Math IB Revision Calcus

1.

Find the equation of the normal to the curve with equation 3

y=x +1 at the point (1,2). Working:

Answers: ………………………………………….. (Total 4 marks)

2.

The graph represents the function f: x  p cos x, p ∈

.

y 3

π 2

–

x

3

1

Find (a)

the value of p;

(b)

the area of the shaded region.

Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks)

3.

Differentiate with respect to x

3 − 4x

(a) (b)

e

sin x

Working:

Answers: (a) ………………………………………….. (b) …………………………………….......... (Total 4 marks)

2

4.

The function f is given by

2x + 1 F(x) = x − 3 , x ∈ (a)

(i)

, x ≠ 3.

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)

−7 2 Show that f′ (x) = ( x − 3) 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. (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) (b) (c)

2

Show that f ′ (x) = x – 2x – 3 and hence find f(x). Find f (0), f (–1) and f′ (–1).

(6)

(3)

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)

3

x

6.

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

y x 2

y =

e

P l n (a)

x

2

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)

7.

Show that this volume is π .

(5) (Total 11 marks)

2

The parabola shown has equation y = 9x. y

y

2

P

9

x

M Q

(a)

=

x

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)

4

 9 , 0  S is the point  4  .

(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 SPM. (3) (Total 16 marks)

8.

Differentiate with respect to x: 2

2

(a)

(x + l)

(b)

1n(3x – 1).

Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks)

9.

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

0

1

a

x

Find the exact value of a. 5

Working:

Answers: ....…………………………………….......... (Total 4 marks)

10.

The diagram shows the graph of the function f given by  π f(x) = A sin  2

x   + B,

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

( 1

, 3

)

( 5

,

3

)

2 ( 0

,

1

) x

0

1

2

3 ( 3

4 ,

–

1

5

)

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)

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

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

(d)

 π x  Show that f'(x) = π cos  2  .

(4)

6

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)

11.

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

(5) (Total 24 marks)

(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)

d d Show that x (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

1

( e ,

0

1

2

1

)

x

3

Use the result of part (b) to show that the area of this region is

1 2

e – 1. (4) (Total 10 marks)

7

12.

2

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

∫

2

0

(ii)

x( x − 4) 2 dx.

Explain, using your answer to part (a), why the value of this integral is greater than 0 but less than 40. (3) (Total 15 marks)

13.

2

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

Answers: ....…………………………………….......... (Total 4 marks)

8

14.

 π  If f ′ (x) = cos x, and f  2  = – 2, find f(x).

Working:

Answers: ....…………………………………….......... (Total 4 marks)

15.

3

Let f (x) = x .

(a)

f (5 + h) − f (5) h Evaluate for h = 0.1.

(b)

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

Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks)

9

16.

The main runway at Concordville airport is 2 km long. An aeroplane, 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. P

A

B

T 2

k

m

Not to scale As the aeroplane slows down, its distance, s, from A, is given by 2

s = c + 100t – 4t , where t is the time in seconds after touchdown, and c metres is the distance of T from A. (a)

The aeroplane touches down 800 m from A, (i.e. c = 800). (i)

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

(ii)

Write down an expression for the velocity of the aeroplane at time t seconds after touchdown, and hence find the velocity after 5 seconds. (3) –1

The aeroplane passes the marker at P with a velocity of 36 m s . 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 aeroplane touches down before reaching the point P, it can stop before reaching the northern end, B, of the runway. (5) (Total 15 marks)

(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.

17.

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)

10

(c)

Find the indefinite integral

∫ (π sin x − x) dx and hence, or otherwise, calculate the area of the region enclosed by the graph, the x-axis and the line x = 1. (4) (Total 10 marks)

18.

The diagram shows part of the graph of the curve with equation 2x

y = e cos x. y

P a( ,

0

(a)

b

)

x

dy 2x Show that dx = e (2 cos x – sin x). (2)

(b)

d2 y 2 Find dx . (4)

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

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

(i)

3 tan a = 4 ; (3)

(ii)

2x

the gradient of the curve at P is e . (5) (Total 14 marks)

11

19.

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:

Answers: ....…………………………………….......... (Total 4 marks)

20.

3

Given that f(x) = (2x + 5) find (a)

f′ (x);

(b)

∫ f ( x)dx.

Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks)

12

21.

