3. differentiate with respect to x 1. 3 4start.sd34.bc.ca/mkeeley/wp-content/uploads/2015/03/...11...

1

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

π2

x

y

Find

(a) the value of p;

(b) the area of the shaded region.

Working:

Answers:

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

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

(Total 4 marks)

2

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) = 312

−+

xx , 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

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.

3

(5)

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

(Total 24 marks)

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

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

(a) Show that f! (x) = x2 – 2x – 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 graph cuts the x-axis.)

(Total 13 marks)

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

x

y

Pln2

y = ex2

(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 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)

4

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

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)

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.PS

(3) (Total 16 marks)

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

x

y

0

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

5

(b) Find the area of the shaded region.

Working:

Answers:

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

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

(Total 4 marks)

9. Differentiate with respect to x:

(a) (x2 + l)2.

(b) 1n(3x – 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.

6

0

y

x1 a

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 !"#$

%& x2π + 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)

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

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 !"#$

%& x2π .

(4)

The line y = k – πx 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)

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 xdd (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 is 21 e – 1.


8

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)

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

∫ −4

0

2 .d)4( xxx

(ii) Explain, using your answer to part (a), why the value of this integral is greater than 0 but less than 40.


14. Find the coordinates of the point on the graph of y = x2 – x at which the tangent is parallel to the line y = 5x.

Working:

Answer:

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

(Total 4 marks)

9

15. If f %(x) = cos x, and f !"#$

%&2π = – 2, find f (x).

Working:

Answer:

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

(Total 4 marks)

16. Let f (x) = x3.

(a) Evaluate h

fhf )5()5( −+ for h = 0.1.

(b) What number does h

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

Working:

Answers:

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

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

(Total 4 marks)

10

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.

TP

A B

2 km

Not to scale

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

s = c + 100t – 4t2,

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 after touchdown, 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 before reaching the northern end, B, of the runway.


11

18. (a) Sketch the graph of y = π sin x – x, –3 ≤ x ≤ 3, on millimetre square paper, using a scale of 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)

(c) Find the indefinite integral

∫ −π xxx )d sin (

and hence, or otherwise, calculate the area of the region enclosed by the graph, the x-axis and the line x = 1.


19. The diagram shows part of the graph of the curve with equation

y = e2x cos x.

P( )a, b

y

x0

(a) Show that xydd = e2x (2 cos x – sin x).

(2)

12

(b) Find 2

2

ddxy

.

(4)

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

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

(i) tan a = ;43

(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) = –2x + 3. Find the equation of the curve.

Working:

Answer:

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

(Total 4 marks)

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

(a) f! (x);

(b) .d)(∫ xxf

Working:

Answers:

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

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

13

22. The diagram shows the graph of the function y = 1 + x1 , 0 < x ≤ 4. Find the exact value of the

area of the shaded region.

4

3

2

11 13

y = 1+ 1x–

0 1 2 3 4

Working:

Answer:

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

(Total 4 marks)

23. A rock-climber slips off a rock-face and falls vertically. At first he falls freely, but after 2 seconds a safety rope slows him down. The height h metres of the rock-climber after t seconds of the fall is given by:

h = 50 – 5t2, 0 ≤ t ≤ 2

h = 90 – 40t + 5t2, 2 ≤ t ≤ 5

(a) Find the height of the rock-climber when t = 2. (1)

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

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(c) Find thdd 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)

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 maximum point 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 by using the following information.

xdd (x2 sin x + 2x cos x – 2 sin x) = x2 cos x.


15

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

xO

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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(c) (i) Find f! (x).

(ii) Hence show that at the point A, cos x = 31 .

(iii) Find the exact maximum value. (9)

(d) Find the exact value of the x-coordinate at the point B. (1)

(e) (i) Find ∫ )(xf dx.

