A LevelMathsEdexcelPureExam QuestionsDifferentiationFurther Differentiation

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

Exam code: 9MA0

5 hours49 questions
Easy
Medium
Hard
Very Hard
1a
4 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.

How did you do?

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

How did you do?

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.

How did you do?

3a
2 marks

The function \text{f} is defined by

\text{f}(x) = x^2 \qquad x \in \mathbb{R}

Use differentiation from first principles to show that

\text{f}'(x) = \lim_{h \to 0}\left(\frac{x^2 + 2xh + h^2 - x^2}{h}\right)

How did you do?

3b
3 marks

Hence prove that \text{f}'(x) = 2x.

How did you do?

4a
2 marks

The curve C has equation

y = 5\text{e}^{-2x} \qquad x \in \mathbb{R}

Find \dfrac{\text{d}y}{\text{d}x}.

How did you do?

4b
3 marks

(i) Find the gradient of the tangent to C at the point where x = 1, giving your answer in the form -a\text{e}^{-2} where a is a positive integer to be found.

(ii) Hence show that the gradient of the normal to C at the point where x = 1 is \dfrac{1}{10}\text{e}^{2}.

How did you do?

5
4 marks

Find \dfrac{\text{d}y}{\text{d}x} for each of the following:

(i) y = \text{sin}(3x^2)

(ii) y = 2\ln(x^3), x > 0, giving your answer in simplest form.

How did you do?

6
4 marks

The curve C has equation

y = \text{e}^{x^2 - 9} \qquad x \in \mathbb{R}

The point P(-3, 1) lies on C.

(i) Find \dfrac{\text{d}y}{\text{d}x}.

(ii) Find the equation of the tangent to C at the point P, giving your answer in the form y = mx + c, where m and c are constants to be found.

How did you do?

7a
3 marks

Given that

y = (x^3 - 2x)\ln x \qquad x > 0

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

7b
3 marks

Given that

y = \text{e}^x \text{cos} \; 2x \qquad x \in \mathbb{R}

find \dfrac{\text{d}y}{\text{d}x}.

How did you do?

8
3 marks

Given that

y = \frac{2x^2 - 3x + 4}{\text{sin} \; 3x} \qquad 0 < x < \tfrac{\pi}{3}

find \dfrac{\text{d}y}{\text{d}x}.

How did you do?

9
2 marks

Write down \dfrac{\text{d}y}{\text{d}x} for each of the following:

(i) y = \text{sec} \; 5x

(ii) y = \text{cosec} \; 3x

How did you do?

1
5 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.

How did you do?

2a
4 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

How did you do?

2b
3 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

How did you do?

3a
1 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

How did you do?

3b
4 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

How did you do?

4
5 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

How did you do?

5a
3 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.

How did you do?

5b
3 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.

How did you do?

6
5 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

How did you do?

7a
1 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.

How did you do?

7b
3 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.

How did you do?

8a
4 marks

Given that \text{f}(x) = \text{sin} \; x, where x is measured in radians,

use differentiation from first principles to show that

\text{f}'(x) = \lim_{h \to 0}\left(\text{sin} \; x\left(\frac{\text{cos} \; h - 1}{h}\right) + \text{cos} \; x\left(\frac{\text{sin} \; h}{h}\right)\right)

How did you do?

8b
3 marks

Hence prove that \text{f}'(x) = \text{cos} \; x.

How did you do?

9
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4 marks

A curve C has equation

y = \text{e}^{-3x} + \ln x \qquad x > 0

Find the gradient of the normal to C at the point (1, \text{e}^{-3}), giving your answer to 3 decimal places.

How did you do?

10a
4 marks

Given that

y = \text{cos}(x^2 - 3x + 7) + \text{sin}(\text{e}^x) \qquad x \in \mathbb{R}

find \dfrac{\text{d}y}{\text{d}x}.

How did you do?

10b
3 marks

Given that

y = \ln(2x^3) \qquad x > 0

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

11
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4 marks

The curve C has equation

y = \text{e}^{3x^2 + 5x - 2} \qquad x \in \mathbb{R}

The point P(-2, 1) lies on C.

Find the equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0, where a, b and c are integers to be found.

How did you do?

12a
3 marks

Given that

y = (4\,\text{cos} \; x - 3\,\text{sin} \; x)\,\text{e}^{3x - 5} \qquad x \in \mathbb{R}

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

12b
3 marks

Given that

y = (x^3 - 4x^2 + 7)\ln x \qquad x > 0

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

13a
3 marks

Given that

y equals fraction numerator 5 x squared minus 10 over denominator 2 x plus 1 end fraction space comma space space x not equal to 0.5

Show that fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 10 x squared plus 10 x plus 20 over denominator open parentheses 2 x plus 1 close parentheses squared end fraction.

