IB DP Mathematics: Applications and Interpretation (AI) — SL and HL
GDC Bootcamp — TI-Nspire CX II Essentials
IB DP Mathematics: Applications and Interpretation (AI) — SL and HL
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Start Here — How This Course Works
Read this first: how to use the five mock sets, exam rules for AI SL (both papers use the GDC), how to self-mark with the IB-style keys, and your 10-day revision plan.
The GDC is allowed in BOTH AI SL papers — fluency is marks. Press-to-Test exam mode, and the core TI-Nspire CX II skills used throughout the marking keys: nSolve, graphing and intersections, regression, financial solver (TVM), statistics and distribution menus.
Full step-by-step notes with Prof. Kitso for AI 1.1, 1.6: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 1.5: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Learn 1 · Arithmetic & geometric sequences and series (AI 1.2, 1.3) — SL & HL
Full step-by-step notes with Prof. Kitso for AI 1.2, 1.3: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 1.4, 1.7: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 1.8: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
HL · Learn 1 · Laws of logarithms, rational exponents, infinite series & log scales (AI 1.9–1.11, 2.10) — HL
Full step-by-step notes with Prof. Kitso for AI 1.9–1.11, 2.10: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 1.12, 1.13: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 1.14, 1.15: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 2.1–2.4: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 2.5, 2.6: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 2.7–2.9: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 3.1–3.3: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 3.4–3.6: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 3.7–3.9: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 3.10–3.13: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 3.14–3.16: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Step-by-step teaching notes from the very beginning: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL 4.1–4.4 (HL students study these too).
Full step-by-step notes with Prof. Kitso for AI 4.5, 4.6: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Learn 4 · Probability distributions: discrete, binomial & normal (AI 4.7–4.9) — SL & HL
Full step-by-step notes with Prof. Kitso for AI 4.7–4.9: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 4.10, 4.11: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 4.14, 4.15, 4.17, 4.19: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 4.12, 4.13, 4.16, 4.18: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 5.1–5.4: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 5.6, 5.7: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Learn 5 · Integration, area & the trapezoidal rule (AI 5.5, 5.8) — SL & HL
Full step-by-step notes with Prof. Kitso for AI 5.5, 5.8: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. SL and HL students.
Full step-by-step notes with Prof. Kitso for AI 5.9, 5.10: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 5.11–5.13: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full step-by-step notes with Prof. Kitso for AI 5.14–5.18: every idea explained simply, worked examples with TI-Nspire CX II screens, exam-style challenge questions and try-it-yourself activities. HL only.
Full mock: Paper 1 (short questions, 80 marks, 90 min) and Paper 2 (extended questions, 80 marks, 90 min), both with the GDC, each with a full IB-style marking key. Paper 1 and its key are FREE to preview. Sit the paper under timed conditions before opening the key.
Second full mock (Paper 1 + Paper 2 with marking keys) covering a fresh rotation of the AI SL syllabus. Re-drill anything you dropped marks on in Set 1 before you sit it.
Third full mock (Paper 1 + Paper 2 with marking keys). By the end of this set you should have met every AI SL topic at least once — use your score log to find remaining weak areas.
Fourth full mock (Paper 1 + Paper 2 with marking keys). Sit both papers back-to-back with only a short break — train the stamina the real exam session demands.
Final full mock (Paper 1 + Paper 2 with marking keys). Treat this as your dress rehearsal: full exam conditions, formula booklet only, and strict timing.
What to do (and not do) in the last two days before each exam: a 48-hour checklist, formula booklet final pass, GDC battery-and-settings check, and the exam-day routine.
Module 2 of 47Lesson 1 of 1· GDC Bootcamp — TI-Nspire CX II Essentials
AI SL & HL · GDC BOOTCAMP
TI-Nspire CX II — fluency is marks
Every AI paper is a calculator paper (SL: Papers 1 and 2; HL: Papers 1, 2 and 3) — GDC fluency is worth real marks in every question.
Paper 1: GDC ✓Paper 2: GDC ✓
🎯 What “Analyze Graph” earns you
Menu → 6: Analyze Graph → 4: Intersection. Read BOTH coordinates, quote them to 3 sf, and mark the point on your sketch — that is usually M1 A1 in one move.
Loans, investments, depreciation. Money out = negative. Write N, I%, PV, PMT, FV in your working.
4. Statistics Lists & Spreadsheet
1-var stats, regression (linear, exponential…), r and r². Always plot before you trust a model.
