Calculus practice questions (IB Maths AA)

Sample questions

All 2 free IB Maths AA questions on this topic, from the free IB Maths AA SL Paper 1 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 16 marks · Differentiation: tangents & normals
   The curve $y = f(x)$ where $f(x) = x^3 - 3x + 1$ passes through the point $P(2, 3)$.
   $f(x) = x^3 - 3x + 1$
   (a)

   Find $f'(x)$.

   [2]
   (b)

   Find the equation of the tangent to the curve at $P$.

   [2]
   (c)

   Find the equation of the normal to the curve at $P$.

   [2]
   Show answerFrom the free IB Maths AA SL Paper 1 Practice Exam →
2. Question 214 marks · Calculus of a cubic curve
   The function $f$ is defined by $f(x) = x^3 - 6x^2 + 9x$ for $x \in \mathbb{R}$. The curve $y = f(x)$ is studied throughout this question.
   $f(x) = x^3 - 6x^2 + 9x$
   (a)

   Find $f'(x)$, and hence find the coordinates of the two stationary points of the curve.

   [4]
   (b)

   Justify the nature of each stationary point.

   [2]
   (c)

   Find $f''(x)$, and hence find the coordinates of the point of inflexion.

   [2]
   (d)

   Find the exact area of the region enclosed by the curve and the $x$-axis.

   [4]
   (e)

   Find the equation of the tangent to the curve at the point of inflexion.

   [2]
   Show answerFrom the free IB Maths AA SL Paper 1 Practice Exam →

Key terms

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

Derivative as a limit

The gradient of the tangent, defined as the limit of the gradient of a chord as the chord shrinks. "From first principles" questions expect this definition written out.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Rules of differentiation

Power, chain, product and quotient rules, all in the formula booklet. Most Paper 1 errors come from choosing the wrong rule for a quotient or forgetting the inner derivative in the chain rule.

$$\frac{d}{dx}x^n = nx^{n-1}, \quad \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}, \quad (uv)' = u'v + uv', \quad \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$

Tangent and normal

The tangent at $x = a$ has gradient $f'(a)$; the normal is perpendicular to it, so its gradient is the negative reciprocal.

$$y - f(a) = f'(a)(x - a), \qquad m_{\text{normal}} = -\frac{1}{f'(a)}$$

Stationary points and the second derivative

Stationary points occur where $f'(x) = 0$. The second derivative classifies them: positive means a local minimum, negative a local maximum; a point of inflexion is where $f''$ changes sign.

$$f'(x) = 0, \qquad f''(x) > 0 \Rightarrow \text{minimum}, \quad f''(x) < 0 \Rightarrow \text{maximum}$$

Definite integral and area

The definite integral is the difference of the antiderivative at the limits. Area between two curves integrates the upper minus the lower function between their intersections; area below the axis needs the sign handled.

$$\int_a^b f(x)\,dx = F(b) - F(a), \qquad A = \int_a^b \big(f(x) - g(x)\big)\,dx$$

Kinematics

Velocity is the derivative of displacement and acceleration the derivative of velocity. Displacement is the integral of velocity; total distance integrates the speed $|v|$.

$$v = \frac{ds}{dt}, \quad a = \frac{dv}{dt}, \quad s = \int v\,dt, \quad \text{distance} = \int |v|\,dt$$

Integration by substitution and by parts (HL)

Substitution reverses the chain rule; integration by parts reverses the product rule and is needed for products such as $x e^x$ or $x \ln x$.

$$\int f\big(g(x)\big)g'(x)\,dx = \int f(u)\,du, \qquad \int u\,dv = uv - \int v\,du$$

Frequently asked questions

Other IB Maths AA topics

• Number and algebra
• Functions
• Geometry and trigonometry
• Statistics and probability

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