IB Mathematics Applications & Interpretation Topic AHL 5.9—differentiating Standard Functions and Derivative Rules Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.9—differentiating Standard Functions and Derivative Rules Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.9—differentiating Standard Functions and Derivative Rules with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.9—differentiating Standard Functions and Derivative Rules and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

Let $y = cos^{6} (3 x)$ .

Consider the function $h (x) = 4 - 3 sin (3 x) cos^{5} (3 x)$ , defined for $0 \leq x \leq \frac{π}{6}$ .

The region $R$ is bounded by the graph of $h$ , the $x$ -axis, the $y$ -axis, and the line $x = \frac{π}{6}$ .

Determine the derivative $\frac{d y}{d x}$ .

[3]

Using your result from the previous part, find $\int sin (3 x) cos^{5} (3 x) d x$ .

[3]

State a definite integral that represents the area of $R$ .

[2]

Determine the precise area of $R$ .

[6]

Question 2

HLPaper 2

Let $y = sin^{5} (2 x)$ .

Consider the function $g (x) = 3 - 2 sin^{4} (2 x) cos (2 x)$ , defined for $0 \leq x \leq \frac{π}{4}$ .

The region $S$ is bounded by the graph of $g$ , the $x$ -axis, the $y$ -axis, and the line $x = \frac{π}{4}$ .

Determine the derivative $\frac{d y}{d x}$ .

[3]

Using your result from the previous part, find $\int sin^{4} (2 x) cos (2 x) d x$ .

[3]

State a definite integral that represents the area of $S$ .

[2]

Determine the precise area of $S$ .

[6]

Question 3

HLPaper 2

A tank in the shape of a cone with height 4 metres and base radius 2 metres is filled with water. The height of the water is $h (t)$ (in metres) at time $t$ minutes and satisfies the differential equation $\frac{d h}{d t} = - 0.01 h$ , where $0.01$ has units $m^{1/2} min^{- 1}$ .

Solve the differential equation to find $h (t)$ .

[5]

Given that $h (0) = 4$ , find the time $T$ when the tank is empty ( $h = 0$ ).

[3]

Find the rate of change of the volume of water when $h = 1$ .

[4]

Question 4

HLPaper 1

A particle moves in a plane such that its position vector (in metres) at time $t$ seconds, for $t \geq 0$ , is given by $r = (t^{2} t^{2} - 4 t)$ .

Find the velocity vector, $v$ , of the particle in terms of $t$ .

[2]

Find the position vector of the particle at the instant its speed is at a minimum.

[5]

Question 5

HLPaper 1

On a transport graph, a drone has position vector (in metres) relative to vertex $O$ at time $t$ seconds, for $0 \leq t \leq π$ , given by $r = (4 sin t 7 t + 4 cos t)$ .

Find the velocity vector, $v$ , in terms of $t$ .

[2]

Find the position vector of the drone at the instant its speed is at a minimum.

[5]

IB Mathematics Applications & Interpretation AHL 5.9—differentiating Standard Functions and Derivative Rules Questionbank

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IB Mathematics Applications & Interpretation AHL 5.9—differentiating Standard Functions and Derivative Rules Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.9—differentiating Standard Functions and Derivative Rules Questionbank

Unlock all 56 questions with a free account