Applications of Differentiation (Edexcel A Level Maths: Pure): Exam Questions

4 hours42 questions
12 marks

straight f open parentheses x close parentheses equals x cubed plus 2 x squared minus 8 x plus 5

Find straight f apostrophe apostrophe open parentheses x close parentheses

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2
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2 marks

The curve C has equation y equals straight f open parentheses x close parentheses

The curve

  • passes through the point P open parentheses 3 comma space minus 10 close parentheses

  • has a turning point at P

Given that

fraction numerator straight d y over denominator straight d x end fraction equals 2 x cubed minus 9 x squared plus 5 x plus k

where k is a constant,

show that k equals 12.

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31 mark
Graph of a positive cubic curve C which intersects the x axis at the origin and a minimum point touches the x axis at the point x=6. The maximum point with coordinates (2, 8) is marked on the graph.
Figure 1

Figure 1 shows a sketch of a curve C with equation y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis is a cubic expression in x.

The curve

  • passes through the origin

  • has a maximum turning point at left parenthesis 2 comma space 8 right parenthesis

  • has a minimum turning point at left parenthesis 6 comma space 0 right parenthesis

Write down the set of values of x for which

straight f apostrophe left parenthesis x right parenthesis less than 0

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4a
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2 marks

Find an expression for  fraction numerator d y over denominator d x end fractionwhen  y equals 3 x squared minus 2 x.

4b
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2 marks

Find the gradient of  y equals 3 x squared minus 2 x space at the points where

(I) x equals 3,

(ii) x equals negative 2.

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5
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5 marks

(i) Find the gradient of the tangent at the point (2 , 3) on the graph of y equals 2 x cubed minus 3 x squared minus 1.

(ii) Hence find the equation of the tangent at the point (2 , 3).

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6
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6 marks

(i) Find an expression for straight f 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.

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7a
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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.

7b
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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 space 1 third.

(ii) What does your answer to part (b) tell you about curve C at the point where  x equals fraction numerator space 1 over denominator 3 end fraction?

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8a
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2 marks

For the graph with equation  y equals 3 x minus space 1 half space x squared, find the gradient of the tangent at the point where x equals 5.

8b
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3 marks

(i) Find the gradient of the normal at the point where  x equals 5.

(ii) Hence find the equation of the normal at the point where  x equals 5.

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9
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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.

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10
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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.

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11
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5 marks

Show that the point (2 , 1) is a (local) maximum point on the curve with equation

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

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1a2 marks

A curve has equation

y equals 2 over 3 x cubed minus 7 over 2 x squared minus 4 x plus 5

Find fraction numerator straight d y over denominator straight d x end fraction writing your answer in simplest form.

1b
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4 marks

Hence find the range of values of x for which y is decreasing.

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2a3 marks

A curve C has equation

y equals x squared minus 2 x minus 24 square root of x comma space space space space space space space space x greater than 0

Find

(i) fraction numerator straight d y over denominator straight d x end fraction

(ii) fraction numerator straight d squared y over denominator straight d x squared end fraction

2b
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2 marks

Verify that C has a stationary point when x equals 4.

2c
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2 marks

Determine the nature of this stationary point, giving a reason for your answer.

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3a2 marks
A graph showing a parabola opening upwards, with points P(2, 10) and Q on the curve labelled. On the parabola, point Q is above and to the right of point P. Axes are marked with origin O, x and y directions.
Figure 1

Figure 1 shows part of the curve with equation y equals 3 x squared minus 2

The point P open parentheses 2 comma space 10 close parentheses lies on the curve.

Find the gradient of the tangent to the curve at P.

3b3 marks

The point Q with x coordinate 2 plus h also lies on the curve.

Find the gradient of the line P Q, giving your answer in terms of h in simplest form.

3c1 mark

Explain briefly the relationship between part (b) and the answer to part (a).

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45 marks

In this question your must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve C has equation

table row cell y equals 1 third x squared minus 2 square root of x plus 3 end cell blank cell x greater or equal than 0 end cell end table

The point P lies on C and has x coordinate 4

The line l is tangent to C at P.

Show that l has equation

13 x minus 6 y minus 26 equals 0

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5
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3 marks

Find the values of x for whichspace straight f open parentheses x close parentheses equals negative 9 x squared plus 5 x minus 3 space is an increasing function.

