calculus - a copia

8/18/2019 Calculus - A Copia

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Calculus – Maths SL

1. Find the equation of the normal to the curve with equation

y = x3 + 1

at the point (1, 2).

Working:

Answer :

.........................................................................

(Total 4 marks)

2. The graph represents the function

f : x ! p cos x, p ∈ .

3

–3

π

2 x

y


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Find

(a) the value of p;

(b) the area of the shaded region.

Working:

Answers:

(a) ..................................................................(b) ..................................................................

(Total 4 marks)


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3. Differentiate with respect to x

(a) x43−

(b) esin x

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

4. The function f is given by

f ( x) =3

12

−

+

x

x , x ∈ , x ≠ 3.

(a) (i) Show that y = 2 is an asymptote of the graph of y = f ( x).(2)

(ii) Find the vertical asymptote of the graph.(1)

(iii) Write down the coordinates of the point P at which the asymptotes intersect.(1)

(b) Find the points of intersection of the graph and the axes. (4)

(c) Hence sketch the graph of y = f ( x), showing the asymptotes by dotted lines.(4)

(d) Show that f ′ ( x) =2

)3(

7

−

−

x and hence find the equation of the tangent at


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the point S where x = 4.(6)

(e) The tangent at the point T on the graph is parallel to the tangent at S .

Find the coordinates of T .

(5)

(f) Show that P is the midpoint of [ST ].(l)

(Total 24 marks)

5. The function f is such that f ″ ( x) = 2 x – 2.

When the graph of f is drawn, it has a minimum point at (3, –7).

(a) Show that f ′ ( x) = x2 – 2 x – 3 and hence find f ( x).(6)

(b) Find f (0), f (–1) and f ′ (–1).(3)

(c) Hence sketch the graph of f , labelling it with the information obtained in part (b).(4)

(Note: It is not necessary to find the coordinates of the points where the graphcuts the x-axis.)

(Total 13 marks)

6. The diagram shows part of the graph of y = .2e x

x

y

P

ln2

y = e x2

(a) Find the coordinates of the point P, where the graph meets the y-axis.(2)

The shaded region between the graph and the x-axis, bounded by x = 0


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and x = ln 2, is rotated through 360° about the x-axis.

(b) Write down an integral which represents the volume of the solid obtained.(4)

(c) Show that this volume is π.(5)

(Total 11 marks)

7. The parabola shown has equation y2 = 9 x.

x

yP

Q

M

y x = 92

(a) Verify that the point P (4, 6) is on the parabola.(2)

The line (PQ) is the normal to the parabola at the point P, and cuts the x-axis at Q.

(b) (i) Find the equation of (PQ) in the form ax + by + c = 0.(5)

(ii) Find the coordinates of Q.(2)


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S is the point .0,49 !

" #$

% &

(c) Verify that SP = SQ.(4)

(d) The line (PM ) is parallel to the x-axis. From part (c), explain why

(QP) bisects the angle M.P̂S (3)

(Total 16 marks)

8. The diagram shows part of the graph of y = 12 x2(1 – x).

x

y

0

(a) Write down an integral which represents the area of the shaded region.


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(b) Find the area of the shaded region.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)


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9. Differentiate with respect to x:

(a) ( x2 + l)2.

(b) 1n(3 x – 1).

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

10. The diagram shows part of the graph of y = x1 . The area of the shaded region is 2 units.

0

y

x1 a


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Find the exact value of a.

Working:

Answer:

......................................................................

(Total 4 marks)

11. The diagram shows the graph of the function f given by

f ( x) = A sin ! " #$

% & x

2π + B,

for 0 ≤ x ≤ 5, where A and B are constants, and x is measured in radians.

0 1 2 3 4 5

2

y

x

(0, 1)

(1,3)

(3, –1)

(5, 3)

The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph.

(a) Write down the values of f (1) and f (5).(2)

(b) Show that the period of f is 4.(2)


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The point (3, –1) is a minimum point of the graph.

(c) Show that A = 2, and find the value of B.(5)

(d) Show that f ′ ( x) = π cos ! " #$

% & x

2π .

(4)

The line y = k – π x is a tangent line to the graph for 0 ≤ x ≤ 5.

(e) Find

(i) the point where this tangent meets the curve;

(ii) the value of k .

(6)

(f) Solve the equation f ( x) = 2 for 0 ≤ x ≤ 5.(5)

(Total 24 marks)


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12. (a) Find the equation of the tangent line to the curve y = ln x at the point (e, 1), and verify

that the origin is on this line.(4)

(b) Show that xd

d ( x ln x – x) = ln x.

(2)

(c) The diagram shows the region enclosed by the curve y = ln x, the tangent line in part (a),

and the line y = 0.

y

x

1

0 1 2 3

(e, 1)

Use the result of part (b) to show that the area of this region is21 e – 1.

(4)

(Total 10 marks)

13. A curve has equation y = x( x – 4)2.

(a) For this curve find

(i) the x-intercepts;

(ii) the coordinates of the maximum point;

(iii) the coordinates of the point of inflexion.(9)

(b) Use your answers to part (a) to sketch a graph of the curve for 0 ≤ x ≤ 4, clearly indicating

the features you have found in part (a).

(3)


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(c) (i) On your sketch indicate by shading the region whose area is given by the following

integral:

' −4

0

2 .d)4( x x x

(ii) Explain, using your answer to part (a), why the value of this integral is greater than0 but less than 40.

(3)

(Total 15 marks)

14. Find the coordinates of the point on the graph of y = x2 – x at which the tangent is parallel to the

line y = 5 x.

Working:

Answer:

......................................................................

