IB Mathematics Applications & Interpretation Topic SL 5.3—introduction to Derivatives Questionbank (SL/HL)

IB Mathematics Applications & Interpretation Topic SL 5.3—introduction to Derivatives Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic SL 5.3—introduction to Derivatives with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.3—introduction to Derivatives and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 2

A technology company is analyzing the performance of its new software product. Let $x$ be the number of months after launch, and let $f (x)$ be the profit in thousands of dollars. The function representing the profit from the software is given by:

$f (x) = 2 x^{3} - 9 x^{2} + 12 x + 1$

Find the derivative $f^{'} (x)$ .

[2]

Solve $f^{'} (x) = 0$ to find the critical points.

[3]

Determine the intervals where the function is increasing and decreasing.

[3]

Determine the $x$ -coordinates of the local maximum and minimum points.

[2]

Sketch the graph of the function, clearly labeling the critical points and behavior.

[3]

Question 2

SLPaper 2

Consider the function $g$ defined by $g (x) = x^{3} - 12 x + 1$ .

Find the derivative $g^{'} (x)$ .

[2]

Solve $g^{'} (x) = 0$ to find the critical points.

[3]

Determine the intervals where the function is increasing and decreasing.

[3]

Determine the $x$ -coordinates of the local maximum and minimum points.

[2]

Sketch the graph of the function, clearly labeling the critical points and behavior.

[3]

Question 3

SLPaper 1

The temperature $T (t)$ of a chemical mixture, measured in $^{\circ} C$ , is modelled by the function $T (t) = 2 t^{3} - 15 t^{2} + 36 t$ , where $t$ is measured in minutes.

Calculate $\frac{d T}{d t}$ .

[2]

Solve $\frac{d T}{d t} = 0$ .

[2]

Determine the intervals on $0 \leq t \leq 3$ for which $T (t)$ is increasing and for which $T (t)$ is decreasing.

[2]

State whether $T (t)$ is increasing, decreasing, or neither at $t = 1$ .

[2]

Find the maximum value of $T (t)$ on the interval $0 \leq t \leq 3$ .

[2]

Question 4

SLPaper 1

A recording engineer models the change in resonance level of a microphone against equalizer adjustment $q$ from a neutral setting by the function $R (q) = q (q^{2} - 1) (q^{2} - 4)$ .

Expand and simplify $R (q)$ , then find the derivative $R^{'} (q)$ .

[4]

Solve $R^{'} (q) = 0$ for $q$ .

[3]

Determine whether each stationary point is a local maximum or a local minimum.

[2]

Question 5

SLPaper 1

An engineer models the cross-section of a reflector by $r (x) = \frac{3}{2} (x - 6)^{2} + 7$ , where $x$ and $r (x)$ are measured in centimetres.

Calculate the derivative $r^{'} (x)$ .

[2]

Find the slope of the tangent line at $x = 8$ .

[2]

Find the equation of the normal line to the function at $x = 8$ .

[2]

Determine the value of $x$ for which the slope of the tangent is $0$ .

[2]

State the domain and range of the function $r (x)$ .

[2]

IB Mathematics Applications & Interpretation SL 5.3—introduction to Derivatives Questionbank

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IB Mathematics Applications & Interpretation SL 5.3—introduction to Derivatives Questionbank

IB Mathematics Applications & Interpretation (AI) Topic SL 5.3—introduction to Derivatives Questionbank

Unlock all 34 questions with a free account