5. Distributions menu → 5: Probability → 5: Distributions
Normal cdf/inverse, binomial, Poisson. Draw the shaded curve in your answer for the R mark.
6. χ² & t-tests menu → 6: Statistics → 7: Stat Tests
GOF and independence tests, 2-sample t. Write hypotheses, p-value, and the comparison to α.
7. Trapezoidal check by hand + GDC
The trapezoidal rule is asked ON PAPER — but verify your answer with a definite integral on the GDC.
🔒 Press-to-Test — do this the night before
Charge to 100% and pack a spare-battery plan.
Back up any documents you care about — exam mode clears them.
Hold esc + home while powering on → choose your exam restrictions → confirm the flashing LED.
Verify the clock/angle settings: degrees for AI SL topics in degrees, radians for AHL sinusoidal work — know where the setting lives.
Habit: after every mock, list any GDC step that cost you more than 60 seconds — drill exactly those before the next set.
📸 TI-Nspire Step-by-Step: Worked Examples
In AI (SL and HL) the GDC is allowed on every paper — the exam is written assuming you use it constantly. Examples 1–10 cover the GDC skills every student needs; Examples 11–20 add the Higher Level skills. Every screen below is drawn exactly as it appears on the TI-Nspire CX II. Follow the key presses, copy the screens, then practise until your fingers know the menus without looking.
💻 Student Software & exam mode. Every screen below is drawn exactly as the TI-Nspire CX Student Software shows it, and every step works on the non-CAS calculator in IB exam mode (Press-to-Test). On a laptop: click menu on the on-screen keypad (same numbers as here); Enter = enter, Tab = next box; after ^ press → to leave the power; new page Ctrl+I; change page Ctrl+←/→; var Ctrl+L. Exam mode has no dot-product, cross-product or eigenvalue keys and no Diff Eq graphs — these examples show the allowed way.
EXAMPLE 1EXPONENTIAL MODELSPaper 1 & 2
📝 EXAM-STYLE QUESTION
The population of a town is modelled by P = 12500e0.032t, where t is in years. Find how long it takes the population to reach 20 000. [3 marks]
STEP 1 — Open Numerical Solve
menu3: Algebra1: Numerical Solve On a Calculator page, this gives you nSolve( ) — your single most-used exam tool.
The Algebra menu — choose 1Type the equation, then the variableThe answer appears instantly
STEP 2 — Read & round
The GDC gives 14.6876. The question wants 3 significant figures → 14.7 years.
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
12500e0.032t = 20000 M1 t = 14.6876… A1 t ≈ 14.7 years (3 sf) A1
⚠️ EXAMINER TRAP
Never write only “14.7”. Write the equation you solved first — if you mistype on the GDC, the M1 method mark survives. And keep the unrounded value on screen for part (b).
EXAMPLE 2LINEAR REGRESSIONPaper 2
📝 EXAM-STYLE QUESTION
Hours studied (x) and test score (y) were recorded for 8 students: (2, 34), (3, 41), (5, 50), (6, 58), (8, 63), (9, 71), (11, 80), (12, 84). Find the equation of the regression line y on x, and the value of r. [4 marks]
STEP 1 — Enter the data in Lists & Spreadsheet
ctrldoc4: Add Lists & Spreadsheet Type a name at the top of each column (e.g. hours, score), then fill the values.
STEP 2 — Run the regression
menu4: Statistics1: Stat Calculations3: Linear Regression (mx+b) Set X List: hours, Y List: score, then OK.
Named columns with the dataChoose Linear Regression (mx+b)Read m, b and r from the output
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
y = 4.92x + 25.7 A1A1 r = 0.996 A2 (Interpret: strong positive linear correlation — say “strong”, “positive” AND “linear” if asked.)
⚠️ EXAMINER TRAP
Use the variables of the question — write y = 4.92x + 25.7, not “m = 4.92, b = 25.7”. Copying the raw GDC output without forming the equation loses a mark every year.
EXAMPLE 3NORMAL DISTRIBUTIONPaper 1 & 2
📝 EXAM-STYLE QUESTION
The heights of students are normally distributed with mean 171 cm and standard deviation 8.2 cm. (a) Find the probability a student is taller than 180 cm. (b) Find the minimum height of the tallest 5% of students. [5 marks]
menu5: Probability5: Distributions3: Inverse Normal Area is to the LEFT: tallest 5% → Area = 0.95, μ = 171, σ = 8.2
(a) Normal Cdf dialog(a) result(b) Inverse Normal — area 0.95
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
(a) P(X > 180) = 0.136 (3 sf) M1A1 — sketch a bell curve and shade the right tail A1 (b) P(X < h) = 0.95 M1 → h = 184.488… ≈ 184 cmA1
⚠️ EXAMINER TRAP
Inverse Normal always takes the area to the left. For the tallest 5% you must enter 0.95, not 0.05 — entering 0.05 gives 157.5 cm, which is obviously the shortest students. Sanity-check against the mean!
