IB Maths AA SL · Unit 5: Calculus
IB Maths AA SL Applications of Differentiation Questions
Exam-style IB Maths AA SL applications of differentiation questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 41 questions
- Paper 1: 21
- Paper 2: 20
- 4 easy
- 6 medium
- 22 hard
- 3 very hard
- 6 starter
- 3 worked examples
Practise Applications of Differentiation questions →
AA SL formula booklet
What's examined in AA SL applications of differentiation
The question bank covers these applications of differentiation question types (number of questions in brackets):
- Optimisation Problems (21)
- Tangents, Normals & Extrema (16)
- Rates of Change (4)
Key formulas
- Chain rule
- \(\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}\)
- Product rule
- \((uv)' = u'v + uv'\)
- Quotient rule
- \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}\)
- Derivative of x^n
- \(\dfrac{d}{dx}\bigl(x^n\bigr) = n x^{\,n-1}\)
- Derivatives of standard functions
- \(\dfrac{d}{dx}(\sin x) = \cos x,\ \dfrac{d}{dx}(\cos x) = -\sin x,\ \dfrac{d}{dx}(\tan x) = \sec^2 x\)
- Exponential & log derivatives
- \(\dfrac{d}{dx}(e^x) = e^x,\ \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}\)
In the same notation as the IB formula booklet. All AA SL formulas →
Applications of Differentiation worked examples
Worked example 1: Finding the equation of a tangent · easy
The equation of a curve is given by $f(x) = x^2 - 4x + 5$. Find the exact gradient of the curve at $x = 3$, and hence find the equation of the tangent to the curve at $x = 3$. Give your answer in the form $y = mx + c$.
1. Differentiate the function to find the gradient function: $f'(x) = 2x - 4$.
2. Substitute $x = 3$ into the derivative to find the tangent gradient: $m_t = f'(3) = 2(3) - 4 = 2$.
3. Calculate the $y$-coordinate at the point of tangency: $f(3) = 3^2 - 4(3) + 5 = 9 - 12 + 5 = 2$. The point is $(3, 2)$.
4. Substitute the gradient and coordinates into the point-slope formula: $y - y_1 = m_t(x - x_1) \implies y - 2 = 2(x - 3)$.
5. Expand and rearrange into the required form: $y = 2x - 6 + 2$.
6. State the final equation: $\mathbf{y = 2x - 4}$.
Examiner tip: A common error is substituting the $x$-coordinate into the original function $f(x)$ when looking for the gradient. Always remember: substitute into $f'(x)$ for the gradient, and $f(x)$ for the $y$-coordinate.
Worked example 2: Horizontal tangents on a cubic · medium
The curve $C$ is given by the equation $y = -x^3 + 6x^2 - 5x + 4$. The tangent to the curve is parallel to the $x$-axis at two specific points. Find the exact $x$-coordinates of these two points.
1. Recognise that a tangent parallel to the $x$-axis is horizontal, so $\frac{dy}{dx} = 0$.
2. Differentiate: $\frac{dy}{dx} = -3x^2 + 12x - 5$.
3. Set the derivative to zero: $-3x^2 + 12x - 5 = 0$.
4. Multiply by $-1$ (optional): $3x^2 - 12x + 5 = 0$.
5. Apply the quadratic formula: $x = \frac{12 \pm \sqrt{(-12)^2 - 4(3)(5)}}{2(3)} = \frac{12 \pm \sqrt{84}}{6}$.
6. Simplify the surd: $x = \frac{12 \pm 2\sqrt{21}}{6} = \mathbf{\frac{6 \pm \sqrt{21}}{3}}$.
Examiner tip: "Parallel to the $x$-axis", "horizontal tangent", and "stationary point" all mean exactly the same thing mathematically: set the first derivative equal to zero.
Worked example 3: Optimisation of an open-topped box · hard
An open-topped box is made from a $20\text{ cm} \times 20\text{ cm}$ square piece of cardboard by cutting out identical squares of side length $x\text{ cm}$ from each corner and folding up the sides. Write an expression for the volume $V(x)$, then use your GDC to find the value of $x$ that maximises the volume and the maximum volume itself, to 3 s.f.
1. Determine the folded dimensions: base is $(20 - 2x)$ by $(20 - 2x)$, height is $x$.
2. Construct the volume function: $V(x) = x(20 - 2x)^2$.
3. Identify the practical domain: $x > 0$ and $20 - 2x > 0$, so $0 < x < 10$.
4. Graph $V(x)$ on the GDC (CG50: MENU 5, Y1 = x(20 - 2x)^2).
5. Use the maximum tool (G-Solv $\to$ MAX) on the valid domain.
6. State: $\mathbf{x = 3.33\text{ cm}}$ (3 s.f.) and maximum volume $\mathbf{593\text{ cm}^3}$ (3 s.f.).
Examiner tip: In Paper 2 optimisation questions do NOT differentiate manually unless the question says "Use calculus to...". Graph and use G-Solv — much faster with less risk of algebra error.
Try these IB Maths AA SL applications of differentiation questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 2
Let \(y = 0.5x^3 - 2x + 1\).
Use the numerical derivative feature on your graphic display calculator to evaluate the gradient of the curve at \(x = 2\).
Hence, find the equation of the normal to the curve at the point where \(x = 2\), giving your answer in the form \(y = mx + c\).
Question 2 · medium · 5 marks · Paper 1
Consider the curve \(y = 2x^2 - 5x\).
Find the gradient of the tangent to the curve at \(x = 1\).
Find the gradient of the normal to the curve at \(x = 1\).
Find the equation of the normal line at \(x = 1\), giving your answer in the form \(ax + by + d = 0\), where \(a, b,\) and \(d\) are integers.
Question 3 · hard · 5 marks · Paper 1
The equation of a curve is \(y = x^3 - 4x^2 + kx\), where \(k\) is a constant. The equation of the tangent to the curve at the point where \(x = 2\) is \(y = -3x\).
State the gradient of the tangent line.
Find \(\frac{dy}{dx}\) in terms of \(x\) and \(k\).
By substituting \(x = 2\) into your derivative, find the exact value of \(k\).
All 41 applications of differentiation questions with mark schemes →
FAQ
How many IB Maths AA SL applications of differentiation questions are there?
There are 41 exam-style applications of differentiation questions in the AA SL question bank (Paper 1: 21 · Paper 2: 20), graded 4 easy, 6 medium, 22 hard, 3 very hard, 6 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is applications of differentiation on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 21 · Paper 2: 20. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA SL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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