Question 1

If , the value of x at which the gradient of the curve is 1 is:

A. 3

B.

C.

D.

E. 0

Question 2

The maximum height (in metres) of a ball thrown in the air with the height at time t seconds

given by: is:

A. 2 m

B. 12 m

C. 1 m

D. 14 m

E. 13 m

Question 3

The average rate of change of between x = 1 and x = 3 is:

A. 4

B. 1

C. 2

D. -1

E. 3

Question 4

The equation of the tangent to the curve at x = 2 is:

A.

B.

C.

D.

E.

MULTIPLE CHOICE REVISION: CAS ALLOWED

Question 5

The gradient of the normal to the curve: y = at is:

A.

B.

C.

D.

E.

Question 6

is equal to:

A. 7

B. 1

C. 0

D. -3

E. cannot be evaluated

Question 7

If then is equal to:

A.

B.

C.

D.

E.

Questions 8,and 9 refer to the graph shown in the diagram below. The function f has domain

Question 8

The is equal to:

A. 4

B. 0

C. 1

D. 3

E. is undefined

Question 9

is equal to :

A. 1

B. 2

C. 3

D. 0

E. is undefined

Question 10

For the graph shown, f(x) is not differentiable for:

A.

B.

C. only

D.

E.

Question 11

The instantaneous rate of change of the function when is:

A. 2

B. -2

C. 12

D. 3

E. 28

Question 12

The average rate of change of the function between and is:

A. 1

B. 9

C. 5

D. 7

E. 3

Question 14

When , the graph of

A. is increasing

B. is decreasing

C. has a local maximum turning point

D. has a local minimum turning point

E. has a stationary point of inflection

Question 15

The local minimum value of

occurs when:

A.

B.

C.

D.

E.

Question 16

The graph below shows the derivative function, of a function

From this graph, we can conclude that:

A. The graph of has a stationary point at

B. The graph of has a maximum turning point at

C. The graph of has a minimum turning point at

D. The graph of has a maximum turning point at

E. The graph of has a stationary point of inflection at

Question 17

A bushfire burns out A hectares of land, t hours after it has started according to the rule:

At time t = 10, the fire is burning at a rate of:

A. 900 ha/hour

B. 0 ha/hour

C. 6000 ha/hour

D. 500 ha/hour

E. 300 ha/hour

Question 18

Question 19

The new Ballarat pipeline has just started pumping water into Ballarat’s water reservoirs The

height h m of the water at one location t days after pumping has begun is given by the function: 2( ) 30 150h t t t + 100 where 0 5t . At exactly 3 days after pumping begins, the height

of the water in the reservoir is:

A. falling at the rate of 30 m/day

B. increasing at the rate of 30m/day

C. increasing at the rate of 180m/day

D. increasing at the rate of 56.7m/day

E. falling at the rate of 56.7m/day.

Question 20

2

7

49lim

7x

x

x

is equal to

A. 0

0

B. 14

C. 7

D. 1

E. 0

Question 21

The graph of the function y = x4 +5x3 +3x2 +2x+1 has n stationary points.

The value of n is

A. 0

B. 1

C. 2

D. 3

E. 4

Question 22

If

g(x) x(x3 2x) then g' (x) is

A.

3x22

B.

4x21

C.

4x31

D.

4x32

E.

4x34x

Question 23

Consider the function :

Which of the following statements is true about this function?

A. is not continuous at

B. has a stationary point at

C.

D. The derivative of f is not defined at x = 0.

E.

SHORT ANSWER (NO CAS !!)

Question 1

The position of a particle moving in a horizontal line relative to an origin O at time t is given

by:

a. Find the velocity of the particle at time t.

b. For what values of t is the particle moving to the left?

c. State the values of t for which the particle is instantaneously at rest.

d. Determine the position of the particle when it is instantaneously at rest.

e. Find the acceleration of the particle at time t = 3.

Question 2

Find: a.

b. where

Question 3

The temperature Celsius on a mountain is related to the height h metres by the rule:

a. Calculate the average rate of change of temperature over the first 10m

b. Calculate the rate of change of temperature at h = 10

Question 4

a. Find the equation of the normal to the curve

at

b. Hence, find the co-ordinates of the point B, where the normal intersects the curve

again.

Question 5

a. Find the gradient of the chord PQ to function if the x-co-ordinates of

P and Q are and respectively.

b. Hence, find the gradient of the tangent to the curve at

Question 6

The position of a particle moving in a straight line is given by the rule:

where x is in cm and t is in seconds.

a. What is the initial position of the particle?

b. Find the velocity of the particle when t = 4

c. Is the particle moving to the right or to the left when t = 3?

d. When is the particle instantaneously at rest?

e. Determine its position when it is instantaneously at rest.

f. Calculate the distance covered by the particle in the first 3 seconds.

Question 7

Use first principles to find the derivative of the function:

Question 8

The volume of a liquid inside a vat during a manufacturing process is given by:

, where .

a. Determine when the maximum volume of liquid is in the vat.

b. Determine the maximum volume.

c. Determine the minimum volume of liquid in the vat, and when this occurs.

Question 9

A company’s income each week is : dollars, where n is the number of

employees. The company spends $760 per employee for wages and materials.

a. Write and expression for the company weekly profit,

b. Determine the number of employees required for maximum weekly profit.

c. Calculate the maximum weekly profit.

Question 10

The total surface area of a cylindrical water tank is square units. If its radius is r and its

height is h,

a. Show that

b. Hence, show that the volume of the cylinder is given by:

c. Find the domain for which the function V(r) is defined.

d. Determine the value of r for which the volume of the tank will be a maximum (there is

no need to justify that you have found a maximum).

Question 11

If has a stationary point at (2, - 3) calculate:

a. The values of a and b

b. The co-ordinates of any other stationary points.

Question 12

a. Find the co-ordinates of the point on the curve: where the tangent

is parallel to the line with equation:

b. Find the co-ordinates of the point on the curve where the gradient

of the tangent is equal to the gradient of the function

for the same

value of x.

Question 13

A function has a stationary point at (2,20).

Find the value of