(Total 4 marks)


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15. If f ′( x) = cos x, and f ! " #$

% &

2π = – 2, find f ( x).

Working:

Answer:

......................................................................

(Total 4 marks)


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16. Let f ( x) = x3.

(a) Evaluateh

f h f )5()5( −+ for h = 0.1.

(b) What number doesh

f h f )5()5( −+ approach as h approaches zero?

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

17. The main runway at Concordville airport is 2 km long. An airplane, landing at Concordville,touches down at point T, and immediately starts to slow down. The point A is at the southern

end of the runway. A marker is located at point P on the runway.

T

PA B

2 km

Not to scale


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As the airplane slows down, its distance, s, from A, is given by

s = c + 100t – 4t 2,

where t is the time in seconds after touchdown, and c metres is the distance of T from A.

(a) The airplane touches down 800 m from A, (ie c = 800).

(i) Find the distance travelled by the airplane in the first 5 seconds after touchdown.(2)

(ii) Write down an expression for the velocity of the airplane at time t seconds aftertouchdown, and hence find the velocity after 5 seconds.

(3)

The airplane passes the marker at P with a velocity of 36 m s–1

. Find

(iii) how many seconds after touchdown it passes the marker;(2)

(iv) the distance from P to A.(3)

(b) Show that if the airplane touches down before reaching the point P, it can stop beforereaching the northern end, B, of the runway.

(5)

(Total 15 marks)

18. (a) Sketch the graph of y = π sin x – x, –3 ≤ x ≤ 3, on millimetre square paper, using a scaleof 2 cm per unit on each axis.

Label and number both axes and indicate clearly the approximate positions of the

x-intercepts and the local maximum and minimum points.(5)

(b) Find the solution of the equation

π sin x – x = 0, x > 0.(1)


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There is an inflexion point at P (a, b).

(c) Use the results from parts (a) and (b) to prove that:

(i) tan a = ;4

3

(3)

(ii) the gradient of the curve at P is e2a

.(5)

(Total 14 marks)

20. A curve with equation y =f ( x) passes through the point (1, 1). Its gradient function is

f ′ ( x) = –2 x + 3.

Find the equation of the curve.

Working:

Answer:

......................................................................

(Total 4 marks)


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21. Given that f ( x) = (2 x + 5)3 find

(a) f ′ ( x);

(b) .d)(

' x x f

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)


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22. The diagram shows the graph of the function y = 1 + x1 , 0


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(b) Sketch a graph of h against t for 0 ≤ t ≤ 5.(4)

(c) Findt

h

d

d for:

(i) 0 ≤ t ≤ 2

(ii) 2 ≤ t ≤ 5(2)

(d) Find the velocity of the rock-climber when t = 2.(2)

(e) Find the times when the velocity of the rock-climber is zero.(3)

(f) Find the minimum height of the rock-climber for 0 ≤ t ≤ 5.(3)

(Total 15 marks)


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24. In this question you should note that radians are used throughout.

(a) (i) Sketch the graph of y = x2 cos x, for 0 ≤ x ≤ 2 making clear the approximate

positions of the positive x-intercept, the maximum point and the end-points.

(ii) Write down the approximate coordinates of the positive x-intercept, the maximumpoint and the end-points.

(7)

(b) Find the exact value of the positive x-intercept for 0 ≤ x ≤ 2. (2)

Let R be the region in the first quadrant enclosed by the graph and the x-axis.

(c) (i) Shade R on your diagram.

(ii) Write down an integral which represents the area of R. (3)

(d) Evaluate the integral in part (c)(ii), either by using a graphic display calculator, or byusing the following information.

xdd

( x2 sin x + 2 x cos x – 2 sin x) = x

2 cos x.

(3)

(Total 15 marks)


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25. In this part of the question, radians are used throughout.

The function f is given by

f ( x) = (sin x)2 cos x.

The following diagram shows part of the graph of y = f ( x).

A

C

B

y

x

O

The point A is a maximum point, the point B lies on the x-axis, and the point C is a point of

inflexion.

(a) Give the period of f .(1)

(b) From consideration of the graph of y = f ( x), find to an accuracy of one significant

figure the range of f .(1)


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27. Given the function f ( x) = x2 – 3bx + (c + 2), determine the values of b and c such that f (1) = 0

and f ′ (3) = 0.

Working:

Answer:

....................................................................

(Total 4 marks)

(a) Give the period of f .(1)

(b) From consideration of the graph of y = f ( x), find to an accuracy of one significantfigure the range of f .

(1)

(c) (i) Find f ′ ( x).

(ii) Hence show that at the point A, cos x =31 28. Note: Radians are

used throughout this question.

(a) Draw the graph of y = π + x cos x, 0 ≤ x ≤ 5, on millimetre square graph paper, using ascale of 2 cm per unit. Make clear

(i) the integer values of x and y on each axis;

(ii) the approximate positions of the x-intercepts and the turning points.(5)


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(b) Without the use of a calculator, show that π is a solution of the equation

π + x cos x = 0.(3)

(c) Find another solution of the equation π + x cos x = 0 for 0 ≤ x ≤ 5, giving your answer to

six significant figures.(2)

(d) Let R be the region enclosed by the graph and the axes for 0 ≤ x ≤ π. Shade R on yourdiagram, and write down an integral which represents the area of R .

(2)

(e) Evaluate the integral in part (d) to an accuracy of six significant figures. (If you consider

it necessary, you can make use of the result .)cos)cossin(d

d x x x x x

x=+

(3)

(Total 15 marks)

29. A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the

ground after t seconds is given by

h = 2 + 20t – 5t 2, t ≥ 0

(a) Find the initial height above the ground of the ball (that is, its height at the instant when

it is released).(2)

(b) Show that the height of the ball after one second is 17 metres.(2)

(c) At a later time the ball is again at a height of 17 metres.