EXAMPLE 4FINANCE SOLVER (TVM)Paper 1 & 2
📝 EXAM-STYLE QUESTION
P5000 is invested at 4.5% per annum, compounded monthly. Find the value of the investment after 6 years. [3 marks]
STEP 1 — Open the Finance Solver
menu8: Finance1: Finance Solver Fill every field, then put the cursor on FV and press enter
STEP 2 — The sign convention
Money you pay in is negative (PV = −5000), money you receive comes out positive. N is in payment periods: 6 years × 12 = 72.
Finance Solver filled inCursor on FV, press enter
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
N = 72, I = 4.5%, PV = −5000, PpY = CpY = 12 M1A1 FV = P6546.52A1 Money answers: always 2 decimal places, not 3 sf.
⚠️ EXAMINER TRAP
The most common zero-mark error in the whole course: CpY left at 1. Compounded monthly means CpY = 12. And write the TVM values you used on paper — that IS your method mark.
EXAMPLE 5GRAPHING & INTERSECTIONPaper 2
📝 EXAM-STYLE QUESTION
A company’s weekly cost is C(x) = 120 + 18x + 0.5x² and revenue is R(x) = 40x, for x items. Find the break-even values of x. [4 marks]
STEP 1 — Graph both functions
ctrldoc2: Add Graphs f1(x) = 120+18x+0.5x², f2(x) = 40x. Set the window with menu4: Window (x: 0 to 45, y: 0 to 1600).
STEP 2 — Find the intersections
menu6: Analyze Graph4: Intersection Click a lower bound left of the crossing and an upper bound right of it. Repeat for the second crossing.
Analyze Graph menuFirst crossing marked on screenSecond crossing
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
Break-even when C(x) = R(x) M1 Sketch of both curves with intersections labelled A1 x = 6.38 and x = 37.6 (3 sf) A1A1
⚠️ EXAMINER TRAP
A “sketch” question means: copy the GDC screen onto paper with axes labelled, intersection coordinates marked, and the window values on the axes. An empty pair of axes gets zero.
🎯 More Worked Examples — Full Variety Pack
Five more examples covering the rest of the AI SL GDC toolbox: statistics from frequency tables, the two hypothesis tests, loans, and series. Master all ten and there is no GDC task left on the paper that can surprise you.
EXAMPLE 6STATISTICS FROM A TABLEPaper 1 & 2
📝 EXAM-STYLE QUESTION
The table shows quiz scores of 40 students: score 5 (freq 3), 6 (7), 7 (12), 8 (9), 9 (6), 10 (3). Find the mean, standard deviation, median and interquartile range. [5 marks]
STEP 1 — Two named columns
Lists & Spreadsheet page: column A = score (5–10), column B = freq (3, 7, 12, 9, 6, 3). Never type 40 separate numbers — the frequency column does it.
STEP 2 — One-Variable Statistics
menu4: Statistics1: Stat Calculations1: One-Variable Statistics X1 List: score Frequency List: freq — this field is the one everyone misses!
Frequency List set to freqScroll the output for everything
The output shows TWO standard deviations: sx (sample, 1.357) and σx (population, 1.340). IB questions want σx unless they explicitly say “unbiased estimate”. Picking the wrong row costs the mark.
EXAMPLE 7χ² TEST FOR INDEPENDENCEPaper 2
📝 EXAM-STYLE QUESTION
120 students were asked their favourite subject: Maths (Boys 18, Girls 12), Science (22, 28), Art (10, 30). Test at the 5% level whether subject choice is independent of gender. [6 marks]
STEP 1 — Put the table in a matrix
Calculator page: menu7: Matrix & Vector1: Create1: Matrix — 2 rows × 3 columns, fill in the observed values. Store it: ctrlvar as obs.
STEP 2 — Run the test
menu6: Statistics7: Stat Tests8: χ² 2-way Test — Observed Matrix: obs.