(ii) Find the area of the shaded region in the diagram. (4)

(f) Given that f" (x) = 9(cos x)3 – 7 cos x, find the x-coordinate at the point C. (4)

(Total 20 marks)

26. Let f! (x) = 1 – x2. Given that f (3) = 0, find f (x).

Working:

Answer:

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

(Total 4 marks)

17

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)

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 a scale 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)

(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 your diagram, and write down an integral which represents the area of R .

(2)

18

(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(dd

xxxxxx

=+


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 – 5t2, 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)

(d) (i) Find thdd

.

(ii) Find the initial velocity of the ball (that is, its velocity at the instant when it is released).

(iii) Find when the ball reaches its maximum height.

(iv) Find the maximum height of the ball. (7)

(Total 15 marks)

19

30. The function f is given by

212–1)(xxxf

+=

(a) (i) To display the graph of y = f (x) for –10 ≤ x ≤ 10, a suitable interval for y, a ≤ y ≤ b must be chosen. Suggest appropriate values for a and b .

(ii) Give the equation of the asymptote of the graph. (3)

(b) Show that .)1(2–2)( 22

2

xx

xf+

=!

(4)

(c) Use your answer to part (b) to find the coordinates of the maximum point of the graph. (3)

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

∫ dxxf )(

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


20

31. The point P ( 0,21 ) lies on the graph of the curve of y = sin (2x –1).

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

Working:

Answer:

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

(Total 4 marks)

32. Find

(a) ∫ + ;d)73(sin xx

(b) ∫ xe xd4– .

Working:

Answers:

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

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

(Total 4 marks)

21


Let f (x) = sin (1 + sin x).

(a) (i) Sketch the graph of y = f (x), for 0 ≤ x ≤ 6.

(ii) Write down the x-coordinates of all minimum and maximum points of f, for 0 ≤ x ≤ 6. Give your answers correct to four significant figures.

(9)

(b) Let S be the region in the first quadrant completely enclosed by the graph of f and both coordinate axes.

(i) Shade S on your diagram.

(ii) Write down the integral which represents the area of S.

(iii) Evaluate the area of S to four significant figures. (5)

(c) Give reasons why f (x) ≥ 0 for all values of x. (2)

(Total 16 marks)

34. The function f is given by .0,2n1)( >= xxx

xf

(a) (i) Show that .2n1–1)( 2xx

xf =!

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)

22

(b) By showing that the second derivative 3

3–2n12)(xx

xf =!! 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 line through 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 your answer by drawing a diagram or otherwise.

(3)

(ii) Find ∫ xxf d)( , by using the substitution u = ln 2x, or otherwise.

(4)

(iii) Using ∫ xxf 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.


23

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

(a) Find f %(x).

(b) When x = 3π

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

Working:

Answers:

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

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

(Total 4 marks)

24

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

.152–: 23 xxxxf ++!

40

–20–15–10–5

5101520253035

–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 ),–()–(–: bxaxxxf ! 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)

25

(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)

37. 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)

26

(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.


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

Consider the function .5–22–3

xx

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) Findyxdd

, 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)

27

39. The derivative of the function f is given by f %(x) = x+11 – 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)

28

40. Figure 1 shows the graphs of the functions f1, f2, f3, f4.

Figure 2 includes the graphs of the derivatives of the functions shown in Figure 1, eg the derivative of f1 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

xx

x x

x x

x x

x

O

O

OO

O O

OO

O

29

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)

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

(a) Show that ∫1

0

– de xkx = k1

(1 – e–k).

(3)

(b) Let k = 0.5

(i) Sketch the graph of y = e–0.5x, 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)

30

(c) (i) Find xydd 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)

42. Let the function f be defined by f (x) = 312x+

, 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–xx

+, show that the second derivative

f" (x) = ( )33

3

)1(1–212

xxx

+.

