**1. ** (a) *Differentiate the following function with respect to x:*
*f (x) = 2x – 9 – 25x–1*

(b) *Calculate the x-coordinates of the points on the curve where the gradient of the tangent to*
the curve is equal to 6.

**(Total 6 marks)**

**2. ** (a) *Differentiate the function y = x*2* + 3x* – 2.

(b) *At a certain point (x, y) on this curve the gradient is 5. Find the co-ordinates of this point.*

**(Total 6 marks)**

**3.**

*f x*( )

A

B

C D

E

F

G H

K L

*y* = ( )*f x*

*x*

*Given the graph of f (x) state*

(a) *the intervals from A to L in which f (x) is increasing.*

**(1)**

(b) *the intervals from A to L in which f (x) is decreasing.*

**(1)**

(c) a point that is a maximum value.

**(1)**

(d) a point that is a minimum value.

(e) the name given to point K where the gradient is zero.

**(1)**
**(Total 5 marks)**

**4. ** (a) Write 3_{2} * in the form 3xa where a*∈ .

(b) *Hence differentiate y =* 3_{2} giving your answer in the form — where c∈ +.

*x* *xc*

*b*

**(Total 6 marks)**

**5. ** *Consider the function f (x) =* 3_{2} * + x* – 4.
*x*

(a) *Calculate the value of f (x) when x = 1.*

**(2)**

(b) *Differentiate f (x).*

**(4)**

(c) *Find f '(l).*

**(2)**

(d) *Explain what f '(l) represents.*

**(2)**

(e) *Find the equation of the tangent to the curve f (x) at the point where x = 1.*

**(3)**

(f) *Determine the x-coordinate of the point where the gradient of the curve is zero.*

**(3)**
**(Total 16 marks)**

**6. ** *The diagram shows a part of the curve y = f(x).*

2

1

–1

–2

–3 –2 –1 0 1 2 3 4 5 6 7 8 *x*

*y*

(a) *For what values of x is f '(x) = 0?*

(b) *Solve g*′*(x) = 0.*

**(2)**

(c) (i) *Calculate the values of g'(x) when*

(a) *x = –3;*

(b) *x = 0.*

(ii) Hence state whether the function is increasing or decreasing at

(a) *x = –3;*

(b) *x = 0.*

**(4)**
**(Total 9 marks)**

**8. ** *The height (cm) of a daffodil above the ground is given by the function h (w) = 24w* – 2.4w2,
*where w is the time in weeks after the plant has broken through the surface (w*≥ 0).

(a) Calculate the height of the daffodil after two weeks.

**(2)**

(b) (i) Find the rate of growth, .

*w*
*h*
d

d

**(2)**

(ii) *The rate of growth when w = k is 7.2 cm per week. Find k.*

**(3)**

(iii) When will the daffodil reach its maximum height? What height will it reach?

**(4)**

(c) Once the daffodil has reached its maximum height, it begins to fall back towards the ground. Show that it will touch the ground after 70 days.

**9.**

*f x*( )

A

B

C D

E

F

G H

K L

*y* = ( )*f x*

*x*

*Given the graph of f (x) state*

(a) *the intervals from A to L in which f (x) is increasing.*

**(1)**

(b) *the intervals from A to L in which f (x) is decreasing.*

**(1)**

(c) a point that is a maximum value.

**(1)**

(d) a point that is a minimum value.

(e) the name given to point K where the gradient is zero.

**(1)**
**(Total 5 marks)**

**10. ** *A, B, C, D and E are points on the curve y = f(x) shown in the diagram below.*

*y*

B

D

E

(b) *D has coordinates (a, f(a)), and the x-coordinate at E is a + 4. Write an expression for the*
gradient of the line segment [DE].

**(3)**
**(Total 6 marks)**

**11. ** *Consider the function f (x) = 2x*3 – 3x2 – 12x + 5.
(a) (i) *Find f ' (x).*

(ii) *Find the gradient of the curve f (x) when x = 3.*

**(4)**

(b) *Find the x*-coordinates of the points on the curve where the gradient is equal to –12.

**(3)**

(c) (i) *Calculate the x-coordinates of the local maximum and minimum points.*

(ii) Hence find the coordinates of the local minimum.

**(6)**

(d) *For what values of x is the value of f (x) increasing?*

**(2)**
**(Total 15 marks)**

**12. ** *The function g is defined as follows*

*g : x* *px*2* + qx + c,* *p, q, c*∈
(a) *Find g*′*(x)*

**(1)**

(b) *If g*′*(x) = 2x + 6, find the values of p and q.*

**(2)**

(c) *g(x) has a minimum value of –12 at the point A. Find*

(i) *the x-coordinate of A;*

(ii) *the value of c.*

**13. ** *A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from*
each corner of the card as shown in the diagram below. What is left is then folded to make an
**open box, of length l cm and width w cm.**

*x*
*x*

9 cm

24 cm

(a) *Write expressions, in terms of x, for*

(i) *the length, l;*

(ii) *the width, w.*

**(2)**

(b) *Show that the volume (B m*3*) of the box is given by B = 4x*3 – 66x2* + 216x.*

**(1)**

(c) Find .
*x*
*B*
d
d

**(1)**

(d) (i) *Find the value of x which gives the maximum volume of the box.*

(ii) Calculate the maximum volume of the box.

