Further Differentiation (Edexcel A Level Maths: Pure): Exam Questions

5 hours48 questions
1a4 marks

table row cell y equals fraction numerator 5 x squared plus 10 x over denominator open parentheses x plus 1 close parentheses squared end fraction end cell blank cell x not equal to negative 1 end cell end table

Show that fraction numerator straight d y over denominator straight d x end fraction equals A over open parentheses x plus 1 close parentheses to the power of n, where A and n are constants to be found.

1b
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1 mark

Hence deduce the range of values for x for which fraction numerator straight d y over denominator straight d x end fraction less than 0

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2
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4 marks
Graph of a curve with x and y axes. The curve crosses the x-axis at point α and later has a local maximum marked P in the fourth quadrant. Origin is labelled O.
Figure 2

Figure 2 shows a sketch of part of the curve with equation y equals straight f open parentheses x close parentheses where

straight f open parentheses x close parentheses equals 8 sin open parentheses 1 half x close parentheses minus 3 x plus 9 space space space space space space space space space space space space space x greater than 0

and x is measured in radians.

The point P, shown in Figure 2, is a local maximum point on the curve.

Using calculus and the sketch in Figure 2, find the x coordinate of P, giving your answer to 3 significant figures.

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3a
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2 marks

Given thatspace straight f open parentheses x close parentheses equals x squared

Use differentiation from first principles to show that

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of open parentheses fraction numerator x squared plus 2 h x plus h squared minus x squared over denominator h end fraction close parentheses space.

3b
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3 marks

Hence prove that

straight f apostrophe open parentheses x close parentheses equals 2 x.

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4a
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2 marks

A curve has the equation space y equals 5 e to the power of negative 2 x end exponent.

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

4b
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3 marks

(i) Find the gradient of the tangent at the point where x equals 1, giving your answer in the form negative a e to the power of negative 2 end exponent where a is a positive integer to be found.

(ii) Hence show that the gradient of the normal to the curve at the point where space x equals 1 space is fraction numerator space 1 over denominator 10 end fraction e squared.

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5
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4 marks

Find  fraction numerator d y over denominator d x end fractionfor

(i) y equals sin space open parentheses 3 x squared close parentheses,

(ii) y equals 2 ln space open parentheses x cubed close parentheses .

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6
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4 marks

The curve with equation  y equals e to the power of x squared minus 9 end exponent  passes through the point with coordinates (-3 , 1).

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

(ii) Find the equation of the tangent to the curve at the point
(-3 , 1).

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7a
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3 marks

Differentiate  space left parenthesis x cubed minus 2 x right parenthesis ln space x spacewith respect to x.

7b
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3 marks

Differentiate  e to the power of x cos space 2 x space space with respect to x.

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8a
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3 marks

Differentiate  fraction numerator cos space x over denominator space sin space x end fraction   with respect to x

8b
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3 marks

Differentiate  fraction numerator space 2 x squared minus 3 x plus 4 over denominator sin space 3 x space end fraction with respect to x.

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9
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2 marks

Write down  fraction numerator d y over denominator d x end fraction  when

(i) y equals sec space 5 x space comma

(ii) y equals cosec space 3 x space.

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10a
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4 marks

The function space straight f left parenthesis x right parenthesis  is defined as

straight f open parentheses x close parentheses equals open parentheses x squared minus 4 x plus 4 close parentheses ln open parentheses space x close parentheses space comma space space space space space space x greater than 0

Show that the graph of space y equals straight f open parentheses x close parentheses  intercepts the x-axis at the points (1 , 0) and (2 , 0).

10b
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4 marks

Find space straight f apostrophe open parentheses x close parentheses.

10c
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2 marks

Find the gradient of the tangent at the point (1 , 0).

10d
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2 marks

Hence find the equation of the tangent at the point (1 , 0), giving your answer in the form  a x plus b y plus c equals 0, where a, b and c are integers to be found.

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15 marks

Given that

y equals fraction numerator 3 sin theta over denominator 2 sin theta space plus space 2 cos theta end fraction space space space space space space space space minus pi over 4 less than theta less than fraction numerator 3 pi over denominator 4 end fraction

show that

fraction numerator straight d y over denominator straight d theta end fraction equals fraction numerator A over denominator 1 plus sin 2 theta end fraction space space space space space space space space minus pi over 4 less than theta less than fraction numerator 3 pi over denominator 4 end fraction

where A is a rational constant to be found.

