Calculus (AI SL) — IB Math AI SL AI SL Study Guide

For: IB Math AI SL candidates sitting IB Math: Applications & Interpretation SL.

Covers: All core IB AI SL calculus subtopics: gradient approximation with technology, contextual optimisation, trapezoidal rule for area under curves, and calculator-derived definite integrals, with exam-focused worked examples and common mistake guidance.

You should already know: IGCSE / pre-DP math.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Math AI SL style for educational use. They are not reproductions of past IBO papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official IBO mark schemes for grading conventions.

1. What Is Calculus (AI SL)?

Calculus is the branch of mathematics focused on measuring rates of change (differential calculus) and accumulated quantities (integral calculus). For IB Math AI SL, the focus is entirely on applied, technology-enabled calculus, rather than abstract analytical computation: you will use your graphic display calculator (GDC) for most numerical calculations, with marks awarded for correct interpretation of results and application to real-world contexts. This topic makes up 15-20% of your total exam marks, and is tested heavily on Paper 2 (calculator-allowed) with occasional non-calculator trapezoidal rule questions on Paper 1.

2. Approximating gradient with technology

The gradient of a non-linear curve at a point is equal to the gradient of the tangent line drawn to the curve at that point. Unlike linear functions, you cannot calculate this value manually for all non-linear functions quickly, so your GDC is designed to approximate this value accurately. To calculate the gradient at a point $x=a$ on most standard GDC models (TI-84 Plus CE, Casio fx-CG50):

1. Input the function $y=f(x)$ into your graphing menu
2. Navigate to the dy/dx (gradient) tool in the graph analysis menu
3. Enter the $x$-value $a$ you are evaluating, and the GDC will return the approximate gradient

Worked Example

Find the gradient of $f(x)=0.5x^3-2x+1$ at $x=2$.

1. Graph the function, select the dy/dx tool, and input $x=2$
2. The GDC will return a value very close to 4.00 (the exact analytical value is 4, small decimal discrepancies from approximation are acceptable)
3. Round your final answer to 3 significant figures: 4.00

Examiner tip: You do not need to show analytical differentiation working if the question explicitly states to use technology. Just state that you used the GDC gradient tool, and give the final rounded answer.

3. Optimisation in context

Optimisation is the process of finding the maximum or minimum value of a function (called the objective function) for a given real-world scenario, for example maximum profit for a business, minimum material required to build a container, or maximum height of a projectile. Follow these standard steps for all AI SL optimisation questions:

1. Identify the objective function you need to optimise, and any variables given
2. Note the valid domain of the independent variable (e.g. number of units sold cannot be negative, cannot exceed maximum production capacity)
3. Use your GDC to find critical points (points where the gradient of the function is 0) and evaluate the objective function at all critical points and the endpoints of the domain
4. Select the maximum or minimum value from the valid results
5. Interpret the result in context, including correct units

Worked Example

A café's daily profit function is $P(x)=-0.08x^2+16x-300$, where $x$ is the number of iced coffees sold per day, with a valid domain $0\le x\le 180$. Find the maximum daily profit, and the number of iced coffees required to reach it.

1. Graph $P(x)$ on your GDC, use the maximum point tool to find the vertex of the parabola at $x=100$, $P=500$
2. Verify $x=100$ is within the domain $0\le x\le 180$
3. Check endpoints to confirm: $P(0)=-300$, $P(180)=252$, so $x=100$ is indeed the maximum
4. Final answer: Maximum profit of $500 when 100 iced coffees are sold

Examiner tip: You will lose 1 mark if you omit units from your final answer, or give an $x$-value that falls outside the given domain, even if your calculation is technically correct.