(i) Write down an equation that t must satisfy when the ball is at a height of 17 metres.

(ii) Solve the equation algebraically.(4)


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(d) (i) Either by inspection or by using an appropriate substitution, find

' dx x f )(

(ii) Hence find the exact area of the region enclosed by the graph of f , the x-axis andthe y-axis.

(8)

(Total 18 marks)

31. The point P ( 0,2

1) lies on the graph of the curve of y = sin (2 x –1).

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

Working:

Answer:

.......................................................................

(Total 4 marks)


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32. Find

(a) ' + ;d)73(sin x x

(b) ' xe xd4– .

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

33. The function f is given by .0,2n1

)( >= x x

x x f

(a) (i) Show that .2n1–1

)(2

x

x x f =′

Hence

(ii) prove that the graph of f can have only one local maximum or minimum point;

(iii) find the coordinates of the maximum point on the graph of f .(6)


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(b) By showing that the second derivative3

3–2n12)(

x

x x f =′′ or otherwise,

find the coordinates of the point of inflexion on the graph of f .(6)

(c) The region S is enclosed by the graph of f , the x-axis, and the vertical linethrough the maximum point of f , as shown in the diagram below.

y

x0

y f x= ( )

h

(i) Would the trapezium rule overestimate or underestimate the area of S ? Justify youranswer by drawing a diagram or otherwise.

(3)

(ii) Find ' x x f d)( , by using the substitution u = ln 2 x, or otherwise.(4)

(iii) Using ' x x f d)( , find the area of S . (4)

(d) The Newton–Raphson method is to be used to solve the equation f ( x) = 0.

(i) Show that it is not possible to find a solution using a starting value of

x1 = 1.

(3)

(ii) Starting with x1 = 0.4, calculate successive approximations x2 , x3, ...

for the root of the equation until the absolute error is less than 0.01.

Give all answers correct to five decimal places.(4)

(Total 30 marks)


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34. Consider the function f ( x) = k sin x + 3 x, where k is a constant.

(a) Find f ′( x).

(b) When x = 3

π

, the gradient of the curve of f ( x) is 8. Find the value of k.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)


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35. The diagram below shows part of the graph of the function

.152–:23

x x x x f ++!

40

–20–15

–10

–5

5

10

15

20

25

30

35

–3 –2 –1 1 2 3 4 5

y

xA

P

B

Q

The graph intercepts the x-axis at A(–3, 0), B(5, 0) and the origin, O. There is a minimum point

at P and a maximum point at Q.

(a) The function may also be written in the form ),–()–(–: b xa x x x f !

where a < b. Write down the value of

(i) a;

(ii) b.(2)

(b) Find

(i) f ′( x);

(ii) the exact values of x at which f '( x) = 0;

(iii) the value of the function at Q.(7)


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(c) (i) Find the equation of the tangent to the graph of f at O.

(ii) This tangent cuts the graph of f at another point. Give the x-coordinate of this point.(4)

(d) Determine the area of the shaded region.(2)

(Total 15 marks)

36. A ball is dropped vertically from a great height. Its velocity v is given by

v = 50 – 50e–0.2t , t ≥ 0

where v is in metres per second and t is in seconds.

(a) Find the value of v when

(i) t = 0;

(ii) t = 10.(2)

(b) (i) Find an expression for the acceleration, a, as a function of t .

(ii) What is the value of a when t = 0?(3)

(c) (i) As t becomes large, what value does v approach?

(ii) As t becomes large, what value does a approach?

(iii) Explain the relationship between the answers to parts (i) and (ii).(3)


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(d) Let y metres be the distance fallen after t seconds.

(i) Show that y = 50t + 250e–0.2t + k , where k is a constant.

(ii) Given that y = 0 when t = 0, find the value of k .

(iii) Find the time required to fall 250 m, giving your answer correct to four significant

figures.(7)

(Total 15 marks)

37. Radian measure is used, where appropriate, throughout the question.

Consider the function .5–2

2–3

x

x y =

The graph of this function has a vertical and a horizontal asymptote.

(a) Write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.(2)

(b) Find y

x

d

d, simplifying the answer as much as possible.

(3)

(c) How many points of inflexion does the graph of this function have?(1)

(Total 6 marks)


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38. The derivative of the function f is given by f ′( x) = x+1

1 – 0.5 sin x, for x ≠ –1.

The graph of f passes through the point (0, 2). Find an expression for f ( x).

Working:

Answer :

......................................................................

(Total 6 marks)


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39. Figure 1 shows the graphs of the functions f 1, f 2, f 3, f 4.

Figure 2 includes the graphs of the derivatives of the functions shown in Figure 1, eg thederivative of f 1 is shown in diagram (d).

Figure 1 Figure 2

f (a)

f (b)

f (c)

f (d)

(e)

1

2

3

4

y y

y y

y y

y y

y

x

x

x x

x x

x x

x

O

O

OO

OO

O

O

O


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Complete the table below by matching each function with its derivative.

Function Derivative diagram

f 1 (d)

f 2

f 3

f 4

Working:

(Total 6 marks)

40. Consider functions of the form y = e –kx

(a) Show that '1

0

– de xkx =k

1 (1 – e

–k ).

(3)

(b) Let k = 0.5

(i) Sketch the graph of y = e–0.5 x, for –1 ≤ x ≤ 3, indicating the coordinates of the

y-intercept.