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

31

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

x f (x) 2f (x)

1 1 2

1.4 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∫ xxf

(ii) Given that ( )∫3

1dxxf = 0.637599, use a diagram to explain why your answer is

greater than this. (5)

(Total 15 marks)

32

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

(a) f! (x);

(b) ∫ .d)( xxf

Working:

Answers:

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

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

(Total 6 marks)

44. The graph of y = x3 – 10x2 +12x + 23 has a maximum point between x = –1 and x = 3. Find the coordinates of this maximum point.

Working:

Answer:

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

(Total 6 marks)

33

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

the lines y = 0 and x = .4π3

x

y

43π π

Given that sin 4π3 =

22 and cos

4π3

= – 22 , calculate the exact area of the shaded region.

Working:

Answer:

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

(Total 6 marks)

34

46. The diagram below shows a sketch of the graph of the function y = sin (ex) 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 maximum point 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 xydd .

(iii) Hence, show that at B, x = ln 2π .

(6)

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

(ii) Evaluate this integral. (5)

35

(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)

47. 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 = tsdd = 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 from the 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 = tsdd = 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)

36

48. Let g (x) = x4 – 2x3 + x2 – 2.

(a) Solve g (x) = 0. (2)

Let f (x) = 1)(

2 3

+xgx . A part of the graph of f (x) is shown below.

y

x0A B

C

(b) The graph has vertical asymptotes with equations x = a and x = b where a < b. Write down the values of

(i) a;

(ii) b. (2)

(c) The graph has a horizontal asymptote with equation y = l. Explain why the value of f (x) approaches 1 as x becomes very large.

(2)

(d) The graph intersects the x-axis at the points A and B. Write down the exact value of the x-coordinate at

(i) A;

(ii) B. (2)

37

(e) The curve intersects the y-axis at C. Use the graph to explain why the values of f! (x) and f!! (x) are zero at C.


49. The diagram below shows the shaded region R enclosed by the graph of y = 2x 21 x+ , the x-axis, and the vertical line x = k.

R

x

y

k

y x x = 2 1+ 2

(a) Find xydd .

(3)

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

∫ + 212 xx dx = 32 (1 + x2) 2

3

+ c.

(3)

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

(Total 9 marks)

38

50. 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). Write the equation in the form y = mx + c.

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

51. Let f (x) = 3ex

+ 5 cos 2x. Find f! (x).

Working:

Answer:

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

(Total 6 marks)

39

52. Given that ∫3

1d)( xxg = 10, deduce the value of

(a) ∫3

1;d)(

21

xxg

(b) ∫ +3

1.d)4)(( xxg

Working:

Answers:

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

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

(Total 6 marks)

40

53. The diagram shows part of the graph of the curve y = a (x – h)2 + k, where a, h, k ∈ .

6

5

10

15

20

1 2 3 4 5

y

x0

P(5, 9)

.

(a) The vertex is at the point (3, 1). Write down the value of h and of k. (2)

(b) The point P (5, 9) is on the graph. Show that a = 2. (3)

(c) Hence show that the equation of the curve can be written as

y = 2x2 – 12x + 19. (1)

(d) (i) Find .ddxy

A tangent is drawn to the curve at P (5, 9).

(ii) Calculate the gradient of this tangent.

(iii) Find the equation of this tangent. (4)

(Total 10 marks)

41

54. 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)

(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 rest when 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)

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

(a) (i) Show that f (–4π ) = 0.

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

42

The diagram shows the graph of y = ex (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 xydd

.

(3)

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

(d) Show that at D, y = 4π

e2 . (5)

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

(Total 17 marks)

43

56. It is given that xydd

= x3+2x – 1 and that y = 13 when x = 2.

Find y in terms of x.

Working:

Answer:

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

(Total 6 marks)

44

57. (a) Find ∫(1 + 3 sin (x + 2))dx.

(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

xxf0

d)( .

y

4

4

2

20

–2

–2–4 x

Find the value of a.