Observed matrix stored as obsTest output
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
H₀: choice independent of gender; H₁: not independent A1 χ² = 8.83, df = 2 A2 p = 0.0121 < 0.05 R1 Reject H₀ — subject choice is NOT independent of gender A1A1
⚠️ EXAMINER TRAP
The conclusion sentence must compare p with the significance level and be in context. “Reject H₀” alone loses the final mark — say what it means for boys, girls and subjects. And df = (rows−1)(cols−1) = 2: quote it, it is often a mark.
EXAMPLE 8TWO-SAMPLE t-TESTPaper 2
📝 EXAM-STYLE QUESTION
Plant heights (cm) with fertiliser A: 23.1, 24.5, 22.8, 25.0, 23.9, 24.2 and fertiliser B: 21.9, 22.5, 23.0, 21.4, 22.8, 22.1, 22.6. Test at the 5% level whether the mean heights differ. [6 marks]
STEP 1 — Data into two lists
Lists & Spreadsheet: columns fa and fb.
STEP 2 — 2-Sample t Test, Pooled: Yes
menu4: Statistics4: Stat Tests4: 2-Sample t Test List 1: fa, List 2: fb, Alternate Hyp: μ1 ≠ μ2, Pooled: Yes (IB always pools).
Dialog — note Pooled: YesTest output
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂ A1 p = 0.00182 (3 sf) A2 0.00182 < 0.05, reject H₀ R1 — the mean heights with the two fertilisers differ A1A1
⚠️ EXAMINER TRAP
“Do the means differ?” = two-tailed (≠). “Is A better than B?” = one-tailed (>). Choosing the wrong Alternate Hyp changes p and usually the conclusion. Underline the question’s wording before opening the dialog.
EXAMPLE 9FINANCE — LOAN REPAYMENTSPaper 1 & 2
📝 EXAM-STYLE QUESTION
Thabo borrows P85 000 to buy a car, at 11% per annum compounded monthly, repaid over 4 years with equal monthly payments. Find the monthly repayment and the total interest paid. [4 marks]
STEP 1 — Loans flip the signs
menu8: Finance1: Finance Solver The bank gives Thabo money now: PV = +85000. He pays it back: Pmt will come out negative. FV = 0 — the loan ends fully paid.
Total interest = (number of payments × payment) − amount borrowed. Students who try to compute interest with a percentage formula here always get it wrong — amortised loans do not work that way.
EXAMPLE 10SEQUENCES & SERIESPaper 1 & 2
📝 EXAM-STYLE QUESTION
Naledi deposits P200 at the start of each year into an account. Each deposit’s value grows by 5% per year. Show the total after the 15th deposit is a geometric series and find its sum. [4 marks]
STEP 1 — Use the sum template
Calculator page: templates then pick the Σ template — or menu4: Calculus3: Sum. Fill: k from 0 to 14, expression 200·1.05^k.
STEP 2 — Check against the formula
This is geometric: u₁ = 200, r = 1.05, n = 15. S₁₅ = 200(1.05¹⁵ − 1)0.05. The template and the formula must agree — instant self-check.
Sum templateFormula gives the same
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
Geometric with u₁ = 200, r = 1.05 A1 S₁₅ = 200(1.05¹⁵ − 1)1.05 − 1M1A1 = P4315.71A1
⚠️ EXAMINER TRAP
15 deposits growing from k = 0 to k = 14 — not 1 to 15. Off-by-one in the counter is the classic series error; the GDC sum makes it visible because you can recount the terms on screen.
HIGHER LEVEL ONLY
Examples 11–20: the HL calculator skills
Sitting Standard Level? You can stop at Example 10. HL students: these ten examples cover the extra Paper 2 and Paper 3 calculator work — all done the exam-mode way on the non-CAS TI-Nspire.
EXAMPLE 11SYSTEMS OF EQUATIONSPaper 1, 2 & 3
📝 EXAM-STYLE QUESTION
Solve the system: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2. [4 marks]
STEP 1 — Open the linear-system solver
menu3: Algebra2: Solve System of Linear Equations Set Number of equations: 3, Variables: x,y,z
STEP 2 — Fill in the template
A 3-row template appears — type one equation per row exactly as written in the question.
Choose the linear-system solverOne equation per rowThe solution triple
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
Solving the system (GDC) M1 x = 1, y = 2, z = 3 A3 In context questions, finish with a sentence: “so the prices are P1, P2 and P3.”
⚠️ EXAMINER TRAP
The output {1,2,3} is in the ORDER of your variable list {x,y,z}. If you typed {z,y,x} you will assign every value to the wrong unknown and lose all the accuracy marks.