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2a4 marks
Graph in the first quadrant of convex (i.e. "concave up") curve C,  with a minimum turning point marked at point P . Axes are labelled x and y, with origin O at their intersection.
Figure 1

Figure 1 shows a sketch of the curve C with equation

y equals fraction numerator 4 x squared plus x over denominator 2 square root of x end fraction minus 4 ln x space space space space space space space space space space space x greater than 0

Show that

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 12 x squared plus x minus 16 square root of x over denominator 4 x square root of x end fraction

2b3 marks

The point P, shown in Figure 1, is the minimum turning point on C.

Show that the x coordinate of P is a solution of

x equals open parentheses 4 over 3 minus fraction numerator square root of x over denominator 12 end fraction close parentheses to the power of 2 over 3 end exponent

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3a1 mark
Graph showing velocity (v) against time (t). The curve rises from origin, peaks, then falls back to the axis at time, T.
Figure 2

A car stops at two sets of traffic lights.

Figure 2 shows a graph of the speed of the car, v space ms to the power of negative 1 end exponent, as it travels between the two sets of traffic lights.

The car takes T seconds to travel between the two sets of traffic lights.

The speed of the car is modelled by the equation

v equals left parenthesis 10 minus 0.4 t right parenthesis ln left parenthesis t plus 1 right parenthesis space space space space space space space space space space 0 less or equal than t less or equal than T

where t seconds is the time after the car leaves the first set of traffic lights.

According to the model, find the value of T

3b4 marks

Show that the maximum speed of the car occurs when

t equals fraction numerator 26 over denominator 1 plus ln open parentheses t plus 1 close parentheses end fraction minus 1

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45 marks

Given that theta is measured in radians, prove, from first principles, that

fraction numerator d over denominator d theta end fraction open parentheses cos theta close parentheses equals negative sin theta

You may assume the formula for cos open parentheses A plus-or-minus B close parentheses and that as h rightwards arrow 0, fraction numerator sin h over denominator h end fraction rightwards arrow 1 and fraction numerator cos h minus 1 over denominator h end fraction rightwards arrow 0

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5a3 marks

The function straight f is defined by

straight f open parentheses x close parentheses equals fraction numerator straight e to the power of 3 x end exponent over denominator 4 x squared plus k end fraction

where k is a positive constant.

Show that

straight f to the power of apostrophe open parentheses x close parentheses equals open parentheses 12 x squared minus 8 x plus 3 k close parentheses straight g open parentheses x close parentheses

where straight g open parentheses x close parentheses is a function to be found.

5b3 marks

Given that the curve with equation y equals straight f left parenthesis x right parenthesis has at least one stationary point, find the range of possible values of k.

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65 marks

y equals sin space x

where x is measured in radians.

Use differentiation from first principles to show that

fraction numerator straight d y over denominator straight d x end fraction equals cos space x

You may

  • use without proof the formula for sin open parentheses A plus-or-minus B close parentheses

  • assume that as h rightwards arrow 0, fraction numerator sin space h over denominator h end fraction rightwards arrow 1 and fraction numerator cos space h minus 1 over denominator h end fraction rightwards arrow 0

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7a1 mark

The function straight g is defined by

straight g open parentheses x close parentheses equals fraction numerator 3 ln open parentheses x close parentheses minus 7 over denominator ln open parentheses x close parentheses minus 2 end fraction space space space space space space space space space space x greater than 0 space space space space space space space space space space x not equal to k

where k is a constant.

Deduce the value of k.

7b3 marks

Prove that

straight g to the power of apostrophe left parenthesis x right parenthesis greater than 0

for all values of x in the domain of straight g.

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8a
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4 marks

Given that straight f open parentheses x close parentheses equals sin space x

Show that

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of open parentheses sin space x space open parentheses fraction numerator cos space h space minus 1 over denominator h end fraction close parentheses plus cos space x space open parentheses fraction numerator sin space h over denominator h end fraction space close parentheses close parentheses

8b
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3 marks

Hence prove that straight f apostrophe open parentheses x close parentheses equals cos space x .

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9
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4 marks

A curve has the equation  y equals e to the power of negative 3 x end exponent plus ln space x space comma space space x greater than 0.

Find the gradient of the normal to the curve at the point open parentheses 1 comma space e to the power of negative 3 end exponent close parentheses, giving your answer correct to 3 decimal places.

