CIE AS Maths: Pure 1

Topic Questions

5.2 Applications of Differentiation

1a
Sme Calculator
2 marks

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

1b
Sme Calculator
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.

Did this page help you?

2
Sme Calculator
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.

Did this page help you?

3
Sme Calculator
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).

Did this page help you?

4a
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.

4b
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 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?

Did this page help you?

5a
Sme Calculator
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.

5b
Sme Calculator
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.

Did this page help you?

6
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?

7
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?

8
Sme Calculator
5 marks

Show that the point open parentheses 2 space comma space 1 close parentheses is a (local) maximum point on the curve with equation

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

Did this page help you?

9a
Sme Calculator
3 marks

In a computer animation, the side length, , of a square is increasing at a constant rate of 2 millimetres per second.

(i)
Write down the value of fraction numerator d s over denominator d t end fraction, where t is time and measured in seconds.
(ii)
Write down a formula for the area, A space m m squared, of the square and hence find fraction numerator d A over denominator d s end fraction.
9b
Sme Calculator
3 marks

Use the chain rule to find an expression for fraction numerator d A over denominator d t end fraction in terms of s and hence find the rate at which the area is increasing when s space equals space 10.

Did this page help you?

10a
Sme Calculator
4 marks

Find the value of fraction numerator d y over denominator d x end fraction and  fraction numerator d to the power of 2 y end exponent 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.

10b
Sme Calculator
1 mark

Explain why x space equals space 2 is not a stationary point.

Did this page help you?

11a
Sme Calculator
3 marks

The side length, x space c m, of a cube increases at a constant rate of 0.1 space c m space s to the power of negative 1 end exponent .

(i)
Write down the value of fraction numerator d x over denominator d t end fraction, where t is time and measured in seconds.
(ii)
Write down a formula for the volume, V space c m cubed, of the cube and hence find fraction numerator d V over denominator d x end fraction.
11b
Sme Calculator
3 marks

Use the chain rule to find an expression for fraction numerator d V over denominator d t end fraction in terms of x and hence find the rate at which the volume is increasing when the side length of the cube is 4 space c m.

Did this page help you?

12a
Sme Calculator
2 marks

The rate at which the radius, r space c m, of a sphere increases over time open parentheses t space seconds close parentheses is directly proportional to the temperature open parentheses T space space to the power of degree space end exponent C close parentheses of its immediate surroundings.
Write down an equation linking fraction numerator d r over denominator d t end fraction, T and the constant of proportionality, k.

12b
Sme Calculator
3 marks

When the surrounding temperature is 20 degree space C, the radius of the sphere is increasing at a rate of 0.4 space c m space straight s to the power of negative 1 end exponent .
Find the value of k.

Did this page help you?

13a
Sme Calculator
5 marks
(i)
Write down a formula for the volume of a cube, V space c m cubed, and the surface area,s space c m squared, of a cube, in terms of the side length of a cube, x space c m.
(ii)
Show that fraction numerator d x over denominator d V end fraction equals 1 over 3 x squared and find an expression for fraction numerator d S over denominator d x end fraction.
13b
Sme Calculator
5 marks

The volume of a cube is decreasing at a constant rate of 0.6 space c m cubed space s to the power of negative 1 end exponent.

(i)
Explain why fraction numerator d V over denominator d t end fraction , where t is time in seconds, has the value of negative space 0.6.
(ii)
Use the chain rule to find an expression for fraction numerator d s over denominator d t end fraction in terms of fraction numerator d S over denominator d x end fraction, fraction numerator d x over denominator d V end fraction and fraction numerator d V over denominator d t end fraction.
(iii)
Hence write fraction numerator d S over denominator d t end fraction in terms of x and find the rate at which the area of the cube is decreasing at the instant when its side length is 5 space cm.

Did this page help you?

1
Sme Calculator
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.

Did this page help you?

2
Sme Calculator
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.

Did this page help you?

3
Sme Calculator
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.

Did this page help you?

4
Sme Calculator
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.

Did this page help you?

5a
Sme Calculator
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

5b
Sme Calculator
2 marks

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

Did this page help you?

6
Sme Calculator
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.

Did this page help you?

7a
Sme Calculator
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.

7b
Sme Calculator
4 marks

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

Did this page help you?

8a
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:

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.

8b
Sme Calculator
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.

8c
Sme Calculator
4 marks

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

8d
Sme Calculator
1 mark

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

Did this page help you?

9
Sme Calculator
4 marks

Did this page help you?

10
Sme Calculator
4 marks

Did this page help you?

11
Sme Calculator
6 marks

Did this page help you?

12
Sme Calculator
6 marks

Did this page help you?

13
Sme Calculator
6 marks

Did this page help you?

14
Sme Calculator
6 marks

Did this page help you?

1
Sme Calculator
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.

Did this page help you?

2
Sme Calculator
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.

Did this page help you?

3a
Sme Calculator
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.

3b
Sme Calculator
3 marks

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

3c
Sme Calculator
2 marks

Find an equation of the tangent to C at P.

Did this page help you?

4a
Sme Calculator
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.

4b
Sme Calculator
3 marks

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

4c
Sme Calculator
2 marks

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

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

Did this page help you?

5a
Sme Calculator
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

5b
Sme Calculator
2 marks

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

Did this page help you?

6a
Sme Calculator
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.

6b
Sme Calculator
3 marks

Determine the coordinates of the local minimum of the curve.

Did this page help you?

7a
Sme Calculator
5 marks

7b
Sme Calculator
2 marks

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

Did this page help you?

8a
Sme Calculator
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.

8b
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?

9a
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.

9b
Sme Calculator
2 marks

Find the nature of the stationary points found in part (a).

Did this page help you?

