International A Level Maths: Pure 2)

Exam Questions

2 hours23 questions
1
Sme Calculator
6 marks
[i]

Find an expression for straight f to the power of apostrophe left parenthesis x right parenthesis when straight f left parenthesis x right parenthesis equals x cubed plus x squared minus 5 x.

[ii]
Solve the equation 3 x squared plus 2 x minus 5 equals 0
[iii]

Hence, or otherwise, find the values of x for which straight f left parenthesis x right parenthesis is a decreasing function.

Did this page help you?

2a
Sme Calculator
3 marks

The curve C has equation y equals 3 x cubed plus 6 x squared minus 5 x plus 1

Find expressions for fraction numerator d y over denominator d x end fraction  and  fraction numerator d squared y over denominator d x squared end fraction.

2b
Sme Calculator
4 marks

[i]   Evaluate  fraction numerator d y over denominator d x end fraction and fraction numerator d squared y over denominator d x squared end fraction when x equals 1 third.

[ii]   What does your answer to part [b] tell you about curve C at the point where x equals 1 third?

Did this page help you?

3
Sme Calculator
3 marks

Find the values of x for which straight f left parenthesis x right parenthesis equals 2 x squared minus 16 x is an increasing function.

Did this page help you?

4
Sme Calculator
4 marks

Find the x-coordinates of the stationary points on the curve with equation

            y equals 1 third x cubed plus 5 over 2 x squared minus 6 x plus 2

Did this page help you?

5
Sme Calculator
5 marks

Show that the point left parenthesis 2 space comma 1 right parenthesis is a [local] maximum point on the curve with equation

            y space equals space 2 x squared minus 2 over 3 x cubed minus 5 over 3

Did this page help you?

6a
Sme Calculator
4 marks

Find the value of  fraction numerator d y over denominator d x end fraction and fraction numerator d squared y over denominator d x squared end fraction at the point where x equals 2 for the curve with equation y space equals space x cubed minus 6 x squared plus 9 x plus 4.

6b
Sme Calculator
1 mark

Explain why x equals 2  is not a stationary point.

Did this page help you?

1
Sme Calculator
3 marks

Find the values of x for which straight f left parenthesis x right parenthesis equals negative 9 x squared plus 5 x minus 3 is an increasing function.

Did this page help you?

2
Sme Calculator
3 marks

Show that the function straight f left parenthesis x right parenthesis equals x cubed minus 3 x squared plus 6 x minus 7 is increasing for all x element of straight real numbers.

Did this page help you?

3a
Sme Calculator
3 marks

A curve has the equation y equals x cubed minus 12 x plus 7

Find expressions for fraction numerator d y over denominator d x end fraction and fraction numerator d squared y over denominator d x squared end fraction.

3b
Sme Calculator
3 marks

Determine the coordinates of the local minimum of the curve.

Did this page help you?

4a
Sme Calculator
5 marks

The diagram below shows part of the curve with equation y equals x cubed plus 11 x squared plus 35 x plus 25. The curve touches the x-axis at A and cuts the x-axis at C. The points A and B are stationary points on the curve.

7-1-sq-4a-medium-edexelmodelling-with-sequences-and-series-

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

4b
Sme Calculator
2 marks

Show that left parenthesis negative 1 comma 0 right parenthesis is a point on the curve and explain why those must be the coordinates of point C.

Did this page help you?

5a
Sme Calculator
2 marks

A company manufactures food tins in the shape of cylinders which must have a constant volume of 150 pi space cm cubed.  To lessen material costs the company would like to minimise the surface area of the tins. 

By first expressing the height h of the tin in terms of its radius r, show that the surface area of the cylinder is given by  S equals 2 pi r squared plus fraction numerator 300 straight pi over denominator r end fraction .

5b
Sme Calculator
4 marks

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

Did this page help you?

6a
Sme Calculator
3 marks

Find the x-coordinates of the stationary points on the graph with equation y equals x cubed minus 6 x squared plus 9 x minus 1.

6b
Sme Calculator
2 marks

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

Did this page help you?