How did you do?

13b
2 marks

Hence show that y is an increasing function for all defined values of x.

How did you do?

14
4 marks

Given that

y = \frac{5x^7}{\text{sin} \; 2x} \qquad 0 < x < \tfrac{\pi}{2}

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

15a
5 marks

Show that if y = \text{cosec} \; 2x, then

\frac{\text{d}y}{\text{d}x} = -2\,\text{cosec} \; 2x\,\text{cot} \; 2x

How did you do?

15b
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1 mark

Hence find the exact gradient of the tangent to the curve y = \text{cosec} \; 2x at the point with coordinates \left(\dfrac{\pi}{3},\, \dfrac{2\sqrt{3}}{3}\right).

How did you do?

16a
4 marks
Sketch of a curve $y = \text{f}(x)$ with three $x$-axis intercepts labelled $A$, $B$ and $C$, left to right.
Figure 1

Figure 1 shows a sketch of part of the curve C with equation y = \text{f}(x), where

\text{f}(x) = (x^2 - 1)\ln(x + 3) \qquad x > -3

The curve C crosses the x-axis at the points A, B and C, as shown in Figure 1.

Find \text{f}'(x).

How did you do?

16b
2 marks

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

How did you do?

16c
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3 marks

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

How did you do?

17
3 marks

Given that

y = \ln(ax^n)

where a > 0 is a real constant and n \geqslant 1 is an integer,

show that

\frac{\text{d}y}{\text{d}x} = \frac{n}{x}

How did you do?

18a
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2 marks

The curve C has equation y = \text{f}(x), where

\text{f}(x) = (x^2 - 4x + 4)\ln x \qquad x > 0

Show that C meets the x-axis at the points (1, 0) and (2, 0).

How did you do?

18b
3 marks

Find \text{f}'(x).

How did you do?

18c
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1 mark

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

How did you do?

18d
2 marks

Hence find the equation of the tangent to C at the point (1, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers to be found.

How did you do?

19
3 marks

Given that

y = \frac{\text{f}(x)}{\text{g}(x)}

by writing y = \text{f}(x)[\text{g}(x)]^{-1} and using the product and chain rules, show that

\frac{\text{d}y}{\text{d}x} = \frac{\text{g}(x)\,\text{f}'(x) - \text{f}(x)\,\text{g}'(x)}{(\text{g}(x))^2}

How did you do?

1a
3 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.

How did you do?

1b
3 marks

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

How did you do?

1c
3 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.

How did you do?

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.

How did you do?

2b
3 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.

How did you do?

3
4 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.

How did you do?

4a
5 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.

How did you do?

4b
2 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

How did you do?

5
5 marks

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

\frac{\text{d}}{\text{d}\theta}(\text{cos} \; \theta) = -\text{sin} \; \theta

You may assume the formula for \text{cos}(A \pm B) and that as h \to 0, \dfrac{\text{sin} \; h}{h} \to 1 and \dfrac{\text{cos} \; h - 1}{h} \to 0.

How did you do?

6
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6 marks

The curve C has equation

y = \text{e}^{-3x} + \ln x \qquad x > 0

Show that the equation of the tangent to C at the point where x = 1 is

y = \left(\frac{\text{e}^3 - 3}{\text{e}^3}\right)x + \frac{4 - \text{e}^3}{\text{e}^3}

How did you do?

7
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4 marks

The curve C has equation

y = 5\,\text{cos}\!\left(\text{e}^x - \tfrac{\pi}{2}\right) \qquad x \in \mathbb{R}

Find the gradient of the normal to C at the point where x = 0, giving your answer to 3 decimal places.

How did you do?

8a
3 marks

Given that

y = (2\,\text{sin} \; 3x - \text{cos} \; 3x)\,\text{e}^{6 - x} \qquad x \in \mathbb{R}

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

8b
3 marks

Given that

y = (x^2 - x)^2 \ln 5x \qquad x > 0

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

9a
2 marks

Given that

x = \text{sec} \; 7y \qquad 0 < y < \tfrac{\pi}{14}

find \dfrac{\text{d}y}{\text{d}x} in terms of y.

How did you do?

9b
4 marks

Hence show that

\frac{\text{d}y}{\text{d}x} = \frac{1}{7x\sqrt{x^2 - 1}}

How did you do?