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

4 1 1= –+ x

y 3 2 1

1

1

3

0

1

3

2

4

Working:

Answers: ....…………………………………….......... (Total 4 marks)

22.

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: 2

0≤ t≤ 2

H = 50 – 5t , 2

H = 90 – 40t + 5t , (a)

2≤ t≤ 5

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

(b)

Sketch a graph of h against t for 0 ≤ t ≤ 5.

(4)

13

(c)

(d)

dh Find dt for: (i)

0≤ t≤ 2

(ii)

2≤ t≤ 5

(2)

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)

23.

Find the minimum height of the rock-climber for 0 ≤ t ≤ 5.

(3) (Total 15 marks)

In this question you should note that radians are used throughout. 2

(a) (i) Sketch the graph of y = x cos x, for 0 ≤ x ≤ 2 making clear the approximate positions of the positive intercept, the maximum point and the endpoints. (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) (ii)

Shade R on your diagram.

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.

d dx (x2 sin x + 2x cos x – 2 sin x) = x2 cos x. (3) (Total 15 marks)

24.

In this part of the question, radians are used throughout. The function f is given by 2

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

14

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

A

C

B

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)

(c)

(i)

Find f′ (x).

(ii)

Hence show that at the point A, cos x =

(iii)

Find the exact maximum value.

1 3. (9)

(d)

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

(e)

(i) (ii)

Find

∫ f (x) dx.

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

15

(f)

25.

3

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

(4) (Total 20 marks)

2

Let f′ (x) = 1 – x . Given that f(3) = 0, find f(x). Working:

Answers: ………………………………………….. (Total 4 marks)

26.

2

Given the function f(x) = x – 3bx + (c + 2), determine the values of b and c such that f(i) = 0 and f′ (3) = 0. Working:

Answers: ………………………………………….. (Total 4 marks)

27.

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)

16

(b)

(c)

Without the use of a calculator, show that π is a solution of the equation π + x cos x = 0.

(3)

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)

(e)

Evaluate the integral in part (d) to an accuracy of six significant figures. (If you consider it d ( x sin x + cos x ) = x cos x .) d necessary, you can make use of the result x (3) (Total 15 marks)

28.

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 2

h = 2 + 20t – 5t , 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)

dh Find dt .

(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)

17

29.

The function f is given by

f ( x) =1 –

2x 1+ x 2

(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 .

(a) (ii)

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

f ′ ( x) =

(b)

Show that

2x 2 – 2 (1 + x 2 ) 2

. (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

∫ f ( x) dx (ii)

Hence find the exact area of the region enclosed by the graph of f, the x-axis and the y-axis. (8) (Total 18 marks)

30.

1 ,0 The point P ( 2 ) lies on the graph of the curve of y = sin(2x –1) . Find the gradient of the tangent to the curve at P. Working:

Answers: ………………………………………….. (Total 4 marks)

18

31.

Find (a)

∫ sin (3x + 7) dx;

(b)

∫e

–4 x

dx

.

Working:

Answers: ………………………………………….. (Total 4 marks)

32.

f ( x) =

The function f is given by

(a)

(i)

Show that

1n 2 x , x

f ′ ( x) =

x > 0.

1 – 1n 2 x x2

.

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)

(b)

f ′′ ( x) =

By showing that the second derivative find the coordinates of the point of inflexion on the graph of f.

21n 2 x – 3 x3 or otherwise, (6)

19

(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 y=

0 (i)

f(

x)

x

Would the trapezium rule overestimate or underestimate the area of S? Justify your answer by drawing a diagram or otherwise. (3)

(ii)

Find

∫ f ( x) dx , by using the substitution u = ln 2x, or otherwise. (4)

(iii)

Using

∫ f ( x) dx , 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)

20

33.

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

Find f ′ (x).

(b)

π When x = 3 , the gradient of the curve of f(x) is 8. Find the value of k.

Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks)

34.

The diagram below shows part of the graph of the function f : x  – x 3 + 2 x 2 + 15 x . y 4 3 3 2 2 1 1

A

0 5 0 5 0 5 0

Q

5 –

3

–

2 – P

1– 5 – 1 0 – 1 5 – 2 0

B 1

2

3

4

5

x

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.

21

(a)

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

(c)

(i) (ii)

Find the equation of the tangent to the graph of f at O.

