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

1FoundationSAQ-SDefinition of a derivative (rate of change)7 marksPaper 1~11 min

The volume of water,

V

litres, in a tank at time

t

minutes is given by

V(t) = 100 + 20t - t^2

, for

0 \leq t \leq 10

(a)

Find

V'(t)

. [2 marks]

(b)

Find the rate of change of volume at

t = 5

minutes, and state whether the volume is increasing or decreasing at this instant. [2 marks]

(c)

Determine the time at which the volume in the tank is greatest, and find this maximum volume. Justify your answer. [3 marks]

Solutions

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

A particle moves along a straight line such that its displacement,

s

metres, from a fixed point

O

after

t

seconds is given by

s(t) = 5t^2 - 3t + 2

, for

t \geq 0

(a)

Show that the velocity of the particle at

t = 2

17\,\text{m s}^{-1}

. [2 marks]

(b)

Find the time at which the velocity is

37\,\text{m s}^{-1}

. [2 marks]

(c)

Determine whether the particle ever moves in the negative direction. Justify your answer. [1 mark]

Solutions

3ChallengeSAQ-LDefinition of a derivative (rate of change)5 marksPaper 1~8 min

The graph of

f(x) = x^3 - 4x

has

x

-intercepts at

A(-2,\,0)

B(0,\,0)

, and

C(2,\,0)

(a)

Use the limit definition of the derivative to determine the gradient of the tangent to the graph of

f

x = -1

. [4 marks]

(b)

(i) Show that

f

is an odd function. [1]
(ii) Hence determine the gradient of the tangent to the graph of

f

x = 1

, justifying your reasoning. [2]
(iii) Write down the equation of the tangent to the graph of

f

x = 1

. [1 mark]

Solutions

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