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Derivatives and Their Applications · Paper 1 — Free Maths AI SL Practice Questions

1FoundationSAQ-SDefinition of a derivative (rate of change)5 marksPaper 1~8 min

A water tank is being filled. The volume of water in the tank,

V

litres, after

t

minutes is given by

V(t) = 2t^2 + 5t

, for

t \geq 0

(a)

State what

V'(t)

represents in this context. [1 mark]

(b)

Find

V'(3)

. [2 marks]

(c)

The tank has a maximum capacity of

500

litres. Determine the value of

t

at which the tank is full, and find the rate at which water is entering the tank at that moment. [2 marks]

Solutions

2FoundationSAQ-SDefinition of a derivative (rate of change)5 marksPaper 1~8 min

The graph of

y = f(x)

for

0 \leq x \leq 6

is shown below, with points

A(1,\,2)

B(3,\,8)

C(5,\,4)

marked. A tangent to the curve at

x = 3

passes through

B(3,\,8)

and the point

(0,\,2)

(a)

State the geometric meaning of

f'(3)

. [1 mark]

(b)

Using the tangent line shown, calculate

f'(3)

. [2 marks]

(c)

Using the coordinates of

A(1,\,2)

and

C(5,\,4)

, calculate the gradient of the chord

AC

and explain whether this value is likely to be a good estimate of

f'(3)

. [2 marks]

Solutions

3MasterySAQ-SDefinition of a derivative (rate of change)5 marksPaper 1~8 min

A water tank is being filled. The volume of water,

V

litres, in the tank at time

t

minutes is given by

V(t) = 2t^2 + 10t

, for

0 \leq t \leq 10

(a)

Find the rate of change of volume at

t = 3

minutes. [3 marks]

(b)

Find the time at which the volume of water in the tank is

168

litres. [2 marks]

Solutions

4ChallengeSAQ-LDefinition of a derivative (rate of change)8 marksPaper 1~12 min

A water tank at a desalination plant has a valve that controls the outflow. The volume of water in the tank,

V

litres, at time

t

minutes after the valve is opened, is modelled by

V(t) = 500 + 240t - 6t^2, \quad 0 \leq t \leq 20.

(a)

Calculate the average rate of change of

V

during the first

5

minutes. [2 marks]

(b)

Using the limit definition of the derivative, show that the instaneous rate of change of

V

t = 5

180

litres per minute. [4 marks]

(c)

The average rate of change found in part (a) is greater than the instaneous rate of change at

t = 5

. Explain this difference in the context of the water tank. [2 marks]

Solutions

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