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Derivatives and Their Applications · Paper 3 — 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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1ChallengeLAQDefinition of a derivative (rate of change)10 marksPaper 3~15 min

Consider the function

f(x) = \sqrt{x+1}

defined for

x \geq -1

(a)

Prove, using the limit definition of the derivative, that

f'(x) = \dfrac{1}{2\sqrt{x+1}}

for

x > -1

. [5 marks]

(b)

Determine the equation of the tangent line to

y = f(x)

at the point where

x = 3

. [2 marks]

(c)

Use the tangent line found in part (b) to write down a linear approximation for

f(3 + h)

for small

h

. Hence calculate the approximate value of

f(3.4)

given by this approximation, and show that the true value of

f(3.4)

is less than this approximation. Justify why the approximation always overestimates

f(x)

near

x = 3

. [3 marks]

Solutions

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