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Calculus - IB Questionbank | RevisionDojo

Practice Calculus with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

A sustainable fashion company produces and sells eco-friendly clothing. The profit, in dollars, from selling x units of a particular clothing item is given by:

${P(x) = 100x - 5x^2}$

Determine the number of units x that maximises the profit.

[4]

Calculate the maximum profit.

[3]

Determine whether the function P(x) has a point of inflection, showing all necessary steps.

[8]

Question 2

SLPaper 2

A structural engineer is designing a curved roof for a building. The height of the roof, $h(x)$ , in meters, is modeled by $h(x)=-0.02 x^{3}+0.3 x^{2}+0.4 x$ , for $0 \leq x \leq 10$ , where $x$ is the horizontal distance from one end. The cross-sectional area under the roof is critical for ventilation calculations. The engineer takes measurements at specific points:

$x(\mathrm{~m})$	0	2	4	6	8	10
$h(x)(\mathrm{m})$	0	0.88	1.92	2.52	2.56	2.00

Find $\frac{d h}{d x}$ and determine the maximum height of the roof.

[4]

Use the trapezoidal rule with 5 intervals to estimate the cross-sectional area under the roof.

[3]

Sketch the graph of $h(x)$ from $x=0$ to $x=10$ and shade the region corresponding to the estimate in part (b).

[2]

Write down the integral representing the exact cross-sectional area and evaluate it.

[3]

Calculate the percentage error of the trapezoidal rule estimate compared to the exact area.

[2]

Explain why the trapezoidal rule estimate differs from the exact area, referencing the shape of the curve.

[2]

Question 3

SLPaper 2

A water tank's cross-sectional area, $A(h)$ , in $\mathrm{m}^{2}$ , varies with height $h$ , in meters. The volume is $\int_{0}^{8} A(h) d h$ . Assume $A(h)=a+b h+c h^{2}$ , with measurements:

$h(\mathrm{~m})$	0	2	4	6	8
$A(h)\left(\mathrm{m}^{2}\right)$	20	18	18	20	24

Use the trapezoidal rule with 4 intervals to estimate the volume.

[3]

Sketch the graph of $A(h)$ and shade the region used in part (a).

[2]

Use the points to form a system of equations for $a, b, c$ . Solve for one coefficient.

[2]

Find the exact volume by integrating the quadratic model, assuming $A(h)=24- 0.5 h^{2}+h .

[3]

Calculate the percentage error of the trapezoidal estimate.

[2]

If water flows in at $2 \mathrm{~m}^{3} / \mathrm{min}$ , estimate the filling time using the trapezoidal volume.

[2]

Explain how the number of intervals affects the trapezoidal rule's accuracy.

[2]

Question 4

HLPaper 1

A predator-prey system models the populations of rabbits ( $R$ , in thousands) and foxes ( $F$ , in hundreds) in a forest, given by:

$\frac{\mathrm{d} R}{\mathrm{dt}}=0.4 R-0.02 R F, \quad \frac{\mathrm{~d} F}{\mathrm{dt}}=0.01 R F-0.3 F,$

where $t$ is in years. Initially, $R=10, F=5$ .

Find the non-zero equilibrium point for the system.

[4]

Interpret the equilibrium point in the context of the model.

[2]

Use Euler's method with a step size of 0.2 years to estimate $R$ and $F$ after 1 year.

[5]

Sketch the phase portrait trajectory for $8 \leq R \leq 12,4 \leq F \leq 6$ , over the first 2 years.

[3]

Question 5

HLPaper 1

Given that $\int_{ - 2}^2 {g\left( y \right){\text{d}}y = 10}$ and $\int_0^2 {g\left( y \right){\text{d}}y = 12}$ , find

$\int_{ - 2}^0 {\left( {g\left( y \right){\text{ + 2}}} \right){\text{d}}y}$ .

$\int_{ - 2}^0 {g\left( {y{\text{ + 2}}} \right){\text{d}}y}$ .

Question 6

HLPaper 1

The wind chill index $W$ is a measure of the temperature, in $°C$ , felt when taking into account the effect of the wind.

When Frieda arrives at the top of a hill, the relationship between the wind chill index and the speed of the wind $v$ in kilometres per hour $(km h^{-1})$ is given by the equation

W = 19.34 - 7.405v^{0.16}

Find an expression for $\frac{dW}{dv}$ .

[2]

When Frieda arrives at the top of a hill, the speed of the wind is $10$ kilometres per hour and increasing at a rate of $5$ $\text{km h}^{-1} \text{ minute}^{-1}$ .

Find the rate of change of $W$ at this time.

[5]

Question 7

SLPaper 1

A local bakery is analyzing the production of its signature pastry, which is modeled by the function $f(x) = x^3 - 6x^2 + 9x + 2$ , where $x$ represents the number of hours spent baking.

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the pastry production with respect to time.

[3]

Find the critical points of the function by solving $f^{\prime}(x) = 0$ . These points will help the bakery understand when the production rate is at a maximum or minimum.

[4]

Determine whether the function is increasing or decreasing on the intervals defined by the critical points. This will inform the bakery about the efficiency of production over time.

[4]

Identify any local maximum or minimum points and justify your answer. This analysis will help the bakery optimize its production schedule.

[4]

Calculus Questionbank