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6
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3 marks

Show that the function space straight f open parentheses x close parentheses equals x cubed minus 3 x squared plus 6 x minus 7 spaceis increasing for all x element of straight real numbers.

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7a
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1 mark

The curve C has equationspace y equals 2 x cubed minus 3 x squared plus 4 x minus 3.

Show that the point P(2, 9) lies on C.

7b
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3 marks

Show that the value of  fraction numerator d y over denominator d x end fraction at  P  is  16.

7c
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2 marks

Find an equation of the tangent to C at P.

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8a
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2 marks

The curve C has equation y equals 3 x squared minus 6 plus 4 over x.  The point Popen parentheses 1 comma space 1 close parentheses lies on C.

Find an expression for fraction numerator d y over denominator d x end fraction.

8b
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3 marks

Show that an equation of the normal to C at point P is x plus 2 y equals 3.

8c
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2 marks

This normal cuts the x-axis at the point Q.

Find the length of PQ, giving your answer as an exact value.

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9a
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2 marks

Given that y equals 2 x cubed minus 8 square root of x, find

fraction numerator d y over denominator d x end fraction

9b
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2 marks

fraction numerator straight d squared y over denominator straight d x squared end fraction

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10a
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3 marks

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

Find expressions forfraction numerator d y over denominator d x end fractionand fraction numerator d squared y over denominator d x squared end fraction.

10b
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3 marks

Determine the coordinates of the local minimum of the curve.

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11a
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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.

q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

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

11b
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2 marks

Show that (-1, 0) is a point on the curve and explain why those must be the coordinates of point C.

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12a
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2 marks

A company manufactures food tins in the shape of cylinders which must have a constant volume of 150π cm3. 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 space fraction numerator 300 pi over denominator r end fraction.

12b
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4 marks

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

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1a3 marks

The curve C has equation

table row cell y equals 5 x to the power of 4 minus 24 x cubed plus 42 x squared minus 32 x plus 11 end cell blank cell x element of straight real numbers end cell end table

Find

(i) fraction numerator straight d y over denominator straight d x end fraction

(ii) fraction numerator straight d squared y over denominator straight d x squared end fraction

1b
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4 marks

(i) Verify that C has a stationary point at x equals 1

(ii) Show that this stationary point is a point of inflection, giving reasons for your answer.

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2a
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4 marks

A company makes drinks containers out of metal.

The containers are modelled as closed cylinders with base radius r cm and height h cm and the capacity of each container is 355 cm3

The metal used

  • for the circular base and the curved sides costs 0.04 pence/cm2

  • for the circular top costs 0.09 pence/cm2

Both metals used are of negligible thickness.

Show that the total cost, C pence, of the metal for one container is given by

C equals 0.13 pi r squared plus fraction numerator 28.4 over denominator r end fraction

2b
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4 marks

Use calculus to find the value of r for which C is a minimum, giving your answer to 3 significant figures.

2c
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2 marks

Using fraction numerator straight d squared C over denominator straight d r squared end fraction prove that the cost is minimised for the value of r found in part (b).

2d
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2 marks

Hence find the minimum value of C, giving your answer to the nearest integer.

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3a
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4 marks
Prism with a top face in the shape of a sector. The angle of the sector BAC is 0.8 radians, height h and radius r
Figure 5

A company makes toys for children.

Figure 5 shows the design for a solid toy that looks like a piece of cheese.

The toy is modelled so that

  • face A B C is a sector of a circle with radius r cm and centre A

  • angle B A C equals 0.8 radians

  • faces A B C and D E F are congruent

  • edges A D comma space C F and B E are perpendicular to faces A B C and D E F

  • edges A D comma space C F and B E have length h cm

Given that the volume of the toy is 240 space cm cubed show that the surface area of the toy, S space cm squared, is given by

S equals 0.8 r squared plus 1680 over r

making your method clear.

3b
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4 marks

Using algebraic differentiation, find the value of r for which S has a stationary point.

3c
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2 marks

Prove, by further differentiation, that this value of r gives the minimum surface area of the toy.

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4
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5 marks

Find the values of x for which straight f open parentheses x close parentheses equals x cubed minus 5 x squared plus 3 x minus 2 is a decreasing function.

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5
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3 marks

Show that the function straight f open parentheses x close parentheses 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.