EXAMPLE 12EQUATIONS WITH TWO ROOTSPaper 2 & 3
📝 EXAM-STYLE QUESTION
The concentration of a drug is modelled by C(t) = t e−0.4t. Find both times at which C(t) = 0.5. [4 marks]
STEP 1 — Graph it first — ALWAYS
f1(x) = x·e^(−0.4x) and f2(x) = 0.5. The graph shows the line crosses the curve twice — so one nSolve is not enough.
STEP 2 — nSolve twice, with a guess each time
nSolve returns only the root nearest your guess. Add a starting guess as the third argument: one near 1, one near 6.
The graph reveals TWO crossingsGuess 1 finds the first rootGuess 6 finds the second
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
t e−0.4t = 0.5 M1 Sketch showing both intersections A1 t = 0.648 and t = 6.36 (3 sf) A1A1
⚠️ EXAMINER TRAP
Writing only t = 0.648 loses half the marks. Any model that rises then falls (drug doses, projectiles, logistic-ish curves) hits most values TWICE. The graph is your insurance — draw it before you solve.
EXAMPLE 13POISSON DISTRIBUTIONPaper 1 & 2
📝 EXAM-STYLE QUESTION
Calls arrive at a rate of 4.2 per hour. Find the probability of receiving at least 6 calls in an hour. [3 marks]
STEP 1 — Translate “at least 6”
P(X ≥ 6) = 1 − P(X ≤ 5). The GDC only gives ≤ directly, so the complement does the work.
State the distribution and its parameter (“X ~ Po(4.2)”) before any number. If your GDC entry is wrong, that line still earns the first mark. “More than 6” would be P(X ≥ 7) — read inequality words twice.
EXAMPLE 14NUMERICAL CALCULUSPaper 2 & 3
📝 EXAM-STYLE QUESTION
For f(x) = x e0.5x, find (a) f ′(3) and (b) ∫₀³ f(x) dx. [4 marks]
STEP 1 — Derivative at a point
menu4: Calculus1: Numerical Derivative at a Point Or press the template key templates and pick the d/dx template. Variable x, value 3.
STEP 2 — Definite integral
menu4: Calculus2: Numerical Integral Fill the template: lower 0, upper 3, function, dx.
Calculus menu(a) derivative at x = 3(b) integral from 0 to 3
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
(a) f ′(3) = 11.2 (3 sf) M1A1 (b) ∫₀³ x e0.5x dx = 13.0 (3 sf) M1A1 Method mark = writing the derivative / integral notation, not the GDC keystrokes.
⚠️ EXAMINER TRAP
On paper, 12.9634 rounds to 13.0 — write “13.0”, not “13”: three significant figures means showing the third figure even when it is a zero.
EXAMPLE 15FINANCE — ANNUITYPaper 1 & 2
📝 EXAM-STYLE QUESTION
Naledi saves P500 at the end of every month into an account paying 6% per annum, compounded monthly. Find the value after 5 years. [3 marks]
STEP 1 — Finance Solver with a payment
menu8: Finance1: Finance Solver This time Pmt is not zero: regular deposits go in as Pmt = −500 (money leaving her pocket).
N = 60 months, Pmt negativeCursor on FV, press enter
Annuity questions have TWO money slots: PV (lump sum now) and Pmt (regular payment). Putting 500 into PV instead of Pmt gives a wildly wrong answer that still looks plausible — check which one the story describes.
🎯 More Worked Examples — Full Variety Pack
Five more examples covering the rest of the AI HL toolbox: matrix powers for Markov chains, eigenvalues, goodness of fit, optimisation from a graph, and fitting exponential models to data. These are the Paper 3 workhorses.
EXAMPLE 16MARKOV CHAINS & MATRIX POWERSPaper 2 & 3
📝 EXAM-STYLE QUESTION
Each year 20% of customers switch from network A to B, and 30% switch from B to A. Initially the market is split 50–50. Find the market shares after 10 years. [4 marks]
STEP 1 — Build the transition matrix
Columns = where you ARE, rows = where you GO. Stay at A: 0.8, B→A: 0.3, A→B: 0.2, stay at B: 0.7. Store: ctrlvar as t, and the initial state as v.
STEP 2 — Raise to the power, multiply
Type t^10·v — the GDC does the whole 10-step evolution in one line.