(ii) Shade the region enclosed by this graph, the x-axis, y-axis and the line x = 1.

(iii) Find the area of this region.(5)


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(c) (i) Find x

y

d

d in terms of k , where y = e

–kx.

The point P(1, 0.8) lies on the graph of the function y = e –kx

.

(ii) Find the value of k in this case.

(iii) Find the gradient of the tangent to the curve at P.(5)

(Total 13 marks)

41. Let the function f be defined by f ( x) =3

1

2

x+, x ≠ –1.

(a) (i) Write down the equation of the vertical asymptote of the graph of f .

(ii) Write down the equation of the horizontal asymptote of the graph of f .

(iii) Sketch the graph of f in the domain –3 ≤ x ≤ 3.(4)

(b) (i) Using the fact that f ′( x) =23

2

)1(

6–

x

x

+, show that the second derivative

f ″ ( x) =( )

33

3

)1(

1–212

x

x x

+.

(ii) Find the x-coordinates of the points of inflexion of the graph of f .(6)


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(c) The table below gives some values of f ( x) and 2 f ( x).

x f ( x) 2 f ( x)

1 1 21.4 0.534188 1.068376

1.8 0.292740 0.585480

2.2 0.171703 0.343407

2.6 0.107666 0.215332

3 0.071429 0.142857

(i) Use the trapezium rule with five sub-intervals to approximate the integral

( ) .d3

1' x x f

(ii) Given that ( )'3

1d x x f = 0.637599, use a diagram to explain why your answer is

greater than this.(5)

(Total 15 marks)


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42. Let f ( x) = .3 x Find

(a) f ′ ( x);

(b) ' .d)( x x f Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)

43. The graph of y = x3 – 10 x

2 +12 x + 23 has a maximum point between x = –1 and x = 3. Find the

coordinates of this maximum point.

Working:

Answer :

......................................................................

(Total 6 marks)

44. The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and


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the lines y = 0 and x = .4

!3

x

y

43π π

Given that sin4

!3 =

2

2 and cos

4

!3 = –

2

2, calculate the exact area of the shaded

region.

Working:

Answer :

......................................................................

(Total 6 marks)


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45. The diagram below shows a sketch of the graph of the function y = sin (e x

) where –1 ≤ x ≤ 2,

and x is in radians. The graph cuts the y-axis at A, and the x-axis at C and D. It has a maximumpoint at B.

–1 0 1 2

AB

C D

y

x

(a) Find the coordinates of A.(2)

(b) The coordinates of C may be written as (ln k , 0). Find the exact value of k .(2)

(c) (i) Write down the y-coordinate of B.

(ii) Find x

y

d

d

.

(iii) Hence, show that at B, x = ln2

!.

(6)

(d) (i) Write down the integral which represents the shaded area.

(ii) Evaluate this integral.(5)


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(e) (i) Copy the above diagram into your answer booklet. (There is no need to copy the

shading.) On your diagram, sketch the graph of y = x3.

(ii) The two graphs intersect at the point P. Find the x-coordinate of P.(3)

(Total 18 marks)

46. In this question, s represents displacement in metres, and t represents time in seconds.

(a) The velocity v m s–1

of a moving body may be written as v =t

s

d

d = 30 – at , where a is a

constant. Given that s = 0 when t = 0, find an expression for s in terms of a and t. (5)

Trains approaching a station start to slow down when they pass a signal which is 200 m fromthe station.

(b) The velocity of Train 1 t seconds after passing the signal is given by v = 30 – 5t.

(i) Write down its velocity as it passes the signal.

(ii) Show that it will stop before reaching the station.(5)

(c) Train 2 slows down so that it stops at the station. Its velocity is given by

v =t

s

d

d = 30 – at , where a is a constant.

(i) Find, in terms of a, the time taken to stop.

(ii) Use your solutions to parts (a) and (c)(i) to find the value of a.(5)

(Total 15 marks)


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47. The diagram below shows the shaded region R enclosed by the graph of y = 2 x 21 x+ , the

x-axis, and the vertical line x = k .

R

x

y

k

y x x= 2 1+2

(a) Find x

y

d

d.

(3)

(b) Using the substitution u = 1 + x2 or otherwise, show that

' +2

12 x x d x =3

2(1 + x

2) 2

3

+ c.

(3)

(c) Given that the area of R equals 1, find the value of k .(3)

(Total 9 marks)

48. Consider the function h ( x) = x 51

.

(i) Find the equation of the tangent to the graph of h at the point where x = a, (a ≠ 0). Writethe equation in the form y = mx + c.

(ii) Show that this tangent intersects the x-axis at the point (–4a, 0).(Total 5 marks)


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49. Let f ( x) = 3e

x

+ 5 cos2 x. Find f ′ ( x).

Working:

Answer :

..................................................................

(Total 6 marks)


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50. Given that '3

1d)( x xg = 10, deduce the value of

(a) '3

1;d)(

2

1 x xg

(b) ' +3

1.d)4)(( x xg

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)

51. An aircraft lands on a runway. Its velocity v m s–1

at time t seconds after landing is given by the

equation v = 50 + 50e–0.5t , where 0 ≤ t ≤ 4.

(a) Find the velocity of the aircraft

(i) when it lands;

(ii) when t = 4.(4)

(b) Write down an integral which represents the distance travelled in the first four seconds.(3)


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(c) Calculate the distance travelled in the first four seconds.(2)

After four seconds, the aircraft slows down (decelerates) at a constant rate and comes to restwhen t = 11.