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10a
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4 marks

Find  fraction numerator d y over denominator d x end fraction  for each of the following:

y equals cos open parentheses space x squared minus 3 x plus 7 close parentheses plus sin space open parentheses e to the power of x close parentheses space

10b
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3 marks

Find fraction numerator d y over denominator d x end fraction  for each of the following:

y equals ln space open parentheses 2 x cubed close parentheses space

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11
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4 marks

Find the equation of the tangent to the curve  y equals e to the power of 3 x squared space plus space 5 x space minus space 2 end exponent  at the point open parentheses negative 2 comma space space 1 close parentheses, giving your answer in the formspace a x plus b y plus c equals 0, where a, b and c are integers.

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12a
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3 marks

Differentiate with respect to x, simplifying your answers as far as possible:

open parentheses 4 cos space x space minus 3 sin space x close parentheses space e to the power of 3 x space minus space 5 space end exponent

12b
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3 marks

open parentheses x cubed minus 4 x squared plus 7 close parentheses ln space x

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13
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4 marks

Differentiate  fraction numerator 5 x to the power of 7 over denominator sin space 2 x space space end fraction with respect to x.

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14a
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5 marks

Show that if  y equals cosec 2 x , then 

fraction numerator d y over denominator d x end fraction equals negative 2 cosec space 2 x space cot space 2 x

14b
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1 mark

Hence find the gradient of the tangent to the curve space y equals cos e c space 2 x space at the point with coordinates open parentheses pi over 3 comma fraction numerator 2 square root of 3 over denominator 3 end fraction close parentheses

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15a
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4 marks

The diagram below shows part of the graph of  y equals straight f open parentheses x close parentheses, wherespace straight f open parentheses x close parentheses spaceis the function defined by

straight f open parentheses x close parentheses equals open parentheses x squared minus 1 close parentheses ln open parentheses x plus 3 close parentheses comma space space space space space x greater than negative 3

q8a-7-3-medium-a-level-maths

Points A, B and C are the three places where the graph intercepts the x-axis.

Find straight f apostrophe open parentheses x close parentheses.

15b
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2 marks

Show that the coordinates of point A are (-2, 0).

15c
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3 marks

Find the equation of the tangent to the curve at point A.

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1a3 marks
A single curve labelled “C”. It starts slightly above the x-axis in quadrant 2, dips below to a clear minimum in the third quadrant, intersects the y axis at a negative value, then crosses the x axis again at a positive value, reaching a local maximum in quadrant 1, and then flattening out as x increases (not crossing the x axis again).
Figure 2

Figure 2 shows a sketch of the curve C with equation y equals straight f open parentheses x close parentheses where

straight f open parentheses x close parentheses equals 4 open parentheses x squared minus 2 close parentheses straight e to the power of negative 2 x end exponent space space space space space space space space x element of straight real numbers

Show that straight f apostrophe open parentheses x close parentheses equals 8 open parentheses 2 plus x minus x squared close parentheses straight e to the power of negative 2 x end exponent.

1b3 marks

Hence find, in simplest form, the exact coordinates of the stationary points of C.

1c3 marks

The function straight g and the function straight h are defined by

straight g open parentheses x close parentheses equals 2 straight f open parentheses x close parentheses space space space space space space space space space space space space space space x element of straight real numbers
straight h open parentheses x close parentheses equals 2 straight f open parentheses x close parentheses minus 3 space space space space space space space space x greater or equal than 0

Find

(i) the range of straight g

(ii) the range of straight h.

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2a
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4 marks

fraction numerator 1 plus 11 x minus 6 x squared over denominator open parentheses x minus 3 close parentheses open parentheses 1 minus 2 x close parentheses end fraction identical to A plus fraction numerator B over denominator open parentheses x minus 3 close parentheses end fraction plus fraction numerator C over denominator open parentheses 1 minus 2 x close parentheses end fraction

Find the values of the constants A, B and C.

2b3 marks

straight f open parentheses x close parentheses equals fraction numerator 1 plus 11 x minus 6 x squared over denominator open parentheses x minus 3 close parentheses open parentheses 1 minus 2 x close parentheses end fraction space space space space space x greater than 3

Prove that straight f open parentheses x close parentheses is a decreasing function.

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34 marks

Given that

y equals fraction numerator x minus 4 over denominator 2 plus square root of x end fraction space space space space x greater than 0

show that

fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator A square root of x end fraction space space space space x greater than 0

where A is a constant to be found.

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4a5 marks

A curve has equation y equals straight f open parentheses x close parentheses, where

straight f open parentheses x close parentheses equals fraction numerator 7 x straight e to the power of x over denominator square root of straight e to the power of 3 x end exponent minus 2 end root end fraction space space space space space space space space space space space space space space space space x greater than ln cube root of 2

Show that

straight f to the power of apostrophe open parentheses x close parentheses equals fraction numerator 7 straight e to the power of x open parentheses straight e to the power of 3 x end exponent open parentheses 2 minus x close parentheses plus A x plus B close parentheses over denominator 2 open parentheses straight e to the power of 3 x end exponent minus 2 close parentheses to the power of 3 over 2 end exponent end fraction

where A and B are constants to be found.