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)

35.

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) (ii)

Find an expression for the acceleration, a, as a function of t.

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)

22

(d)

Let y metres be the distance fallen after t seconds. –0.2t

(i)

Show that y = 50t + 250e

+ 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)

36.

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

y= Consider the function

3x – 2 . 2x – 5

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)

dx Find dy , simplifying the answer as much as possible. (3)

(c)

How many points of inflexion does the graph of this function have? (1) (Total 7 marks)

23

37.

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, e.g. the derivative of f1 is shown in diagram (d). Figure 1

Figure 2

y

y

f1

( a )

O

O

x

y

y

f2

( b

O

x

) O

x

y

x

y

f3

( c )

O

x

O

y

y

f4

( d O

x

)

x

x

O

y ( e ) O

x

24

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

Derivative diagram

f1

(d)

f2 f3 f4

Working:

(Total 6 marks)

38.

Consider functions of the form y = e 1

(a)

Show that

∫

0

e – kx dx

–kx

1 –k = k (1 – e ). (3)

(b)

Let k = 0.5 –0.5x

, for –1 ≤ x ≤ 3, indicating the coordinates of the

(i)

Sketch the graph of y = e 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)

25

(c)

dy –kx Find dx in terms of k, where y = e .

(i)

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

Find the value of k in this case.

(iii)

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

–kx

.

(5) (Total 13 marks)

39.

2 3 Let the function f be defined by f(x) = 1 + 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)

– 6x 2 3 2 (i) Using the fact that f′ (x) = (1 + x ) , show that the second derivative 12 x 2 x 3 – 1 3 3 f″ (x) = (1 + x ) .

(b)

(

(ii)

)

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

(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 3

(i)

f ( x ) dx. Use the trapezium rule with five sub-intervals to approximate the integral ∫ 1

26

3

f ( x ) dx Given that ∫ = 0.637599, use a diagram to explain why your answer is

(ii)

1

greater than this. (5) (Total 15 marks)

40.

Let f(x) = (a)

x 3 . Find

f′ (x);

∫

f ( x)dx. (b) Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks)

41.

3

2

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

Answer: ………………………………………….. (Total 6 marks)

27

42.

The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and 3π the lines y = 0 and x = 4 . y

π

3π 4

x

3π 3π 2 2 4 4 2 2 Given that sin = and cos =– , calculate the exact area of the shaded region. Working:

Answer: ………………………………………….. (Total 6 marks)

43.

x

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

B

A

–

(a)

1

0

1

C

D

2

x

Find the coordinates of A. (2) 28

(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)

dy Find dx .

(iii)

π Hence, show that at B, x = ln 2 . (6)

(d)

(i) (ii)

Write down the integral which represents the shaded area.

Evaluate this integral. (5)

(e)

(i) Copy the above diagram into your answer booklet. (There is no need to 3 copy the shading.) On your diagram, sketch the graph of y = x . (ii)

The two graphs intersect at the point P. Find the x-coordinate of P. (3) (Total 18 marks)

44.

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

(a)

ds The velocity v ms of a moving body may be written as v = dt = 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. –1

(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 ds v = dt = 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)

x

45.

3 2 Let f(x) = e + 5cos x. Find f′ (x).

29

Working:

Answer: ………………………………………….. (Total 6 marks)

3

46.

Given that 3

(a)

∫ g ( x ) dx 1

= 10, deduce the value of

1

∫ 2 g ( x)dx; 1

3

∫ ( g ( x) + 4)dx.

1 (b) Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks)

30

47.

–1

An aircraft lands on a runway. Its velocity v ms at time t seconds after landing is given by the –0.5t equation v = 50 + 50e , 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)

48.

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

(a)

(i) (ii)

π Show that f(– 4 ) = 0.

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

(3)

31

x

The diagram shows the graph of y = e (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. y 6

C a ( ,

b

)

4 D 2

–

(b)

2

–

1

1

2

3

x

dy Find dx . (3)

(c)

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

(d)

Show that at D, y =

π 2e 4

. (5)

(e)

Find the area of the shaded region. (2) (Total 17 marks)

32

49.

dy 3 It is given that dx = x +2x – 1 and that y = 13 when x = 2. Find y in terms of x. Working:

Answer: ………………………………………….. (Total 6 marks)

33

50.