(d) Sketch a graph of velocity against time for 0 ≤ t ≤ 11. Clearly label the axes and mark on

the graph the point where t = 4.(5)

(e) Find the constant rate at which the aircraft is slowing down (decelerating) between t = 4

and t = 11.(2)

(f) Calculate the distance travelled by the aircraft between t = 4 and t = 11. (2)(Total 18 marks)

52. Consider the function f ( x) = cos x + sin x.

(a) (i) Show that f (–4

!) = 0.

(ii) Find in terms of π, the smallest positive value of x which satisfies f ( x) = 0. (3)


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The diagram shows the graph of y = e x

(cos x + sin x), – 2 ≤ x ≤ 3. The graph has a maximum

turning point at C(a, b) and a point of inflexion at D.

6

4

2

1 2 3–2 –1

D

C(a, b)

y

x

(b) Find

x

y

d

d.

(3)

(c) Find the exact value of a and of b.(4)

(d) Show that at D, y = 4!

e2 .(5)

(e) Find the area of the shaded region.(2)

(Total 17 marks)


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53. It is given that x

y

d

d= x

3+2 x – 1 and that y = 13 when x = 2.

Find y in terms of x.

Working:

Answer :

..................................................................

(Total 6 marks)


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54. (a) Find '(1 + 3 sin ( x + 2))d x.

(b) The diagram shows part of the graph of the function f ( x) = 1 + 3 sin ( x + 2).

The area of the shaded region is given by 'a

x x f 0

d)( .

y

4

4

2

20

–2

–2–4 x

Find the value of a.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)


51/125


55. The diagram shows the graph of y = f ( x).

y

x0


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On the grid below sketch the graph of y = f ′ ( x).

y

x0

(Total 6 marks)

56. Consider the function f ( x) = 1 + e–2 x

.

(a) (i) Find f ′( x).

(ii) Explain briefly how this shows that f ( x) is a decreasing function for all values of x

(ie that f ( x) always decreases in value as x increases).(2)

Let P be the point on the graph of f where x = –2

1.

(b) Find an expression in terms of e for

(i) the y-coordinate of P;

(ii) the gradient of the tangent to the curve at P.(2)


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(c) Find the equation of the tangent to the curve at P, giving your answer in the form

y = ax + b.(3)

(d) (i) Sketch the curve of f for –1 ≤ x ≤ 2.

(ii) Draw the tangent at x = –21 .

(iii) Shade the area enclosed by the curve, the tangent and the y-axis.

(iv) Find this area.(7)

(Total 14 marks)

57. Note: Radians are used throughout this question.

A mass is suspended from the ceiling on a spring. It is pulled down to point P and then released.

It oscillates up and down.

P

diagram not toscale

Its distance, s cm, from the ceiling, is modelled by the function s = 48 + 10 cos 2! t where t is

the time in seconds from release.

(a) (i) What is the distance of the point P from the ceiling?

(ii) How long is it until the mass is next at P?(5)


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(b) (i) Findt

s

d

d.

(ii) Where is the mass when the velocity is zero?(7)

A second mass is suspended on another spring. Its distance r cm from the ceiling is modelled by

the function r = 60 + 15 cos 4πt . The two masses are released at the same instant.

(c) Find the value of t when they are first at the same distance below the ceiling.(2)

(d) In the first three seconds, how many times are the two masses at the same height?(2)

(Total 16 marks)

58. Consider the function f given by f ( x) =2

2

)1–(

2013–2

x

x x +, x ≠ 1.

A part of the graph of f is given below.

x

y

0

The graph has a vertical asymptote and a horizontal asymptote, as shown.

(a) Write down the equation of the vertical asymptote.(1)


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(b) f (100) = 1.91 f (–100) = 2.09 f (1000) = 1.99

(i) Evaluate f (–1000).

(ii) Write down the equation of the horizontal asymptote.(2)

(c) Show that f ′( x) =( )31–

27–9

x

x, x ≠ 1.

(3)

The second derivative is given by f ″ ( x) =4

)1–(

18–72

x

x, x ≠ 1.

(d) Using values of f ′( x) and f ″ ( x) explain why a minimum must occur at x = 3.(2)

(e) There is a point of inflexion on the graph of f . Write down the coordinates of this point.(2)

(Total 10 marks)


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59. Let f ( x) = x3 – 2 x

2 – 1.

(a) Find f ′ ( x).

(b) Find the gradient of the curve of f ( x) at the point (2, –1).

Working:

Answers:

(a) …………………………………………

(b) …………………………………………(Total 6 marks)

60. A car starts by moving from a fixed point A. Its velocity, v m s–1 after t seconds is given by

v = 4t + 5 – 5e–t

. Let d be the displacement from A when t = 4.

(a) Write down an integral which represents d .

(b) Calculate the value of d .

Working:

Answers:

(a) …………………………………………..

(b) …………………………………………..(Total 6 marks)


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61. (a) Consider the function f ( x) = 2 +1

1

− x. The diagram below is a sketch of part of the

graph of y = f ( x).

5

4

3

2

1

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 21 3 4 50 x

y

Copy and complete the sketch of f ( x).(2)

(b) (i) Write down the x-intercepts and y-intercepts of f ( x).

(ii) Write down the equations of the asymptotes of f ( x).(4)

(c) (i) Find f ′ ( x).

(ii) There are no maximum or minimum points on the graph of f ( x).

Use your expression for f ′ ( x) to explain why.

(3)

The region enclosed by the graph of f ( x), the x-axis and the lines x = 2 and x = 4, is labelled A,

as shown below.

5

4

3

2

1

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 21 3 4 50 x

y

A


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(d) (i) Find ' )( x f d x.

