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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 2

A function $f$ is defined for $-2 \leq x \leq 2$ , with the behavior of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ given below:

$x$	$-2<x<-1$	$x=-1$	$-1<x<1$	$1<x<2$
$f^{\prime}(x)$	Positive	0	Negative	Negative
$f^{\prime \prime}(x)$	Negative	0	Positive	Negative

Identify the x -coordinates of the local extrema of $f$ .

[3]

Determine the intervals where $f$ is concave up.

[2]

Given $f(-2)=3$ , sketch the graph of $f$ , indicating the local extrema and points of inflection.

[4]

Question 5

SLPaper 1

A roof's height, $y(x)$ , in meters, is modeled by $y(x)=-0.1 x^{2}+1.2 x$ , for $0 \leq x \leq 12$ , where $x$ is the horizontal distance from one end. The area under the curve represents the cross-sectional area of the roof space.

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

[3]

Use the trapezoidal rule with 4 intervals to estimate the cross-sectional area.

[3]

Compare the estimate from part (b) with the exact area found by integration.

[3]

Question 6

HLPaper 1

A function $k(x)$ has derivative $k^{\prime}(x)=\frac{3 \cos x}{\sin ^{2} x+1}$ . The graph of $k$ passes through the point $\left(\frac{\pi}{2}, 0\right)$ .

Find $k(x)$ .

[5]

Determine the x -coordinate of the point where the graph of $k$ has a horizontal tangent in the interval $0 \leq x \leq \pi$ .

[2]

Question 7

SLPaper 1

The height of a projectile, $h(t)$ meters, launched at time $t$ seconds is modeled by $h(t)= -4.9 t^{2}+19.6 t+2$ , for $t \geq 0$ .

Find $h^{\prime}(t)$ .

[2]

Determine the time when the projectile reaches its maximum height.

[2]

Find the maximum height and verify it is a maximum using the second derivative.

[4]

Sketch the graph of $h(t)$ for $0 \leq t \leq 5$ , showing the maximum point and intercepts.

[4]

Question 8

HLPaper 2

The velocity of a particle moving in a straight line is given by $v(t)=t^{2} e^{-0.5 t}$ , for $t \geq 0$ , where $t$ is time in seconds and $v$ is in meters per second.

Find the acceleration $a(t)$ .

[3]

Find the time $t$ when the acceleration is zero, for $0<t<10$ .

[3]

Sketch the graph of $a(t)$ for $0 \leq t \leq 10$ , indicating the point where $a(t)=0$ . marks]

[3]

Question 9

HLPaper 2

The slope field for $\frac{d y}{d x}=\frac{y^{2}-x^{2}}{y+x}$ , for $-2 \leq x \leq 2,-2 \leq y \leq 2$ , excluding $x+y=0$ , is shown below.

Find the equation of the curve where $\frac{d y}{d x}=0$ .

[2]

Sketch the curve from part (a) on the slope field.

[2]

Sketch the solution curve through $(1,1)$ .

[2]

Use Euler's method with step size 0.1 to approximate $y$ at $x=1.2$ .

[3]

Determine the coordinates of the intersection of the solution curve through $(1,1)$ with the curve from part (a).

[3]

Analyze the behavior of the solution curve as $x \rightarrow 2$ .

[4]

Question 10

SLPaper 2

A sculpture's height, $s(t)$ , in meters, is modeled by a cubic function over a 9 -meter base, where $t$ is the distance from one end. Heights are:

$t(\mathrm{~m})$	0	3	6	9
$s(t)(\mathrm{m})$	1	2.5	3	2

Use the trapezoidal rule with 3 intervals to estimate the area under the curve.

[3]

Assume $s(t)=a t^{3}+b t^{2}+c t+d$ . Use the given points to find two linear equations in $a, b, c, d$ .

[2]

Sketch the graph of $s(t)$ and shade the region used in part (a).

[2]

Calculus Questionbank