10
5 marks
Sketch of $$y = \text{f}(x)$$ with a maximum turning point labelled $$A$$ in the first quadrant.
Figure 2

Figure 2 shows a sketch of part of the curve with equation y = \text{f}(x), where

\text{f}(x) = \frac{\text{sin} \; x}{1 - \text{e}^x} \qquad x > 0

The point A, shown in Figure 2, is a maximum turning point on the curve.

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

\frac{\text{cos} \; x + \text{e}^x(\text{sin} \; x - \text{cos} \; x)}{\text{e}^{2x} - 2\text{e}^x + 1} = 0

How did you do?

11
4 marks

The curve C has equation

y = 3^x + 2^{-x} \qquad x \in \mathbb{R}

Show that the gradient of the normal to C at the point \left(1,\, \dfrac{7}{2}\right) is

\frac{2}{\ln 2 - 6\ln 3}

How did you do?

12
4 marks

The function \text{f} is defined by

\text{f}(x) = \text{sin} \; \left(\text{cos} \; \left(\ln \frac{1}{x}\right)\right) \qquad x > 0

Find \text{f}'(x).

How did you do?

1a
4 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

How did you do?

1b
2 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.

How did you do?

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.

How did you do?

2a
2 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.

How did you do?

2b
2 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).

How did you do?

2c
3 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.

How did you do?

3a
1 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.

How did you do?

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

How did you do?

3c
4 marks

The rate of growth is a maximum after T months.

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

How did you do?

3d
1 mark

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

State the value of P.

How did you do?

4
9 marks

Given that x is measured in radians, prove, from first principles, that the derivative of \text{tan} \; 3x is 3\,\text{sec}^2 3x.

You may assume the formulae for \text{sin}(A \pm B), \text{cos}(A \pm B) and that as h \to 0, \dfrac{\text{sin} \; h}{h} \to 1 and \dfrac{\text{cos} \; h - 1}{h} \to 0.

How did you do?

5a
4 marks

The curve C has equation

y = 4^{-x^4} \qquad x \in \mathbb{R}

Show that

\frac{\text{d}y}{\text{d}x} = -(\ln 4)\,x^3\, 4^{1 - x^4}

How did you do?

5b
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2 marks

Hence find the equation of the tangent to C at the point \left(1,\, \dfrac{1}{4}\right), giving your answer in the form y = ax + b, where a and b are to be given as exact values.

How did you do?

6a
3 marks

Given that

y equals left parenthesis 5 plus text sin end text squared 3 x right parenthesis   text e  end text to the power of x squared minus 3 x plus 2 end exponent      x element of straight real numbers

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

6b
3 marks

Given that

y = 3^{\sqrt{x}}\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right) \qquad x > 0

find \dfrac{\text{d}y}{\text{d}x}, giving your answer in its simplest form.

How did you do?

7
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6 marks
Sketch of $$y = \text{f}(x)$$ with maximum point $$A$$ in the upper region and minimum point $$B$$ below the $$x$$-axis, right endpoint at $$(2\pi/3, 0)$$.
Figure 1

Figure 1 shows a sketch of part of the curve with equation y = \text{f}(x), where

\text{f}(x) = \frac{\text{sin} \; 3x}{\text{e}^{2x - 3}} \qquad 0 \leqslant x \leqslant \frac{2\pi}{3}

The points A and B, shown in Figure 1, are the maximum and minimum turning points on the curve respectively. The curve crosses the x-axis at the origin and at the point \left(\dfrac{2\pi}{3},\, 0\right).

Find the range of \text{f}(x), giving your answer to 3 decimal places.

How did you do?

8
5 marks

The curve C has equation

y = \text{arctan} \; x \qquad x \in \mathbb{R}

The point A lies on C.

The tangent to C at the point A passes through the point \left(0,\, \dfrac{1}{2}\right).

Show that the x-coordinate of A satisfies the equation

x - \text{tan} \; \left(\frac{(1 + x)^2}{2(1 + x^2)}\right) = 0

How did you do?

9
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5 marks

A sequence of functions u_1, u_2, u_3, \ldots is defined by the recurrence relation

u_{k+1}(x) = \frac{\text{d}}{\text{d}x}(u_k(x)) \qquad k \geqslant 1

where

u_1(x) = \text{sin}(x\sqrt{2})

Based on this sequence, the function \text{f}_n(x) is defined by

\text{f}_n(x) = \sum_{r=1}^{n} u_r(x)

Calculate the exact value of \text{f}_{41}\left(\dfrac{\pi\sqrt{2}}{4}\right).

How did you do?