Working:

Answers:

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

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

45

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

y

x0

46

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

y

x0

(Total 6 marks)

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

(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 = –21 .

(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)

47

(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)


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 2πt 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)

48

(b) (i) Find tsdd .

(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)

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

2

)1–(2013–2

xxx + , 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)

49

(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–

27–9xx , x ≠ 1.

(3)

The second derivative is given by f ((x) = 4)1–(18–72

xx

, 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)

50

62. Let f (x) = x3 – 2x2 – 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)

63. 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)

51

64. (a) Consider the function f (x) = 2 + 11−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

52

(d) (i) Find ∫ )(xf dx.

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

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

(Total 16 marks)

65. Let f (x) = 6 3 2x . Find f % (x).

Working:

Answer:

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

53

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

s = 10t – 0.5t2.

(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)

54

67. The population p of bacteria at time t is given by p = 100e0.05t.

Calculate

(a) the value of p when t = 0;

(b) the rate of increase of the population when t = 10.

Working:

Answers:

(a) …………………………………………..

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

68. The derivative of the function f is given by f % (x) = e–2x + x−11

, x < 1.

The graph of y = f (x) passes through the point (0, 4). Find an expression for f (x).

Working:

Answer:

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

55

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

0=∫ xxf .

(a) Deduce the value of

(i) ;d)(23

0xxf∫

(ii) ( ) .d2)(3

0xxf∫ +

(b) If 8d)2( =−∫ xxfd

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

Working:

Answers:

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

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

(b) c = ......................., d = ....................... (Total 6 marks)

56

70. Part of the graph of the periodic function f is shown below. The domain of f is 0 ≤ x ≤ 15 and the 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)

57

71. 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 6x

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 + 6x – 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, is perpendicular 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)

58

72. Let f (x) = 1 + 3 cos (2x) for 0 ≤ x ≤ π, 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 (2x) – 1, 0 ≤ x ≤ 2π

.

(b) (i) The graph of f may be transformed to the graph of g by a stretch in the x-direction with scale factor 2

1 followed by another transformation. Describe fully this other transformation.

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

(Total 10 marks)

73. 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

59

(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.


74. Let f (x) = 2 11x+

.

(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

) (126

xx+

− .

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

(4)

60

(d) Let R be the region under the graph of f, between x = –21

and x = 21

,

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)

75. 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

–6y g x = ( )’ y g x = ( )’ ’

g x’( ) g x’ ( )’

1 12 23 34 45 56 67 78 8x x

61

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)

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

(a) Find f " (x).

(b) Write down ∫ xxf d)( .

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

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

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

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

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

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

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

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

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

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

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

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

62

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

(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.

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

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

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

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

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

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

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

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

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

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

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

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

78. The function f is defined by .5.225.0–: 2 ++ xxxf !

(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 = –0.5x + 2.5.

(3)

63

Let 5.25.0–: +xxg !

(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)

79. The diagram below shows the graphs of f (x) = 1 + e2x, g (x) = 10x + 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 this occurs.

(6)

64

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.


65

80. Consider the function .1,)1–(2–: 2 ≠x

xx

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

nxxa)1–(

–

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

66

(c) Given that 4)1–(8–2)(

xx

xh =!! , calculate the coordinates of P.

(3) (Total 8 marks)

81. Let f (x) = (3x + 4)5. Find

(a) f %(x);

(b) ∫f (x)dx.

Working:

Answers:

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

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

(Total 6 marks)

67

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

Given that xydd = 3x2 – 5, find y in terms of x.

Working:

Answer:

....……………………………………..........

(Total 6 marks)

83. 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

68

Calculate the following.

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

(b) ( )∫ +3

1d6)( xxg' .

Working:

Answers:

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

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

(Total 6 marks)

84. 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

69

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

(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 xydd .

(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)

85. Let f (x) = 21 sin 2x + cos x for 0 ≤ x ≤ 2π.

(a) (i) Find f %(x).

