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

What this topic covers

Calculus is where AA HL stops rewarding memorised algebra and starts rewarding process. Every mark scheme in this topic is built around specific words — show that, hence, determine, justify — and examiners are trained to withhold marks the instant your working skips a step, even when your final answer is correct. The four subtopics here build on each other in a strict chain: limits define continuity, continuity is the gatekeeper for differentiability, derivatives feed every application (tangents, optimisation, kinematics), and integration — the reverse process — feeds differential equations. Learn it in that order and the HL-only material (implicit differentiation, integration by parts, volumes of revolution, and the whole of differential equations) stops looking like extra content and starts looking like the natural next question.

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1FoundationSAQ-SDefinition of a derivative (rate of change)8 marksPaper 1~12 min

The height of a tree,

h

metres,

t

years after it was planted is modelled by

h(t) = 1.2 + 0.8\ln(t+1), \quad t \geq 0.

(a)

Using the limit definition of the derivative, show that

h'(t) = \frac{0.8}{t+1}.

[3 marks]

(b)

Find the instaneous rate of growth of the tree when

t = 4

, and interpret this value in context. [2 marks]

(c)

The owner claims the tree's rate of growth will eventually fall below

0.05

metres per year. Determine the value of

t

at which this occurs and comment on whether the model remains realistic for large

t

. [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

at time

t

seconds is given by

s(t) = t^3 - 6t^2 + 9t

, for

t \geq 0

(a)

Use the definition of the derivative as a limit to show that

v(t) = s'(t) = 3t^2 - 12t + 9

. [3 marks]

(b)

Hence find the values of

t

at which the particle is instaneously at rest. [2 marks]

Solutions

3ChallengeSAQ-LDefinition of a derivative (rate of change)8 marksPaper 1~12 min

The graph of

y = f(x)

for

0 \leq x \leq 4

is shown below. The function

f

is continuous and differentiable on

(0, 4)

. Let

g(x) = x\,f(x)

(a)

Using the geometric interpretation of the derivative, state the value of

f'(1)

, justifying your answer. [2 marks]

(b)

Determine the value of

g'(1)

. [3 marks]

(c)

(i) Show that

g'(x) = f(x) + x\,f'(x)

. Hence write

g'(3)

in terms of

f(3)

and

f'(3)

. [2]
(ii) Given that

f'(3) = -2

, find the value of

g'(3)

. Hence determine a value of

x

in the interval

(0, 4)

at which

g'(x) = 0

, other than

x = 1

, justifying your answer. [3 marks]

Solutions

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