(ii) Write down an expression that represents the area labelled A.

(iii) Find the area of A.(7)

(Total 16 marks)

62. Let f ( x) = 63 2

x . Find f ′ ( x).

Working:

Answer :

…………………………………………........(Total 6 marks)


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63. The displacement s metres of a car, t seconds after leaving a fixed point A, is given by

s = 10t – 0.5t 2.

(a) Calculate the velocity when t = 0.

(b) Calculate the value of t when the velocity is zero.

(c) Calculate the displacement of the car from A when the velocity is zero.

Working:

Answers:

(a) …………………………………………..

(b) …………………………………………..

(c) …………………………………………..(Total 6 marks)


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66. Let f be a function such that 8d)(3

0=' x x f .

(a) Deduce the value of

(i) ;d)(23

0 x x f '

(ii) ( ) .d2)(3

0 x x f ' +

(b) If 8d)2( =−' x x f d

c, write down the value of c and of d .

Working:

Answers:

(a) (i) ........................................................

(ii) .......................................................

(b) c = ......................., d = .......................

(Total 6 marks)


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67. Part of the graph of the periodic function f is shown below. The domain of f is 0 ≤ x ≤ 15 andthe period is 3.

4

3

2

1

00 1 2 3 4 5 6 7

x

f x( )

8 9 10

(a) Find

(i) f (2);

(ii) f ′ (6.5);

(iii) f ′ (14).

(b) How many solutions are there to the equation f ( x) = 1 over the given domain?

Working:

Answers:

(a) (i) ………………………………………

(ii) ………………………………………

(iii) ………………………………………

(b) ……………………………………………

(Total 6 marks)


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68. The function f ( x) is defined as f ( x) = –( x – h)2 + k . The diagram below shows part of the graph

of f ( x). The maximum point on the curve is P (3, 2).

4

2

–2

–4

–6

–8

–10

–12

–1 1 2 3 4 5 6 x

y

P(3, 2)

(a) Write down the value of

(i) h;

(ii) k .(2)

(b) Show that f ( x) can be written as f ( x) = – x2 + 6 x – 7.(1)

(c) Find f ′ ( x).(2)

The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, isperpendicular to the tangent at Q.

(d) (i) Calculate the gradient of L.

(ii) Find the equation of L.

(iii) The line L intersects the curve again at R. Find the x-coordinate of R.(8)

(Total 13 marks)


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69. Let f ( x) = 1 + 3 cos (2 x) for 0 ≤ x ≤ ! , and x is in radians.

(a) (i) Find f ′ ( x).

(ii) Find the values for x for which f ′ ( x) = 0, giving your answers in terms of π.(6)

The function g ( x) is defined as g ( x) = f (2 x) – 1, 0 ≤ x ≤ 2

! .

(b) (i) The graph of f may be transformed to the graph of g by a stretch in the x-direction

with scale factor21 followed by another transformation. Describe fully this other

transformation.

(ii) Find the solution to the equation g ( x) = f ( x)(4)

(Total 10 marks)

70. Let h ( x) = ( x – 2) sin ( x – 1) for –5 ≤ x ≤ 5. The curve of h ( x) is shown below. There is a

minimum point at R and a maximum point at S. The curve intersects the x-axis at the points(a, 0) (1, 0) (2, 0) and (b, 0).

–5 –4 –3 –2 –1 1 2 3 4 5 x

y

S

( , 0)b( , 0)a

R

4

3

2

1

–1

–2

–3

–4

–5

–6

–7


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(a) Find the exact value of

(i) a;

(ii) b.

(2)

The regions between the curve and the x-axis are shaded for a ≤ x ≤ 2 as shown.

(b) (i) Write down an expression which represents the total area of the shaded regions.

(ii) Calculate this total area.(5)

(c) (i) The y-coordinate of R is –0.240. Find the y-coordinate of S.

(ii) Hence or otherwise, find the range of values of k for which the equation( x – 2) sin ( x – 1) = k has four distinct solutions.

(4)

(Total 11 marks)

71. Let f ( x) =2

1

1

x+.

(a) Write down the equation of the horizontal asymptote of the graph of f .(1)

(b) Find f ′ ( x).(3)

(c) The second derivative is given by f ″ ( x) =32

2

)(1

26

x

x

+

−.

Let A be the point on the curve of f where the gradient of the tangent is a maximum. Findthe x-coordinate of A.

(4)


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(d) Let R be the region under the graph of f , between x = –2

1 and x =

2

1,

as shaded in the diagram below

R

1

2

–1

–1

112

12

– x

y

Write down the definite integral which represents the area of R.(2)

(Total 10 marks)

72. Let y = g ( x) be a function of x for 1 ≤ x ≤ 7. The graph of g has an inflexion point at P, and a

minimum point at M.

Partial sketches of the curves of g′ and g″ are shown below.

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

y g x= ( )’ y g x= ( )’’

g x’( ) g x’ ( )’

1 12 23 34 45 56 67 78 8 x x


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Use the above information to answer the following.

(a) Write down the x-coordinate of P, and justify your answer.(2)

(b) Write down the x-coordinate of M, and justify your answer.

(2)

(c) Given that g (4) = 0, sketch the graph of g. On the sketch, mark the points P and M.(4)

(Total 8 marks)

73. The function f is given by f ( x) = 2sin (5 x – 3).

(a) Find f " ( x).

(b) Write down ' x x f d)( .

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.....................................................................................................................................(Total 6 marks)

74. The velocity v m s–1 of a moving body at time t seconds is given by v = 50 – 10t .


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(a) Find its acceleration in m s–2.