One way of writing f %(x) is –2 sin2 x – sin x + 1.

(ii) Factorize 2 sin2 x + sin x – 1.

(iii) Hence or otherwise, solve f %(x) = 0. (6)

70

The graph of y = f (x) is shown below.

A

B

ab

x

y

0 2π

There is a maximum point at A and a minimum point at B.

(b) Write down the x-coordinate of point A. (1)

(c) The region bounded by the graph, the x-axis and the lines x = a and x = b is shaded in the diagram above.

(i) Write down an expression that represents the area of this shaded region.

(ii) Calculate the area of this shaded region. (5)

(Total 12 marks)

86. Let f (x) = 15

3 2

−xx .

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

(b) Find f ′ (x). Give your answer in the form 2

2

)15( −+

xbxax where a and b ∈ .

(4) (Total 5 marks)

71

87. The function g (x) is defined for –3 ≤ x ≤ 3. The behaviour of g%(x) and g((x) is given in the tables below.

x –3 < x < –2 –2 –2 < x < 1 1 1 < x < 3

g%(x) negative 0 positive 0 negative

x –3 < x < –21 –

21 –

21 < x < 3

g((x) positive 0 negative

Use the information above to answer the following. In each case, justify your answer.

(a) Write down the value of x for which g has a maximum. (2)

(b) On which intervals is the value of g decreasing? (2)

(c) Write down the value of x for which the graph of g has a point of inflexion. (2)

(d) Given that g (–3) = 1, sketch the graph of g. On the sketch, clearly indicate the position of the maximum point, the minimum point, and the point of inflexion.

(3) (Total 9 marks)

72

88. Given ∫ −

k

x3 21

dx = ln 7, find the value of k.

Working:

Answers:

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

89. Let f (x) = (2x + 7)3 and g (x) cos2 (4x). Find

(a) f ′ (x);

(b) g′ (x).

Working:

Answers:

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

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

73

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

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

Working:

Answers:

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

91. A particle moves with a velocity v m s−1 given by v = 25 − 4t2 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.


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

74

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

93. Consider the graph of the function, f , defined by

f (x) = 3x4 – 4x3 – 30x2 – 36x + 112, – 2 ≤ x ≤ 4.5.

(a) Given that f (x) = 0 has one solution at x = 4, find the other solution. (2)

(b) The tangent to the graph of f is horizontal at x = 3 and at one other value of x. Find this other value.

(3)

(c) Find the x-coordinates of both points of inflexion on the graph of f . (4)

75

(d) Write down both coordinates of the point of inflexion on the graph of f where the tangent is horizontal.

(2)

A sketch of the graph of f1 is given below.

(e) Write down the equations of the two vertical asymptotes. (2)

(f) The tangent to the graph of f1 is horizontal at P. Write down the x-coordinate of P.


76

94. (a) Let f (x) = e5x. Write down f ′ (x).

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

(c) Let h (x) = e5x sin 2x. Find h′ (x).

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

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

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

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

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

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

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

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

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

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

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

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

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

77

95. The following diagram shows part of the curve of a function ƒ. The points A, B, C, D and E lie on 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 the given points.

A C E

ƒ′′ (x)

(Total 6 marks)

78

96. The velocity, v m s−1, of a moving object at time t seconds is given by v = 4t3 − 2t. When t = 2, the displacement, s, of the object is 8 metres.

Find an expression for s in terms of t.

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

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

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

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

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

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

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

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

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

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

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

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

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

79

97. Consider the infinite geometric series 405 + 270 + 180 +....

(a) For this series, find the common ratio, giving your answer as a fraction in its simplest form.

(b) Find the fifteenth term of this series.

(c) Find the exact value of the sum of the infinite series.

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

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

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

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

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80

98. 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 < x < b

e < x < ƒ

(b) Complete the table below by noting the points on the graph described by the following conditions.

