Calculus practice questions (IB Maths AI)

Sample questions

The one free IB Maths AI question on this topic, from the free IB Maths AI SL Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 110 marks · Calculus: turning points & trapezoidal rule
   A garden designer models the width of a flower bed. One edge of the bed is a straight path along the x-axis, and the other edge follows the curve $y = w(x)$, where $w(x) = 0.1x^3 - 0.9x^2 + 2.4x$ for $0 \leq x \leq 5$, with $x$ and $w(x)$ in metres. The bed ends with a straight edge at $x = 5$.
   $w(x) = 0.1x^3 - 0.9x^2 + 2.4x, \qquad 0 \leq x \leq 5$
   (a)

   Find $w'(x)$.

   [2]
   (b)

   Find the coordinates of the local maximum point and the local minimum point of the curve.

   [4]
   (c)

   Use the trapezoidal rule with five intervals of width 1 m to estimate the area of the flower bed.

   [3]
   (d)

   Use your GDC to find the actual area of the bed, and state whether the trapezoidal rule gives an underestimate or an overestimate.

   [1]
   Show answerFrom the free IB Maths AI SL Practice Exam →

Key terms

The definitions and formulas IB Maths AI expects for calculus, typeset as they appear on the paper.

Derivative as a rate of change

The gradient of the curve at a point, found by the power rule for polynomial terms. Increasing where $f'(x) > 0$, decreasing where $f'(x) < 0$.

$$\frac{d}{dx}\left(ax^n\right) = anx^{n-1}$$

Tangent and normal

The tangent at $x = a$ has gradient $f'(a)$; the normal is perpendicular, so its gradient is $-\frac{1}{f'(a)}$. Both pass through the point $(a, f(a))$.

$$y - f(a) = f'(a)(x - a)$$

Turning points and optimisation

Local maxima and minima occur where $f'(x) = 0$. In a modelling question (maximum volume, minimum cost) write the quantity as a function of one variable, differentiate, set to zero, and check the nature with the GDC or the sign of $f'$.

$$f'(x) = 0$$

Anti-differentiation with a boundary condition

Reversing differentiation gives a family of curves differing by a constant $c$; a known point on the curve fixes $c$.

$$\int ax^n\,dx = \frac{a x^{n+1}}{n+1} + c, \quad n \ne -1$$

Definite integral and area

The area between a curve and the $x$-axis from $a$ to $b$, evaluated on the GDC. Where the curve dips below the axis, split the integral or integrate the absolute value.

$$A = \int_a^b f(x)\,dx$$

Trapezoidal rule

Approximates a definite integral with $n$ trapezoids of equal width $h$. The approximation is an overestimate where the curve is concave up and an underestimate where it is concave down.

$$\int_a^b y\,dx \approx \frac{h}{2}\Big(y_0 + y_n + 2(y_1 + y_2 + \cdots + y_{n-1})\Big), \qquad h = \frac{b - a}{n}$$

Euler's method (HL)

A numerical step-by-step solution of a differential equation from a starting point, with step length $h$. Smaller steps give better accuracy at the cost of more steps.

$$y_{n+1} = y_n + h\,f(x_n, y_n), \qquad x_{n+1} = x_n + h$$

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