(b) The initial displacement s is 40 metres. Find an expression for s in terms of t .

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.....................................................................................................................................(Total 6 marks)

75. The function f is defined by .5.225.0–: 2 ++ x x x f !

(a) Write down

(i) f ′( x);

(ii) f ′(0).(2)

(b) Let N be the normal to the curve at the point where the graph intercepts the y-axis. Show

that the equation of N may be written as y = –0.5 x + 2.5.(3)


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Let 5.25.0–: + x xg !

(c) (i) Find the solutions of f ( x) = g ( x).

(ii) Hence find the coordinates of the other point of intersection of the normal and the

curve.(6)

(d) Let R be the region enclosed between the curve and N .

(i) Write down an expression for the area of R.

(ii) Hence write down the area of R.(5)

(Total 16 marks)

76. The diagram below shows the graphs of f ( x) = 1 + e2 x, g ( x) = 10 x + 2, 0 ≤ x ≤ 1.5.

f

g

p

x

y

16

12

8

4

0.5 1 1.5

(a) (i) Write down an expression for the vertical distance p between the graphs of f and g.

(ii) Given that p has a maximum value for 0≤

x ≤

1.5, find the value of x at which thisoccurs.(6)


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The graph of y = f ( x) only is shown in the diagram below. When x = a, y = 5.

x

y

16

12

8

4

0.5 1 1.5

5

a

(b) (i) Find f–1( x).

(ii) Hence show that a = ln 2.(5)

(c) The region shaded in the diagram is rotated through 360° about the x-axis. Write down an

expression for the volume obtained.(3)

(Total 14 marks)


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77. Consider the function .1,)1–(

2–:

2 ≠ x

x

x xh !

A sketch of part of the graph of h is given below.

x

yA

B

P

Not to scale

The line (AB) is a vertical asymptote. The point P is a point of inflexion.

(a) Write down the equation of the vertical asymptote.(1)

(b) Find h"( x), writing your answer in the form

n x

xa

)1–(

–

where a and n are constants to be determined.(4)


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79. The curve y = f ( x) passes through the point (2, 6).

Given that x

y

d

d = 3 x2 – 5, find y in terms of x.

Working:

Answer :

....……………………………………..........

(Total 6 marks)

80. The table below shows some values of two functions, f and g, and of their derivatives f ′ and g ′.

x 1 2 3 4

f ( x) 5 4 –1 3

g ( x) 1 –2 2 –5

f ′( x) 5 6 0 7

g′ ( x) –6 –4 –3 4


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Calculate the following.

(a) xdd

( f ( x) + g ( x)), when x = 4;

(b) ( )' +3

1d6)( x xg' .

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)

81. The equation of a curve may be written in the form y = a( x – p)( x – q). The curve intersects the

x-axis at A(–2, 0) and B(4, 0). The curve of y = f ( x) is shown in the diagram below.

4

2

–2

–4

–6

–4 –2 0 2 4 6 x

y

A B

(a) (i) Write down the value of p and of q.


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(ii) Given that the point (6, 8) is on the curve, find the value of a.

(iii) Write the equation of the curve in the form y = ax2 + bx + c.(5)

(b) (i) Find x

y

d

d

.

(ii) A tangent is drawn to the curve at a point P. The gradient of this tangent is 7.

Find the coordinates of P.(4)

(c) The line L passes through B(4, 0), and is perpendicular to the tangent to the curve at

point B.

(i) Find the equation of L.

(ii) Find the x-coordinate of the point where L intersects the curve again.(6)

(Total 15 marks)

82. Let f ( x) =15

3 2

− x

x .

(a) Write down the equation of the vertical asymptote of y = f ( x).(1)

(b) Find f " ( x). Give your answer in the form2

2

)15( −

+

x

bxax where a and b ∈ .

(4)

(Total 5 marks)


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83. The function g ( x) is defined for –3 ≤ x ≤ 3. The behaviour of g′( x) and g″ ( x) is given in thetables below.

x –3


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84. Given ' −k

x3 2

1d x = ln 7, find the value of k .

Working:

Answers:

........................................................

(Total 6 marks)


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85. Let f ( x) = (2 x + 7)3 and g ( x) cos

2 (4 x). Find

(a) f " ( x);

(b) g" ( x).

Working:

Answers:

(a) .................................................

(b) ................................................. (Total 6 marks)


79/125


86. The following diagram shows a rectangular area ABCD enclosed on three sides by 60 m offencing, and on the fourth by a wall AB.

Find the width of the rectangle that gives its maximum area.

Working:

Answers:

........................................................

(Total 6 marks)

87. A particle moves with a velocity v m s−1

given by v = 25 − 4t 2 where t ≥ 0.

(a) The displacement, s metres, is 10 when t is 3. Find an expression for s in terms of t .(6)

(b) Find t when s reaches its maximum value.(3)

(c) The particle has a positive displacement for m ≤ t ≤ n. Find the value of m and the value

of n.(3)

(Total 12 marks)


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88. The graph of y = sin 2 x from 0≤ x ≤ π is shown below.

The area of the shaded region is 0.85. Find the value of k .(Total 6 marks)


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89. (a) Let f ( x) = e5 x

. Write down f " ( x).

(b) Let g ( x) = sin 2 x. Write down g" ( x).

(c) Let h ( x) = e5 x

sin 2 x. Find h" ( x).

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(Total 6 marks)


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90. The following diagram shows part of the curve of a function ƒ. The points A, B, C, D and E lieon the curve, where B is a minimum point and D is a maximum point.

(a) Complete the following table, noting whether ƒ"( x) is positive, negative or zero at the

given points.