Conditions Point

g′ (x) = 0, g′′ (x) < 0

g′ (x) < 0, g′′ (x) = 0

(Total 6 marks)

81

99. A part of the graph of y = 2x – 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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82

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

(a) Write down ƒ′ (x).

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

(b) p;

(c) k.

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83

101. 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 – 4x + 2) e–x. Use this result to find the exact value of the x-coordinate of C.

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84

102. The displacement s metres at time t seconds is given by

s = 5 cos 3t + t2 + 10, for t ≥ 0.

(a) Write down the minimum value of s.

(b) Find the acceleration, a, at time t.

(c) Find the value of t when the maximum value of a first occurs.

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85

103. Let f (x) = –43 x2 + x + 4.

(a) (i) Write down f ′ (x).

(ii) Find the equation of the normal to the curve of f at (2, 3).

(iii) This normal intersects the curve of f at (2, 3) and at one other point P. Find the x-coordinate of P.

(9)

Part of the graph of f is given below.

(b) Let R be the region under the curve of f from x = −1 to x = 2.


(ii) Calculate this area.

(iii) The region R is revolved through 360° about the x-axis. Write down an expression for the volume of the solid formed.

(6)

(c) Find ∫k

xxf1

,d )( giving your answer in terms of k.


104. Consider the functions f and g where f (x) = 3x – 5 and g (x) = x – 2.

(a) Find the inverse function, f −1. (3)

86

(b) Given that g–1 (x) = x + 2, find (g–1 ◦ f) (x). (2)

(c) Given also that (f −1 ◦ g) (x) 33+x

, solve (f −1 ◦ g) (x) = (g–1 ◦ f) (x).

(2)

Let h (x) = )()(xgxf

, 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 expression 353

−

−

xx

may also be written as 3 + 21−x

. Use this to answer the

following.

(i) Find ∫ )(xh dx.

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

3 h (x)dx.

(5)

(f) On your sketch, shade the region whose area is represented by ∫5

3 h (x)dx.


87

105. 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)

88

106. 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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89

107. 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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108. The function f is defined as f (x) = (2x +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)

90

(b) (i) Show that f ′ (x) = (1− 2x) 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 + 2x), 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).


91

109. 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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92

110. 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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93

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

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

(Total 6 marks)

94

111. Let f (x) = x3 − 3x2 − 24x +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 N1 and N2 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 N2;

(ii) the tangent at Q intersects N1.

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95

112. It is given that ∫3

1 f (x)dx = 5.

(a) Write down ∫3

1 2 f (x)dx.

(b) Find the value of ∫3

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

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96

113. Let f ′ (x) = 12x2 − 2.

Given that f (−1) =1, find f (x).

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97

114. 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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98

115. Let f (x) = 3 cos 2x + sin2 x.

(a) Show that f ′ (x) = −5 sin 2x.

(b) In the interval 4π ≤ x ≤

43π , one normal to the graph of f has equation x = k.

Find the value of k.

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99

116. The following diagram shows part of the graph of a quadratic function, with equation in the form y = (x − p)(x − q), where p, q ∈ .

(a) Write down

(i) the value of p and of q;

(ii) the equation of the axis of symmetry of the curve. (3)

(b) Find the equation of the function in the form y = (x − h)2 + k, where h, k ∈ . (3)

(c) Find xydd .

(2)

(d) Let T be the tangent to the curve at the point (0, 5). Find the equation of T. (2)

(Total 10 marks)

100

117. The function f is defined as f (x) = ex sin x, where x is in radians. Part of the curve of f is shown below.

There is a point of inflexion at A, and a local maximum point at B. The curve of f intersects the x-axis at the point C.

(a) Write down the x-coordinate of the point C. (1)

(b) (i) Find f % (x).

(ii) Write down the value of f % (x) at the point B. (4)

(c) Show that f ((x) = 2ex cos x. (2)

(d) (i) Write down the value of f ((x) at A, the point of inflexion.