4b2 marks

Hence show that the x coordinates of the turning points of the curve are solutions of the equation

x equals fraction numerator 2 table row blank blank straight e end table to the power of 3 x end exponent minus 4 over denominator table row blank blank straight e end table to the power of 3 x end exponent plus 4 end fraction

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5
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7 marks

Show from first principles that the derivative of  cos space x is negative sin space x .

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6
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6 marks

A curve has the equation  y equals e to the power of negative 3 x end exponent plus ln space x space comma space space x greater than 0.

Show that the equation of the tangent to the curve at the point with x-coordinate 1 is

y equals open parentheses fraction numerator e cubed minus 3 over denominator e cubed end fraction close parentheses x plus fraction numerator 4 minus e cubed over denominator e cubed end fraction

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7
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3 marks

For space y equals ln space open parentheses a x to the power of n space close parentheses , where a greater than 0 is a real number and space n greater or equal than 1 spaceis an integer, show that

fraction numerator d y over denominator d x end fraction equals n over x

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8
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4 marks

Find the gradient of the normal to the curve space y equals 5 cos space left parenthesis e to the power of x minus pi over 2 right parenthesis spaceat the point with x-coordinate 0.  Give your answer correct to 3 decimal places. 

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9a
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3 marks

Differentiate with respect to x, simplifying your answers as far as possible:

open parentheses 2 sin space 3 x space minus cos space 3 x close parentheses space e to the power of 6 minus x space end exponent

9b
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3 marks

open parentheses x squared minus x close parentheses squared ln space 5 x

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10
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3 marks

By writing  y equals fraction numerator space straight f open parentheses x close parentheses over denominator straight g open parentheses x close parentheses end fraction  as space y equals straight f open parentheses x close parentheses open square brackets straight g open parentheses straight x close parentheses close square brackets to the power of negative 1 space end exponent and then using the product and chain rules, show that

fraction numerator d y over denominator d x end fraction equals fraction numerator straight g open parentheses x close parentheses straight f apostrophe open parentheses x close parentheses minus straight f open parentheses x close parentheses straight g apostrophe open parentheses x close parentheses over denominator open parentheses g open parentheses x close parentheses close parentheses squared end fraction

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11a
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2 marks

Given that x equals sec space 7 y ,

Find  fraction numerator d y over denominator d x end fraction  in terms of y

11b
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4 marks

Hence find fraction numerator d y over denominator d x end fraction in terms of x.

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12
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5 marks

The diagram below shows part of the graph of space y equals straight f open parentheses x close parentheses, where space straight f open parentheses x close parentheses spaceis the function defined by

straight f open parentheses x close parentheses equals fraction numerator sin space x over denominator 1 minus e to the power of x end fraction space comma space   x greater than 0

q7-7-3-further-differentiation-medium-a-level-maths-pure-screenshots

Point A is a maximum point on the graph.

Show that the x-coordinate of A is a solution to the equation

fraction numerator cos space x space plus e to the power of x open parentheses sin space x minus cos space x close parentheses over denominator space e to the power of 2 x end exponent minus 2 e to the power of x plus 1 end fraction space equals 0

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1a4 marks

straight f open parentheses x close parentheses equals 10 straight e to the power of negative 0.25 x end exponent sin space x

Show that the x-coordinates of the turning points of the curve with equation y equals straight f open parentheses x close parentheses satisfy the equation tan space x equals 4

1b2 marks
Graph showing an oscillating curve with amplitude and frequency decreasing over time. The curve intersects the x-axis several times.
Figure 3

Figure 3 shows a sketch of part of the curve with equation y equals straight f open parentheses x close parentheses.

Sketch the graph of H against t where

H open parentheses t close parentheses equals open vertical bar 10 straight e to the power of negative 0.25 t end exponent sin space t close vertical bar

showing the long-term behaviour of this curve.

1c
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3 marks

The function H open parentheses t close parentheses is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked.

Using this model, find the maximum height of the ball above the ground between the first and second bounce.

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2a2 marks

The curve C, in the standard Cartesian plane, is defined by the equation

table row cell x equals 4 space sin space 2 y end cell blank cell negative pi over 4 less than y less than pi over 4 end cell end table

The curve passes through the origin O

Find the value of fraction numerator straight d y over denominator straight d x end fraction at the origin.