10
Sme Calculator
4 marks

In a computer animation, the radius of a circle increases at a constant rate of 1 millimetre per second.  Find the rate, per second, at which the area of the circle is increasing at the time when the radius is 8 millimetres.  Give your answer as a multiple of straight pi.

Did this page help you?

11
Sme Calculator
5 marks

The side length of a cube increases at a rate of 0.1 space c m space s to the power of negative 1 end exponent.
Find the rate of change of the volume of the cube at the instant the side length is 5 cm .
You may assume that the cube remains cubical at all times.

Did this page help you?

12a
Sme Calculator
5 marks

In the production process of a glass sphere, hot glass is blown such that the radius, r cm, increases over time open parentheses straight t space seconds close parentheses in direct proportion to the temperature T degree space C of the glass.
Find an expression, in terms of r and T, for the rate of change of the volume open parentheses V space c m cubed close parenthesesof a glass sphere.

12b
Sme Calculator
3 marks

When the temperature of the glass is 1200 to the power of degree space C, a glass sphere has a radius of 2 space c m and its volume is increasing at a rate of 5 space c m cubed space s to the power of negative 1 end exponent.
Find the rate of increase of the radius at this time.

Did this page help you?

13
Sme Calculator
6 marks

An ice cube, of side length x space c m, is melting at a constant rate of 0.8 space cm to the power of 3 space end exponent straight s to the power of negative 1 end exponent.

Assuming that the ice cube remains in the shape of a cube whilst it melts, find the rate at which its surface area is melting at the point when its side length is 2 space cm.

Did this page help you?

14
Sme Calculator
5 marks

A bowl is in the shape of a hemisphere of radius 8 space c m.

The volume of liquid in the bowl is given by the formula

      V equals 8 pi h squared minus 1 third pi h cubed

where h space c m is the depth of the liquid (ie the height between the bottom of the bowl and the level of the liquid).

Liquid is leaking through a small hole in the bottom of the bowl at a constant rate of 5 space c m cubed space s to the power of negative 1 end exponent.  Find the rate of change of the depth of liquid in the bowl at the instant the height of liquid is 3 space c m.

Did this page help you?

1
Sme Calculator
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.

Did this page help you?

2
Sme Calculator
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.

Did this page help you?

3a
Sme Calculator
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.

3b
Sme Calculator
2 marks

Calculate the area of triangle ABC.

Did this page help you?

4a
Sme Calculator
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.

4b
Sme Calculator
1 mark

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

4c
Sme Calculator
4 marks

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

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

Did this page help you?

5a
Sme Calculator
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.

5b
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?

6a
Sme Calculator
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.

6b
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?

7a
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.

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 .
 

7b
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

7c
Sme Calculator
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.

Did this page help you?

8
Sme Calculator
4 marks

Find the coordinates of the stationary points, and their nature, on the graph with equation y equals e to the power of 0.4 x end exponent open parentheses x squared plus 3 x minus 4 close parentheses. Giving your answer to three significant figures.

Did this page help you?

9
Sme Calculator
6 marks

A plant pot in the shape of square-based pyramid (stood on its vertex) is being filled with soil at a rate of 72 space c m to the power of 3 space end exponent s to the power of negative 1 end exponent .
The plant pot has a height of 1 space straight m and a base length of 40 space cm.
Find the rate at which the depth of soil is increasing at the moment when the depth is 60 space cm
.

(The volume of a pyramid is a third of the area of the base times the height.)

Did this page help you?

10
Sme Calculator
6 marks

An expanding spherical air bubble has radius, r space cm , at a time, t space seconds, determined by the function r open parentheses t close parentheses space equals space 0.3 space plus space 0.1 t squared.

The bubble will burst if the rate of expansion of its volume exceeds 4 t space c m to the power of 3 space end exponent s.

Find, to one decimal place, the length of time the bubble expands for.

Did this page help you?

11
Sme Calculator
7 marks

A small conical pot, stood on its base, is being filled with salt via a small hole at its vertex.  The cone has a height of 6 space c m and a radius of 2 space c m.

Salt is being poured into the pot at a constant rate of 0.3 space c m to the power of 3 space end exponent s to the power of negative 1 end exponent.
Find, to three significant figures, the rate of change in depth of the salt at the instant when the pot is half full by volume.

Did this page help you?

12a
Sme Calculator
5 marks

A large block of ice used by sculptors is in the shape of a cuboid with dimensions x space m space by space 2 straight x space straight m space by space 5 straight x space straight m.  The block melts uniformly with its surface area decreasing at a constant rate of k space m squared space s to the power of negative 1 end exponent. You may assume that as the block melts, the shape remains mathematically similar to the original cuboid.

Show that the rate of melting, by volume, is given by

      fraction numerator 15 k x over denominator 34 end fraction m cubed space s to the power of negative 1 end exponent.

12b
Sme Calculator
3 marks

In the case when k space equals space 0.2 comma the block of ice remains solid enough to be sculpted whilst the rate of melting, by volume, is less than  0.05 space m cubed s to the power of negative 1 end exponent.

Find the value of x for the largest block of ice that can be used for ice sculpting under such conditions, giving your answer as a fraction in its lowest terms.

Did this page help you?

13
Sme Calculator
7 marks

The volume of liquid in a hemispherical bowl is given by the formula

      V space equals space 1 third πh squared open parentheses 3 R space minus h close parentheses

where R is the radius of the bowl and h is the depth of liquid.

(ie the height between the bottom of the bowl and the level of the liquid).

In a particular case, liquid is leaking through a small hole in the bottom of a bowl at a rate directly proportional to the depth of liquid.

When the bowl is full, the rate of volume loss is equal to straight pi.

Show that the rate of change of the depth of the liquid is inversely proportional to R space open parentheses h minus space 2 R close parentheses

Did this page help you?