**(4)**
**(Total 8 marks)**

**14. ** *The velocity, vms*–1*, of a kite, after t seconds, is given by*
*v = t*3 – 4t2* + 4t, * 0≤*t*≤ 4.
(a) What is the velocity of the kite after

(i) one second?

(ii) half a second?

(c) (i) Find * in terms of t. Find the value of t at the local maximum and minimum*
*t*

*v*
d
d

values of the function.

(ii) Explain what is happening to the function at its local maximum point. Write down the gradient of the tangent to its curve at this point.

**(8)**

(d) *On graph paper, draw the graph of the function v = t*3 – 4t2* + 4t, 0*≤*t*≤ 4.

Use a scale of 2 cm to represent 1 second on the horizontal axis and 2 cm to represent 2 ms–1 on the vertical axis.

**(5)**

(e) Describe the motion of the kite at different times during the first 4 seconds. Write down the intervals corresponding to changes in motion.

**(3)**

*The acceleration, a, of a second kite is given by the equation a = 4t* – 3, 0≤*t*≤ 4.

(f) *After 1 second, the velocity, u, of this kite, is 3 ms*–1*. Write the equation for u in terms*
*of t.*

**(4)**
**(Total 24 marks)**

**15. ** *The diagram below is a part of the graph of the function y = 2x*2. P is the point (1, 2) on the
curve and Q is a neighbouring point.

*y*

*x*

O M N

(1, 2)P R Q

*y* = 2*x*2

(a) **Copy and complete the following table**

PR 0.1 0.01 0.001

QN

QR

Gradient of PQ

(b) Show how you can use your table to

(i) *predict the gradient of the curve y = 2x*2 at the point (1,2);

**(1)**

(ii) *deduce the gradient of the curve y = 2x*2 + 3 at the point (1,5).

**(3)**
**(Total 8 marks)**

**16. ** A farmer wishes to enclose a rectangular field using an existing fence for one of the four sides.
Existing fence

*x*

*y*

(a) *Write an expression in terms of x and y that shows the total length of the new fence.*

**(1)**

(b) The farmer has enough materials for 2500 metres of new fence. Show that

*y = 2500 – 2x*

**(1)**

(c) *A(x) represents the area of the field in terms of x.*

(i) Show that

*A(x) = 2500x – 2x*2

**(2)**

(ii) *Find A'(x).*

**(1)**

*(iii) Hence or otherwise find the value of x that produces the maximum area of the field.*
**(3)**

(iv) Find the maximum area of the field.

**17. ** A cylinder is cut from a solid wooden sphere of radius 8 cm as shown in the diagram. The
*height of the cylinder is 2h cm.*

D C

O

A B

E

*h*
*h*

8cm

8 cm

(a) *Find AE (the radius of the cylinder), in terms of h.*

**(2)**

(b) *Show that the volume (V) of the cylinder may be written as*

*V= 2*π*h*(64 –*h*2) cm3.

**(2)**

(c) (i) Determine, correct to three significant figures, the height of the cylinder with the greatest volume that can be produced in this way.

**(5)**

(ii) Calculate this greatest volume, giving your answer correct to the nearest cm3.

**(3)**
**(Total 12 marks)**

**18. ** *The letters A to E are placed at particular points on the curve y = f(x).*

*y*

*x*

A B

C

D

E

(a) *What is the gradient of the curve y =f(x) at the point marked C?*

**(1)**

(b) In passing from point B, through point C, to point D what is happening to ? Is it

*x*
*y*
d
d

decreasing or increasing?

**(2)**
**(Total 3 marks)**

**19. ** *The function f (x) is given by f (x) = x*3 – 3x2* + 3x*, for –1 ≤*x ≤ 3.*
(a) *Differentiate f (x) with respect to x.*

**(2)**

(b) Copy and complete the table below.

*x* –1 0 1 2 3

*f (x)* 0 1 2 9

*f'(x)* 12 0 12

**(3)**

(c) *Use the information in your table to sketch the graph of f (x).*

**(2)**

(d) Write down the gradient of the tangent to the curve at the point (3, 9).

**(1)**
**(Total 8 marks)**

**20. ** The perimeter of a rectangle is 24 metres.

(a) The table shows some of the possible dimensions of the rectangle.
*Find the values of a, b, c, d and e.*

Length (m) Width (m) Area (m2_{)}

1 11 11

*a* 10 *b*