2b2 marks

(i) Use the small angle approximation for sin space 2 y to find an equation linking x and y for points close to the origin.

(ii) Explain the relationship between the answers to (a) and (b)(i).

2c3 marks

Show that, for all points open parentheses x comma space y close parentheses lying on C,

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator a square root of b minus x squared end root end fraction

where a and b are constants to be found.

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3a1 mark

A scientist is studying a population of mice on an island.

The number of mice, N, in the population, t months after the start of the study, is modelled by the equation

N equals fraction numerator 900 over denominator 3 plus 7 straight e to the power of negative 0.25 t end exponent end fraction comma space space space t element of straight real numbers comma space space space t greater or equal than 0

Find the number of mice in the population at the start of the study.

3b
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4 marks

Show that the rate of growth fraction numerator d N over denominator d t end fraction is given by fraction numerator d N over denominator d t end fraction equals fraction numerator N open parentheses 300 minus N close parentheses over denominator 1200 end fraction

3c4 marks

The rate of growth is a maximum after T months.

Find, according to the model, the value of T.

3d1 mark

According to the model, the maximum number of mice on the island is P.

State the value of P.

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4
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9 marks

Show from first principles that the derivative of space tan space 3 x spaceis 3 sec squared 3 x.

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5
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4 marks

A curve has the equation space y equals 3 to the power of x plus 2 to the power of negative x end exponent.

Show that the gradient of the normal to the curve at the point  open parentheses 1 comma space fraction numerator space 7 over denominator 2 end fraction close parentheses space is

fraction numerator 2 over denominator ln space 2 minus 6 ln space 3 space space space end fraction space space

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6
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4 marks

Find the derivative of the function space straight f open parentheses x close parentheses equals sin space open parentheses cos space open parentheses ln space 1 over x close parentheses close parentheses space comma space space x greater than 0.

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7a
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4 marks

Show that the derivative  y equals 4 to the power of negative x to the power of 4 end exponent  is

fraction numerator d y over denominator d x end fraction equals negative open parentheses ln space 4 close parentheses space x cubed 4 to the power of 1 minus x to the power of 4 end exponent

7b
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2 marks

Hence find the equation of the tangent to the curve at the point open parentheses 1 comma 1 fourth close parentheses, giving your answer in the form y equals a x plus b, where a and b are to be given as exact values.

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8a
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3 marks

Differentiate with respect to x, simplifying your answers where possible:

open parentheses 5 plus sin to the power of 2 space end exponent 3 x close parentheses space e to the power of x squared minus 3 x plus 2 end exponent

8b
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3 marks

3 to the power of square root of x end exponent open parentheses square root of x minus fraction numerator 1 over denominator square root of x end fraction close parentheses

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9
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6 marks

The diagram below shows the graph of y equals straight f open parentheses x close parentheses, where straight f open parentheses x close parentheses is the function defined by

space straight f open parentheses x close parentheses equals fraction numerator sin space 3 x over denominator e to the power of 2 x minus 3 end exponent end fraction space comma space   space space space space space space space space space 0 less or equal than x less or equal than fraction numerator 2 pi over denominator 3 end fraction

q8a-7-3-further-differentiation-vh-a-level-maths-pure-screenshots

The points A and B are maximum and minimum points, respectively.

Find the range of straight f open parentheses x close parentheses , giving your answer correct to 3 decimal places.

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10
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5 marks

A is the point on the graph of  space y equals arctan space x spacesuch that the tangent to the graph at A passes through the point open parentheses 0 comma space space 1 half close parentheses
Show that the x-coordinate of A satisfies the equation

x minus tan space open parentheses fraction numerator open parentheses 1 plus x close parentheses squared over denominator 2 open parentheses 1 plus x squared close parentheses space end fraction close parentheses equals 0

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11
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5 marks

A sequence of functions is defined by the recurrence relation

u subscript k plus 1 end subscript open parentheses x close parentheses equals fraction numerator straight d over denominator straight d x end fraction u subscript k open parentheses x close parentheses comma space space space u subscript 1 open parentheses x close parentheses equals sin space open parentheses x square root of 2 close parentheses

Based on that sequence, the functionspace straight f subscript n open parentheses x close parentheses spaceis defined by

straight f subscript n open parentheses x close parentheses equals sum from r equals 1 to n of u subscript r open parentheses x close parentheses

Calculate the value of space straight f subscript 41 space open parentheses fraction numerator straight pi square root of 2 over denominator 4 end fraction close parentheses

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