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6
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5 marks

The curve C has equation y equals 3 x squared minus 6 x plus square root of 2 x end root.  The point P(2,  2) lies on C.

Find an equation of the tangent to C at P.

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7
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6 marks

The curve C has equation y equals space fraction numerator 9 over denominator square root of 3 x end root end fraction minus 3 over x.  The point P open parentheses 3 comma space 2 close parentheses lies on C.

The normal to C at P intersects the x-axis at the point Q.

Find the coordinates of Q.

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8a
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3 marks

Given that y equals space 4 over x minus cube root of 27 over x end root, find

fraction numerator d y over denominator d x end fraction

8b
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2 marks

fraction numerator straight d squared y over denominator straight d x squared end fraction

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9
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5 marks

A curve has the equation y equals x open parentheses x plus 6 close parentheses squared plus 4 open parentheses 3 x plus 11 close parentheses.

The point Popen parentheses x comma space y close parentheses is the stationary point of the curve.

Find the coordinates of P and determine its nature.

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10a
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3 marks

The diagram below shows a part of the curve with equation y equals f open parentheses x close parentheses, where

straight f open parentheses x close parentheses equals 460 minus x cubed over 300 minus 8100 over x comma space space space space space space space space space space space space space space x greater than 0

Point A is the maximum point of the curve.

KTI0dIN4_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

Find straight f apostrophe open parentheses x close parentheses.

10b
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4 marks

Use your answer to part (a) to find the coordinates of point A.

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11a
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1 mark

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

dVG~C3Lv_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

The base of each triangle is 2x 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 space space P over 6.

11b
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4 marks

Show that

    space space 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.

11c
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4 marks

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

11d
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1 mark

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

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1a4 marks
Diagram of a cylindrical shape with a hemispherical top, showing radius as "r m" and height as "h m" with arrows for dimensions.
Figure 9

[A sphere of radius r has volume 4 over 3 pi r cubed and surface area 4 pi r squared]

A manufacturer produces a storage tank.

The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9.

The walls of the tank are assumed to have negligible thickness.

The cylinder has radius r metres and height h metres and the hemisphere has radius r metres.

The volume of the tank is 6 m3.

Show that, according to the model, the surface area of the tank, in m2, is given by

12 over r plus 5 over 3 pi r squared

1b
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4 marks

The manufacturer needs to minimise the surface area of the tank.

Use calculus to find the radius of the tank for which the surface area is a minimum.

1c
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2 marks

Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

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2
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4 marks

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

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3
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4 marks

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

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4a
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7 marks

A curve has equation y equals 5 minus open parentheses x minus 3 close parentheses squared.  

A is the point on the curve with x coordinate 0, and B is the point on the curve with x coordinate 6.  

C is the point of intersection of the tangents to the curve at A and B

Find the coordinates of point C.

4b
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2 marks

Calculate the area of triangle ABC.

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5a
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6 marks

A curve is described by the equation y equals straight f open parentheses x close parentheses, where

straight f open parentheses x close parentheses equals fraction numerator 1 over denominator square root of x end fraction space comma space space space x greater than 0

P is the point on the curve such that the normal to the curve at P also passes through the origin.

Find the coordinates of point P. Give your answer in the form open parentheses 2 to the power of a comma space 2 to the power of b close parentheses, where a and b are rational numbers to be found.

5b
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1 mark

Write down the equation of the normal to the curve at P.

5c
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4 marks

Show that an equation of the tangent to the curve at P is

open parentheses 2 to the power of 1 third end exponent close parentheses x plus open parentheses 2 to the power of 5 over 6 end exponent close parentheses y equals 3

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6a
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3 marks

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

Find straight f apostrophe open parentheses x close parentheses and straight f apostrophe apostrophe open parentheses x close parentheses.

6b
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4 marks

P is the stationary point on the curve.

Find the coordinates of P and determine its nature.

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7a
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3 marks

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

mao-shtQ_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

Show thatspace O P squared equals 9 minus 1 half x squared plus 1 over 16 x to the power of 4.

7b
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8 marks

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

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8a
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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 , where 0 degree less than theta less than 360 degree.

q7a-7-2-applications-of-differentiation-very-hard-a-level-maths-pure

Although r and theta may be varied, it is necessary that the table have a fixed area of  A m2.

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

8b
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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

8c
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5 marks

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

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