(ii) Hence, calculate the coordinates of A. (4)

(e) Let R be the region enclosed by the curve and the x-axis, between the origin and C.


(ii) Find the area of R. (4)

(Total 15 marks)

101

118. The function f (x) is defined as f (x) = 3 + 52

1−x

, x ≠ 25 .

(a) Sketch the curve of f for −5 ≤ x ≤ 5, showing the asymptotes. (3)

(b) Using your sketch, write down

(i) the equation of each asymptote;

(ii) the value of the x-intercept;

(iii) the value of the y-intercept. (4)

(c) The region enclosed by the curve of f, the x-axis, and the lines x = 3 and x = a, is revolved through 360° about the x-axis. Let V be the volume of the solid formed.

(i) Find ( )∫ "

"#

$%%&

'

−+

−+ 252

152

69xx

dx.

(ii) Hence, given that V = !"

#$%

&+ 3ln3

328π , find the value of a.


102

119. Let f (x) = 223qxx

p−

− , where p, q∈ +.

Part of the graph of f, including the asymptotes, is shown below.

(a) The equations of the asymptotes are x =1, x = −1, y = 2. Write down the value of

(i) p;

(ii) q. (2)

(b) Let R be the region bounded by the graph of f, the x-axis, and the y-axis.

(i) Find the negative x-intercept of f.

(ii) Hence find the volume obtained when R is revolved through 360° about the x-axis. (7)

(c) (i) Show that f ′ (x) = ( )( )22

2

1

13

−

+

x

x .

(ii) Hence, show that there are no maximum or minimum points on the graph of f. (8)

103

(d) Let g (x) = f ′ (x). Let A be the area of the region enclosed by the graph of g and the x-axis, between x = 0 and x = a, where a > 0. Given that A = 2, find the value of a.


120. The following diagram shows part of the graph of y = cos x for 0 ≤ x ≤ 2π. Regions A and B are shaded.

(a) Write down an expression for the area of A. (1)

(b) Calculate the area of A. (1)

104

(c) Find the total area of the shaded regions.

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(Total 6 marks)

105

121. Consider the function f (x) = 4x3 + 2x. Find the equation of the normal to the curve of f at the point where x =1.

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106

122. Differentiate each of the following with respect to x.

(a) y = sin 3x (1)

(b) y = x tan x (2)

(c) y = xxln

(3)

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107

123. On the axes below, sketch a curve y = f (x) which satisfies the following conditions.

x f (x) f ′ (x) f ′′ (x)

−2 ≤ x < 0 negative positive

0 –1 0 positive

0 < x <1 positive positive

1 2 positive 0

1 < x ≤ 2 positive negative

(Total 6 marks)

124. Consider the function f (x) e(2x–1) + ( )!

!"

#$$%

&

−125x

, x ≠ 21 .

(a) Sketch the curve of f for −2 ≤ x ≤ 2, including any asymptotes. (3)

108

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

(ii) Write down which one of the following expressions does not represent an area between the curve of f and the x-axis.

∫2

1 f (x)dx

∫2

0 f (x)dx

(iii) Justify your answer. (3)

(c) The region between the curve and the x-axis between x = 1 and x = 1.5 is rotated through 360° about the x-axis. Let V be the volume formed.

(i) Write down an expression to represent V.

(ii) Hence write down the value of V. (4)

(d) Find f ′ (x). (4)

(e) (i) Write down the value of x at the minimum point on the curve of f.

(ii) The equation f (x) = k has no solutions for p ≤ k < q. Write down the value of p and of q.


109

125. A Ferris wheel with centre O and a radius of 15 metres is represented in the diagram below.

Initially seat A is at ground level. The next seat is B, where BOA = 6π .