(a) (b)

Find ∫ (1 + 3 sin(x + 2))dx.

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

0

f ( x ) dx

.

y

4

2

–

4

–

2

0

–

2

4

x

2

Find the value of a. Working:

Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks)

34

51.

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

y

0

x

35

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

y

x

0

(Total 6 marks)

52.

Consider the function f(x) = 1 + e (a)

(i) (ii)

–2x

.

Find f′ (x).

Explain briefly how this shows that f(x) is a decreasing function for all values of x (i.e. that f(x) always decreases in value as x increases). (2)

36

1 Let P be the point on the graph of f where x = – 2 . (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)

(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)

1 Draw the tangent at x = – 2 .

(iii)

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

(iv)

Find this area. (7) (Total 14 marks)

53.

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.

d

i a g r a m s c a l e

n

o t

P

Its distance, s cm, from the ceiling, is modelled by the function s = 48 + 10cos2πt where t is the time in seconds from release. (a)

(i) (ii)

What is the distance of the point P from the ceiling?

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

37

t o

(b)

(i) (ii)

ds Find dt .

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 cos4π t. 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)

54.

2 x 2 – 13x + 20 ( x – 1) 2 Consider the function f given by f(x) = , A part of the graph of f is given below.

x ≠ 1.

y

x

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

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

(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)

38

9 x – 27

(c)

Show that f′ (x) = ( x – 1) , 3

x ≠ 1.

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

(3)

x ≠ 1.

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

(2)

There is a point of inflexion on the graph of f. Write down the coordinates of this point. (2) (Total 10 marks)

55.

Let p = log10x, q = log10 y and r = log10z.  x     y2 z   in terms of p, q and r. Write the expression log10 

Working:

Answer: ………………………………………….. (Total 6 marks)

39

1 (a) Consider the function f(x) = 2 + x − 1 . The diagram below is a sketch of part of the graph of y = f(x).

56.

y 5 4 3 2 1 –

5 –

4 –

3 –

10

2 –

1

–

1

–

2

–

3

–

4

–

5

2

3

4

5

x

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

(b)

(i) (ii)

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

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

(c)

(i) (ii)

Find f′ (x).

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. y 5 4 3 2

A

1 –

5 –

4 –

3 –

10

2 –

1

–

1

–

2

–

3

–

4

–

5

2

3

4

5

x

40

(d)

(i)

Find

∫ f (x) dx.

(ii)

Write down an expression that represents the area labelled A.

(iii)

Find the area of A. (7) (Total 15 marks)

57.

2

2

The diagram below shows part of the graph of f(x) = x sin(x + π ) and the shaded region A. y

A P

0

1

Q 2

x

3 This graph crosses the x-axis at P and Q. The point P has coordinates ( π , 0).

(a)

Find the x-coordinate of Q. (2)

(b) (c)

3

Use the substitution u = x + π to find

∫ f (x) dx. (4)

Hence, using your answer to (b), find the area of the region A. (3) (Total 9 marks)

41

58.

3 2 Let f(x) = 6 x . Find f′ (x).

Working:

Answer: …………………………………………........ (Total 6 marks)

59.

The displacement s metres of a car, t seconds after leaving a fixed point A, is given by 2

s = 10t – 0.5t . (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)

60.

The population p of bacteria at time t is given by p = 100e

0.05t

.

Calculate (a)

the value of p when t = 0;

42

(b)

the rate of increase of the population when t = 10.

Working:

Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks)

61.

The derivative of the function f is given by f′ (x) = e

–2x

1 + 1 − x , 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)

62.

Let f be a function such that (a)

∫

3

0

f ( x ) dx = 8

.

Deduce the value of 3

(i)

∫

(ii)

∫ ( f ( x ) + 2 ) dx .

0

2 f ( x ) dx

;

3

0

43

(b)

∫

d

f ( x − 2) d x = 8

c

, write down the value of c and of d.

Working:

Answers: (a) (i) …………………………………… (ii) …………………………………... (b) c = ......................., d = ....................... (Total 6 marks)

63.

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. f(

x ) 4 3 2 1 0

(a)

0

1

2

3

4

5

6

7

8

9

1

0

x

Find (i)

f(2);

(ii)

f′ (6.5);

(iii)

f′ (14).

44

(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)

64.