Store T and vTen years in one lineCheck the long-run steady state
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
T = [[0.8, 0.3],[0.2, 0.7]], v₀ = [0.5, 0.5]ᵀ A1A1 v₁₀ = T¹⁰v₀ = [0.600, 0.400]ᵀ M1 A has 60.0%, B has 40.0% A1
⚠️ EXAMINER TRAP
Check your matrix columns sum to 1 BEFORE computing — 0.8+0.2 = 1 ✓, 0.3+0.7 = 1 ✓. A transition matrix whose columns don’t sum to 1 is wrong with certainty, and the GDC will happily give you a wrong answer from it.
EXAMPLE 17EIGENVALUES & EIGENVECTORSPaper 2 & 3
📝 EXAM-STYLE QUESTION
Find the eigenvalues and corresponding eigenvectors of M = [[4, 1],[2, 3]]. [5 marks]
STEP 1 — Eigenvalues from the characteristic equation
In IB exam mode the eigenvalue keys are blocked, so use λ² − (a+d)λ + det M = 0. Store M as m, then type det(m) → 10 and polyRoots({1,−7,10}) → {2,5}.
STEP 2 — Eigenvectors by hand, checked on the GDC
Solve (M − λI)v = 0 for each λ: λ = 5 gives (1, 1) and λ = 2 gives (1, −2). Check on the GDC: m·[1;1] → [5;5] = 5·(1, 1) and m·[1;−2] → [2;−4] = 2·(1, −2).
Pair each eigenvector with its own eigenvalue — check it with m·v on the GDC before you write it. Any non-zero multiple is correct, but examiners expect simple whole numbers such as (1, 1) and (1, −2).
EXAMPLE 18χ² GOODNESS OF FITPaper 2 & 3
📝 EXAM-STYLE QUESTION
A die is rolled 60 times: frequencies 8, 12, 9, 11, 6, 14 for faces 1–6. Test at the 5% level whether the die is fair. [6 marks]
menu4: Statistics4: Stat Tests7: χ² GOF — Observed: obs, Expected: exp, Deg of Freedom: 5 (categories − 1).
GOF dialogOutput
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
H₀: the die is fair; H₁: it is not fair A1 Expected values all 10 A1 χ² = 4.20, df = 5, p = 0.521 A2 0.521 > 0.05: insufficient evidence to reject H₀ — the die is consistent with being fair R1A1
⚠️ EXAMINER TRAP
Never write “H₀ is accepted” or “the die IS fair” — a test can only fail to find evidence against fairness. That wording distinction is worth a mark at HL, and Paper 3 loves to probe it.
EXAMPLE 19OPTIMISATION FROM A GRAPHPaper 2 & 3
📝 EXAM-STYLE QUESTION
A firm’s daily profit is P(x) = −0.4x² + 22x − 120 for x items. Find the number of items that maximises profit, and the maximum profit. [3 marks]
STEP 1 — Graph and Analyze
f1(x) = −0.4x²+22x−120, window x: 0 to 60, y: 0 to 220. Then menu6: Analyze Graph3: Maximum — bracket the peak with lower and upper bounds.
Maximum found and labelledCalculus check (optional)
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
Maximum at x = 27.5 from the graph M1 x must be a whole number of items: P(27) = 182.4, P(28) = 182.4 → 27 or 28 items A1 Maximum profit P182.50 at x = 27.5 (if x continuous) A1
⚠️ EXAMINER TRAP
Context decides whether x = 27.5 is acceptable. Items, people, tickets → integers: evaluate both neighbours and say so. A pure “find the maximum of the function” → 27.5 stands. Read the noun in the question.
EXAMPLE 20EXPONENTIAL REGRESSIONPaper 2 & 3
📝 EXAM-STYLE QUESTION
Bacteria counts (thousands) were measured: t = 0, 1, 2, 3, 4 hours; N = 12, 17, 25, 36, 52. Fit a model N = a·bᵗ and estimate N at t = 6. [5 marks]
STEP 1 — Data in, regression out
Lists & Spreadsheet: columns time, count. Then menu4: Statistics1: Stat CalculationsA: Exponential Regression.
STEP 2 — Save the equation, then predict
In the dialog set “Save RegEqn to” f1 — then f1(6) evaluates the model instantly. No retyping, no rounding error.
Regression outputPredict with the saved model
✍️ WHAT YOU WRITE ON PAPER (the marks live here!)
N = 11.9 × 1.45ᵗ A1A1 (r = 0.9999: excellent fit A1) N(6) = 109 (3 sf) → about 109 000 bacteria M1A1
⚠️ EXAMINER TRAP
t = 6 is OUTSIDE the data range (0–4) — that is extrapolation. State it: “this assumes exponential growth continues, which may not be reliable”. That sentence is a standing 1-mark reward in AI papers.