4. Trapezoidal rule for area

The trapezoidal rule is a numerical method to approximate the area under a curve between two points $x=a$ and $x=b$, used when you cannot calculate an exact integral analytically, or when you are given discrete data points instead of a continuous function. The rule works by splitting the area under the curve into $n$ equal-width trapezoids, calculating the area of each trapezoid, and summing the results. The formula is: $$A\approx \frac{h}{2}\left[y_0+2(y_1+y_2+...+y_{n-1})+y_n\right]$$ Where:

• $h=\frac{b-a}{n}$ is the width of each strip
• $y_k$ is the value of the function at $x=a+kh$ for $k=0,1,...,n$

Worked Example

Use the trapezoidal rule with 4 strips to approximate the area under $f(x)=\sqrt{x+1}$ between $x=0$ and $x=4$.

1. Calculate $h=\frac{4-0}{4}=1$
2. Calculate the $y$-values: $y_0=1$, $y_1=\sqrt{2}\approx 1.414$, $y_2=\sqrt{3}\approx 1.732$, $y_3=2$, $y_4=\sqrt{5}\approx 2.236$
3. Substitute into the formula: $$A\approx \frac{1}{2}\left[1+2(1.414+1.732+2)+2.236\right]=0.5\times 13.528=6.76$$
4. Final answer: 6.76 square units (3 sig figs)

Examiner tip: The trapezoidal rule overestimates area if the curve is concave up, and underestimates area if the curve is concave down. Examiners often ask you to state this relationship for 1 extra mark.

5. Definite integrals from calculator

The definite integral of $f(x)$ between $x=a$ and $x=b$ is the exact net area under the curve between those points: area above the $x$-axis is counted as positive, area below the $x$-axis is counted as negative. For AI SL, you are almost never required to calculate definite integrals analytically (by finding antiderivatives) unless the function is extremely simple: instead, you use your GDC's built-in integral tool. To calculate a definite integral on most GDC models:

1. Navigate to the ∫f(x)dx tool in the calculation menu
2. Input the function, lower limit $a$, upper limit $b$, and the variable $x$
3. The GDC will return the exact net integral value

Worked Example

Find the definite integral of $f(x)=\sin (x)+0.5x$ between $x=0$ and $x=\pi$.

1. Ensure your GDC is in radian mode
2. Input the integral ${\int }_0^{\pi }(\sin (x)+0.5x)dx$ into your GDC
3. The GDC will return ~4.47, which matches the analytical value of $2+\frac{{\pi }^2}{4}\approx 4.467$
4. Final answer: 4.47 (3 sig figs)

Examiner tip: If the question asks for total area (not net integral value), you must split the integral at points where $f(x)$ crosses the $x$-axis, take the absolute value of each segment, or calculate the integral of $|f(x)|$ on your GDC to avoid subtracting area below the axis.

6. Common Pitfalls (and how to avoid them)

• Wrong move: Rounding intermediate gradient or integral values to 2 significant figures early, leading to a final answer outside the examiner's acceptable error range. Why it happens: Students think rounding early saves time. Correct move: Store all intermediate values in your GDC's memory, only round the final answer to 3 significant figures (or the number specified in the question).
• Wrong move: Forgetting to check the domain of the objective function in optimisation questions, giving a maximum/minimum that is not practically possible. Why it happens: Students only find the critical point where $dy/dx=0$, ignoring context-based limits on the independent variable. Correct move: Always evaluate the objective function at the endpoints of the domain as well as critical points, then select the maximum/minimum from valid results.
• Wrong move: Forgetting to multiply the sum of the middle $y$-values by 2 in the trapezoidal rule formula. Why it happens: Students mix up the rule with Riemann sum formulas where all $y$-values are multiplied by the same coefficient. Correct move: Memorize the structure: first and last $y$-values are counted once, all middle values are counted twice, multiply the total sum by $h/2$.
• Wrong move: Using degree mode on your GDC when calculating integrals or gradients of trigonometric functions. Why it happens: Students forget to switch back from degree mode after solving geometry or triangle questions. Correct move: Check your GDC's mode indicator before starting any calculus question; 99% of IB AI SL calculus questions use radians unless explicitly stated otherwise.
• Wrong move: Confusing net integral value with total area under a curve. Why it happens: Students assume the definite integral gives total area, rather than net signed area. Correct move: If the question asks for total area, graph the function first to find $x$-intercepts between the limits, split the integral at those points, and take the absolute value of each segment before summing.