(a) Find the length of the arc AB. (2)

(b) Find the area of the sector AOB. (2)

(c) The wheel turns clockwise through an angle of 32π . Find the height of A above the

ground. (3)

The height, h metres, of seat C above the ground after t minutes, can be modelled by the function

h (t) = 15 − 15 cos !"

#$%

&+4π2t .

(d) (i) Find the height of seat C when t = 4π .

(ii) Find the initial height of seat C.

(iii) Find the time at which seat C first reaches its highest point. (8)

110

(e) Find h′ (t). (2)

(f) For 0 ≤ t ≤ π,

(i) sketch the graph of h′;

(ii) find the time at which the height is changing most rapidly. (5)

(Total 22 marks)

126. (a) Find .d32

1x

x∫ +

(2)

(b) Given that xx

d32

1∫ +

= ln P , find the value of P.

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(Total 6 marks)

127. A particle moves along a straight line so that its velocity, v ms−1 at time t seconds is given by v

111

= 6e3t + 4. When t = 0, the displacement, s, of the particle is 7 metres. Find an expression for s in terms of t.

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112

128. Consider f (x) = 31

x3 + 2x2 – 5x. Part of the graph of f is shown below. There is a maximum

point at M, and a point of inflexion at N.

(a) Find f ′ (x). (3)

(b) Find the x-coordinate of M. (4)

(c) Find the x-coordinate of N. (3)

(d) The line L is the tangent to the curve of f at (3, 12). Find the equation of L in the form y = ax + b.


113

129. Let ∫ =5

1.12d)(3 xxf

(a) Show that ∫ −=5

1.4d)( xxf

(2)

(b) Find the value of ( ) ( )∫ ∫ +++5

1

5

2.d)( d)( xxfxxxfx

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(Total 7 marks)

114

130. Let f : x ! sin3 x.

(a) (i) Write down the range of the function f.

(ii) Consider f (x) =1, 0 ≤ x ≤ 2π. Write down the number of solutions to this equation. Justify your answer.

(5)

(b) Find f ′ (x), giving your answer in the form a sinp x cosq x where a, p, q ∈ . (2)

(c) Let g (x) = 21

) (cossin3 xx for 0 ≤ x ≤ 2π . Find the volume generated when the curve of

g is revolved through 2π about the x-axis. (7)

(Total 14 marks)

115

131. The diagram below shows part of the graph of the gradient function, y = f ′ (x).

p q r

y

x

(a) On the grid below, sketch a graph of y = f ( (x), clearly indicating the x-intercept.

y

xp q r

(2)

(b) Complete the table, for the graph of y = f (x).

x-coordinate

(i) Maximum point on f

(ii) Inflexion point on f (2)

(c) Justify your answer to part (b) (ii).

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(Total 6 marks)

116

132. The following diagram shows the graphs of f (x) = ln (3x – 2) + 1 and g (x) = – 4 cos (0.5x) + 2, for 1 ≤ x ≤ 10.

(a) Let A be the area of the region enclosed by the curves of f and g.

(i) Find an expression for A.

(ii) Calculate the value of A. (6)

(b) (i) Find f ′ (x).

(ii) Find g′ (x). (4)

(c) There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x.


117

133. Let f (x) = ex (1 – x2).

(a) Show that f ′ (x) = ex (1 – 2x – x2). (3)

Part of the graph of y = f (x), for – 6 ≤ x ≤ 2, is shown below. The x-coordinates of the local minimum and maximum points are r and s respectively.

(b) Write down the equation of the horizontal asymptote. (1)

(c) Write down the value of r and of s. (4)

(d) Let L be the normal to the curve of f at P(0, 1). Show that L has equation x + y = 1. (4)

(e) Let R be the region enclosed by the curve y = f (x) and the line L.

(i) Find an expression for the area of R.

(ii) Calculate the area of R. (5)

(Total 17 marks)

3. differentiate with respect to x 1. 3 4start.sd34.bc.ca/mkeeley/wp-content/uploads/2015/03/...11...

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