2

The function f(x) is defined as f(x) = –(x – h) + k. The diagram below shows part of the graph of f(x). The maximum point on the curve is P(3, 2). y 4

P

( 3

,

3

4

2

)

2 –

(a)

1

1 –

2

–

4

–

6

–

8

–

1

0

–

1

2

2

5

6

x

Write down the value of (i)

h;

(ii)

k. (2)

(b)

2

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

(c)

Find f′ (x). (2)

45

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)

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

65. (a)

(i) (ii)

Find f′ (x).

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

(6)

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

(i) The graph of f may be transformed to the graph of g by a stretch 1in the x-direction with scale factor followed by another transformation. Describe fully this other transformation. (ii)

Find the solution to the equation g(x) = f(x) (4) (Total 10 marks)

66.

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). y 4 3 2

(a –

5 –

4 –

, 3 –

S

1

0

)

2 –

1 –

1

–

2

–

3

–

4

–

5

–

6

–

7

(b 1

R

2

3

4

, 5

0

)

x

46

(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) (ii)

The y-coordinate of R is –0.240. Find the y-coordinate of S.

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)

67.

1 2 Let f(x) = 1 + x . (a)

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

(b)

Find f′ (x). (3)

(c)

6x 2 − 2 2 3 The second derivative is given by f″ (x) = (1 + x ) .

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)

47

(d)

1 1 Let R be the region under the graph of f, between x = – 2 and x = 2 , as shaded in the diagram below y 2

1

R –

1

–

1 2

–

x

1

1 2

1

Write down the definite integral which represents the area of R. (2) (Total 10 marks)

68.

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. g ’(

g ’’(

x)

6

6

5

5

4

4

3

3

2

2

1

1

0

1

–

1

– –

2

3

5

4

7

6

8

x

0

1

–

1

2

–

2

3

–

3

–

4

–

4

–

5

–

5

–

6

–

6

y = ’

g

(

x)

)x

2

3

4

y = ’ ’ g

5

7

6

(

8

x

x)

Use the above information to answer the following. (a)

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

48

(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)

69.

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

Find f"(x).

(b)

Write down

∫ f ( x)dx .

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

49

70.

–1

The velocity v m s of a moving body at time t seconds is given by v = 50 – 10t. –2

(a)

Find its acceleration in m s .

(b)

The initial displacement s is 40 metres. Find an expression for s in terms of t. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (Total 6 marks)

71.

2 The function f is defined by f : x  – 0.5 x + 2 x + 2.5.

(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)

Let g : x  – 0.5 x + 2.5 (c)

(i) (ii)

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

Hence find the coordinates of the other point of intersection of the normal and the curve. (6)

50

(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)

72.

2x

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

1

6

1

2

f g

p 8

4

0

(a)

. 5

1

1

x . 5

(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)

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

1

6

1

2

8 5 4

0

(b)

(i) (ii)

. a5

1

1

x . 5

–1

Find f (x).

Hence show that a = ln 2. (5) 51

(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)

x–2

h: x  73.

( x – 1) 2

Consider the function

, x ≠ 1.

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

A

P x

N

o t

t o

s c a l e

B 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

a–x ( x – 1) n where a and n are constants to be determined. (4)

h ′′ ( x) = (c)

Given that

2x – 8 ( x – 1) 4 , calculate the coordinates of P. (3) (Total 8 marks)

52

74.

5

Let f(x) = (3x + 4) . Find (a)

f′ (x);

(b)

∫

f(x)dx.

Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks)

75.

The curve y = f(x) passes through the point (2, 6). dy 2 Given that dx = 3x – 5, find y in terms of x.

Working:

Answer: ....…………………………………….......... (Total 6 marks)

53

76.

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

Calculate the following.

(a)

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

(b)

∫ ( g' ( x) + 6) dx .

3

1

Working:

Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks)

54

77.

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. y 4 2 A –

(a)

(i)

4

–

B 0

2

2

–

2

–

4

–

6

6 x

4

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 = ax + bx + c.

2

(5)

(b)

(i) (ii)

dy Find dx .

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)

78.

3x 2 Let f(x) = 5 x − 1 .

(a)

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

(b)

ax 2 + bx Find f(x). Give your answer in the form (5 x − 1) where a and b ∈

. (4) (Total 5 marks)

55

79.

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