1
Sme Calculator
5 marks

Find the values of x for which straight f left parenthesis x right parenthesis equals x cubed minus 5 x squared plus 3 x minus 2 is a decreasing function.

Did this page help you?

2
Sme Calculator
3 marks

Show that the function straight f left parenthesis x right parenthesis equals 7 x squared minus 2 x left parenthesis x squared plus 5 right parenthesis is decreasing for all x element of straight real numbers.

Did this page help you?

3
Sme Calculator
5 marks

A curve has the equation y equals x left parenthesis x plus 6 right parenthesis squared plus 4 left parenthesis 3 x plus 11 right parenthesis

The point P left parenthesis x comma y right parenthesis is the stationary point of the curve. 

Find the coordinates of P and determine its nature.

Did this page help you?

4a
Sme Calculator
3 marks

The diagram below shows a part of the curve with equation y equals straight f left parenthesis x right parenthesis, where

       f open parentheses x close parentheses equals 460 minus x cubed over 300 minus 8100 over x,   x greater than 0

Point A is the maximum point of the curve.

7-1-sq-4a-hard-edexelmodelling-with-sequences-and-series-

Find straight f to the power of apostrophe left parenthesis x right parenthesis.

4b
Sme Calculator
4 marks

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

Did this page help you?

5a
Sme Calculator
1 mark

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

7-1-sq-5a-hard-edexelmodelling-with-sequences-and-series-

The base of each triangle is 2 x metres, and the equal sides are each y metres in length. 

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

Explain why 0 less than x less than P over 6.

5b
Sme Calculator
4 marks

Show that

      A squared equals 4 over 9 P squared x squared minus 16 over 3 P x cubed
where A is the total area of the garden bed.

5c
Sme Calculator
4 marks

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

5d
Sme Calculator
1 mark

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

Did this page help you?

6
Sme Calculator
4 marks

Find the coordinates of the stationary points, and their nature, on the graph with equation y space equals space 4 x minus x squared minus 2 x cubed.

Did this page help you?

1
Sme Calculator
4 marks

Find the values of x for which straight f left parenthesis x right parenthesis equals 4 x plus 3 over x is a decreasing function, where x not equal to 0.

Did this page help you?

2
Sme Calculator
4 marks

Show that the function straight f open parentheses x close parentheses equals square root of x minus fraction numerator 7 over denominator square root of x end fractionx greater than 0,  is increasing for all x in its domain.

Did this page help you?

3a
Sme Calculator
3 marks

A curve is described by the equation y equals straight f left parenthesis x right parenthesis, where straight f left parenthesis x right parenthesis equals 7 minus 2 x squared plus square root of x comma space x greater or equal than 0.

Find straight f to the power of apostrophe left parenthesis x right parenthesis and straight f to the power of apostrophe apostrophe left parenthesis x right parenthesis.

3b
Sme Calculator
4 marks

P is the stationary point on the curve. 

Find the coordinates of P and determine its nature.

Did this page help you?

4a
Sme Calculator
3 marks

The diagram below shows the part of the curve with equation y space equals space 3 minus 1 fourth x squared for which y greater than 0.  The marked point P left parenthesis x comma y right parenthesis lies on the curve. O is the origin.

7-1-sq-4a-very-hard-edexelmodelling-with-sequences-and-series-

Show that O P to the power of 2 space end exponent equals space 9 minus 1 half x squared plus 1 over 16 x to the power of 4.

4b
Sme Calculator
8 marks

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

Did this page help you?

5a
Sme Calculator
2 marks

The top of a patio table is to be made in the shape of a sector of a circle with radius r and central angle theta, where 0 degree less than theta less than 360 degree.

7-1-sq-5a-very-hard-edexelmodelling-with-sequences-and-series-

Although r and theta may be varied, it is necessary that the table have a fixed area of  A space straight m squared

Explain why r greater than square root of A over straight pi end root.

5b
Sme Calculator
2 marks

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

         P equals 2 r plus fraction numerator 2 A over denominator r end fraction

5c
Sme Calculator
5 marks

Show that the minimum possible value for P is equal to the perimeter of a square with area begin mathsize 20px style A end style. Be sure to prove that your value is a minimum.

Did this page help you?