A B E

f " ( x)

(b) Complete the following table, noting whether ƒ""( x) is positive, negative or zero at thegiven points.

A C E

ƒ"" ( x)

(Total 6 marks)


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91. The velocity, v m s−1

, of a moving object at time t seconds is given by v = 4t 3 − 2t .

When t = 2, the displacement, s, of the object is 8 metres.

Find an expression for s in terms of t .

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92. The graph of a function g is given in the diagram below.

The gradient of the curve has its maximum value at point B and its minimum value at point D.The tangent is horizontal at points C and E.

(a) Complete the table below, by stating whether the first derivative g" is positive or negative,and whether the second derivative g"" is positive or negative.

Interval g" g""

a


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93. A part of the graph of y = 2 x – x2 is given in the diagram below.

The shaded region is revolved through 360° about the x-axis.

(a) Write down an expression for this volume of revolution.

(b) Calculate this volume.

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94. Consider the function ƒ : x ! 3 x2 – 5 x + k .

(a) Write down ƒ" ( x).

The equation of the tangent to the graph of ƒ at x = p is y = 7 x – 9. Find the value of

(b) p;

(c) k .

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95. The diagram below shows the graph of ƒ ( x) = x2 e– x for 0 ≤ x ≤ 6. There are points of inflexion

at A and C and there is a maximum at B.

(a) Using the product rule for differentiation, find ƒ" ( x).

(b) Find the exact value of the y-coordinate of B.

(c) The second derivative of ƒ is ƒ"" ( x) = ( x2 – 4 x + 2) e–x. Use this result to find the exact value of the x-coordinate of C.

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(b) Given that g–1

( x) = x + 2, find (g–1

# f ) ( x).(2)

(c) Given also that ( f −1 # g) ( x)3

3+ x, solve ( f −1 # g) ( x) = (g–1 # f ) ( x).

(2)

Let h ( x) =)(

)(

xg

x f , x ≠ 2.

(d) (i) Sketch the graph of h for −3 ≤ x ≤ 7 and −2 ≤ y ≤ 8, including any asymptotes.

(ii) Write down the equations of the asymptotes.(5)

(e) The expression3

53

−

−

x

x may also be written as 3 +

2

1

− x. Use this to answer the

following.

(i) Find ' )( xh d x.

(ii) Hence, calculate the exact value of '5

3 h ( x)d x.

(5)

(f) On your sketch, shade the region whose area is represented by '

5

3 h ( x)d x.

(1)

(Total 18 marks)


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99. The following diagram shows the graph of a function f .

Consider the following diagrams.

Complete the table below, noting which one of the diagrams above represents the graph of

(a) f "( x);

(b) f ""( x).

Graph Diagram

(a) f " ( x)

(b) f " ( x)

(Total 6 marks)


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100. The velocity v in m s−1

of a moving body at time t seconds is given by v = e2t−1

. When t = 0 5.

the displacement of the body is 10 m. Find the displacement when t =1.

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101. The shaded region in the diagram below is bounded by f ( x) = x , x = a, and the x-axis. The

shaded region is revolved around the x-axis through 360°. The volume of the solid formed is

0.845π.

Find the value of a.

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..............................................................................................................................................(Total 6 marks)

102. The function f is defined as f ( x) = (2 x +1) e− x

, 0 ≤ x ≤ 3. The point P(0, 1) lies on the graph of f

( x), and there is a maximum point at Q.

(a) Sketch the graph of y = f ( x), labelling the points P and Q.(3)


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(b) (i) Show that f " ( x) = (1− 2 x) e− x.

(ii) Find the exact coordinates of Q.(7)

(c) The equation f ( x) = k , where k ∈ , has two solutions. Write down the range of values

of k .(2)

(d) Given that f ″ ( x) = e− x

(−3 + 2 x), show that the curve of f has only one point of inflexion.(2)

(e) Let R be the point on the curve of f with x-coordinate 3. Find the area of the region

enclosed by the curve and the line (PR).(7)

(Total 21 marks)


95/125


103. The velocity v of a particle at time t is given by v = e−2t

+ 12t . The displacement of the particle

at time t is s. Given that s = 2 when t = 0, express s in terms of t .

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96/125


104. The graph of the function y = f ( x), 0 ≤ x ≤ 4, is shown below.

(a) Write down the value of

(i) f " (1);

(ii) f " (3).

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97/125


(b) On the diagram below, draw the graph of y = 3 f (− x).

(c) On the diagram below, draw the graph of y = f (2 x).

(Total 6 marks)


98/125


105. Let f ( x) = x3 − 3 x2 − 24 x +1.

The tangents to the curve of f at the points P and Q are parallel to the x-axis, where P is to the

left of Q.

(a) Calculate the coordinates of P and of Q.

Let N 1 and N 2 be the normals to the curve at P and Q respectively.

(b) Write down the coordinates of the points where

(i) the tangent at P intersects N 2;

(ii) the tangent at Q intersects N 1.

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99/125


106. It is given that '3

1 f ( x)d x = 5.

(a) Write down '3

1 2 f ( x)d x.

(b) Find the value of '3

1 (3 x2 + f ( x))d x.

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100/125


107. Let f " ( x) = 12 x2 − 2.

Given that f (−1) =1, find f ( x).

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101/125


108. The velocity, v, in m s−1

of a particle moving in a straight line is given by v = e3t −2

, where t is

the time in seconds.

(a) Find the acceleration of the particle at t =1.

(b) At what value of t does the particle have a velocity of 22.3 m s−1?

(c) Find the distance travelled in the first second.

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calculus - a copia

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