7. Practice Questions (IB Math AI SL Style)

Question 1

A clothing brand's monthly revenue function is $R(x)=12x-0.005x^2$, where $x$ is the number of t-shirts sold per month, with valid domain $0\le x\le 2000$. (a) Use your GDC to find the gradient of $R(x)$ when 500 t-shirts are sold. (2 marks) (b) Find the number of t-shirts the brand needs to sell to maximise monthly revenue, and the maximum revenue value. (3 marks)

Solution 1

(a) Step 1: Graph $R(x)$ on your GDC, use the dy/dx tool at $x=500$. The exact gradient is $12-0.01(500)=7$. Final answer: 7 dollars per t-shirt. (2 marks: 1 for correct GDC method, 1 for correct answer with units) (b) Step 1: Find critical point where $dy/dx=0$: $12-0.01x=0\to x=1200$. Step 2: Verify $x=1200$ is within the domain $0\le x\le 2000$. Step 3: Calculate $R(1200)=12(1200)-0.005(1200{)}^2=7200$. Step 4: Check endpoints: $R(0)=0$, $R(2000)=4000$, so $x=1200$ is the maximum. Final answer: Maximum revenue of $7200 when 1200 t-shirts are sold. (3 marks: 1 for valid critical point, 1 for domain validation, 1 for correct answer with units)

Question 2

The speed of a runner over an 8-second period is recorded in the table below:

Solution 2

Step 1: Identify values: $n=4$ strips, $h=2$, $y_0=0$, $y_1=2.8$, $y_2=4.1$, $y_3=3.7$, $y_4=1.2$. Step 2: Substitute into formula: $A\approx \frac{2}{2}\left[0+2(2.8+4.1+3.7)+1.2\right]=1\times (0+21.2+1.2)=22.4$. Final answer: 22.4 meters. (3 marks: 1 for correct $h$ and $y$ values, 1 for correct formula substitution, 1 for correct answer with units)

Question 3

Use your GDC to calculate the following: (a) The definite integral of $f(x)=e^{-0.1x}+2x$ between $x=2$ and $x=6$. (2 marks) (b) The total area under $f(x)=x^2-9$ between $x=0$ and $x=4$. (3 marks)

Solution 3

(a) Input ${\int }_2^6(e^{-0.1x}+2x)dx$ into your GDC. Final answer: 31.0 (3 sig figs). (2 marks: 1 for correct GDC input, 1 for correct answer) (b) Step 1: Find $x$-intercept of $f(x)$ between 0 and 4: $x^2-9=0\to x=3$. Step 2: Calculate total area as $|{\int }_0^3(x^2-9)dx|+{\int }_3^4(x^2-9)dx=18+\frac{10}{3}\approx 21.3$. Final answer: 21.3 (3 sig figs). (3 marks: 1 for identifying $x=3$ as intercept, 1 for splitting integral/taking absolute value, 1 for correct answer)

8. Quick Reference Cheatsheet

9. What's Next

This calculus foundation connects directly to later core AI SL topics including kinematics (calculating speed, distance and acceleration from motion functions), continuous probability distributions (calculating probabilities from density functions using integrals), and financial modelling (optimising investment returns and accumulated value of recurring payments). It is also one of the highest-weighted skills for Paper 2, where 30-40% of questions involve technology-enabled calculus applications, so mastering these skills will directly boost your overall exam score.

If you struggle with any of the steps in this guide, from GDC operation to trapezoidal rule substitution, you can ask Ollie, our AI tutor, for personalized walkthroughs, extra practice questions, or clarification on any IB Math AI SL topic. You can also find more topic-specific study guides and full mock exams on the homepage to test your knowledge before your official exam.