Formulas & Calculator — Whole-Course Summary (SL)

Module 3 of 47Lesson 1 of 2 · Formulas & Calculator — Whole-Course Summary (SL)

IB DP AI · SL · Prof. Kitso

Formulas & Calculator — your whole course on one page

Every formula you will use in Mathematics: Applications & Interpretation SL, topic by topic — with what to press on the TI-Nspire CX II and what the screen shows.

The calculator is allowed on every AI paper — speed on the TI-Nspire is worth marks.

🗺️ How your course fits together

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Start Herehow the course works

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GDC BootcampTI-Nspire basics

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Formulas & calculatorthis page — the whole course

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Learn unitsstep-by-step lessons

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Topic Revision78 subtopics · 879 questions

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Mock Sets 1–5full papers + mark schemes

📝 Your exam (SL)

Paper 190 min · short questions🔢 GDC

Paper 290 min · long questions🔢 GDC

🧭 The five topics at a glance — tap one to open it

Topic 1🔢 Number & algebra12 formula cards · 16 h Topic 2📈 Functions7 formula cards · 31 h Topic 3📐 Geometry & trigonometry8 formula cards · 18 h Topic 4📊 Statistics & probability10 formula cards · 36 h Topic 5∫ Calculus6 formula cards · 19 h

📖 In the booklet you are given it — learn WHERE it is.
🧠 Learn by heart NOT in the booklet — you must remember it.
[menu] a key to press Statistics a menu item to choose

⚙️ Before every paper: set up your TI-Nspire (tap)

Turn on Press-to-Test when the invigilator tells you (hold [esc] + [on] from OFF).
[doc] → Settings → Document Settings: Angle Degree for triangles & bearings, Radian for calculus & circular functions.
Display Digits Float 6 (default) — then round to 3 s.f. unless told otherwise.
Store long answers: [ctrl][var] (sto→) so you never retype a rounded number.
The TI shows whole numbers with a dot: 10. means 10.
New page for each question: [ctrl][doc] → Add Calculator.

Topic 1 🔢 Number & algebra12 cards

AI 1.1Standard form🧠 Learn by heart

a\times10^{k},\quad 1\le |a|<10,\ k\in\mathbb{Z}

🧮 On your TI-Nspire CX II

type 3.2›[EE]💻 click EE on keypad›type 5

💡 The calculator shows 2.00667E15 — you must write 2.01 × 10¹⁵, never ‘E’.

AI 1.2Arithmetic sequence — nth term📖 In the booklet

u_n=u_1+(n-1)d

🧮 On your TI-Nspire CX II

Calculator page›type the formula with n replaced

💡 Find n? Use nSolve or a table — see ‘Solving equations’ below.

AI 1.2Arithmetic series — sum of n terms📖 In the booklet

S_n=\dfrac{n}{2}\big(2u_1+(n-1)d\big)=\dfrac{n}{2}\big(u_1+u_n\big)

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›4: Calculus›3: Sum (Σ)

⌨️ Type: sum(seq(u₁+(n−1)·d, n, 1, N))

AI 1.3Geometric sequence — nth term📖 In the booklet

u_n=u_1r^{\,n-1}

🧮 On your TI-Nspire CX II

AI 1.3Geometric series — sum of n terms📖 In the booklet

S_n=\dfrac{u_1\left(r^{n}-1\right)}{r-1}=\dfrac{u_1\left(1-r^{n}\right)}{1-r},\quad r\ne1

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›4: Calculus›3: Sum (Σ)

⌨️ Type: sum(seq(u₁·r^(n−1), n, 1, N))

AI 1.4Compound interest📖 In the booklet

FV=PV\times\left(1+\dfrac{r}{100k}\right)^{kn}

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›8: Finance›1: Finance Solver…

💡 Money you pay in is negative (PV = −5000). N = total number of periods = k × n.

AI 1.4Depreciation🧠 Learn by heart

FV=PV\times\left(1-\dfrac{r}{100}\right)^{n}

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›8: Finance›1: Finance Solver…›I(%) negative

AI 1.5Exponents ⇄ logarithms📖 In the booklet

a^{x}=b\iff x=\log_{a}b,\quad a>0,\ b>0,\ a\ne1

🧮 On your TI-Nspire CX II

[ctrl]💻 Ctrl›[10ˣ]💻 type 10^›→ log template›type base and number

⌨️ Type: log(b, a) · ln(x) is [ctrl][eˣ]

AI 1.6Percentage error📖 In the booklet

\varepsilon=\left|\dfrac{v_A-v_E}{v_E}\right|\times100\%

🧮 On your TI-Nspire CX II

type abs( or use the |□| template from the [⊞] key

💡 vₐ = approximate value, vₑ = exact value. Always divide by the EXACT value.

AI 1.6Upper & lower bounds🧠 Learn by heart

\text{a value given to the nearest } u \text{ lies in } \left[x-\dfrac{u}{2},\ x+\dfrac{u}{2}\right)

🧮 On your TI-Nspire CX II

💡 Max of a quotient =

\dfrac{\text{max}}{\text{min}}

. Write bounds with ≤ and <.

AI 1.7Loans & annuities (amortization)🧠 Learn by heart

\text{Use the Finance Solver: } N,\ I\%,\ PV,\ Pmt,\ FV,\ PpY,\ CpY

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›8: Finance›1: Finance Solver…›cursor on Pmt›[enter]💻 Enter

💡 The minus sign on Pmt means money going OUT of your pocket. Total interest = N × |Pmt| − PV.

AI 1.8Solving equations with technology🧠 Learn by heart

\text{systems of linear equations (up to 3 unknowns)} \quad\&\quad \text{polynomial equations}

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›3: Algebra›2: Solve System of Linear Equations… / 3: Polynomial Tools›Find Roots of Polynomial

⌨️ Type: polyRoots({2, −3, −5})

💡 The wizard asks for the number of equations / degree first. Write the answer as x = …, not just the list.

Topic 2 📈 Functions7 cards

AI 2.1Gradient of a line📖 In the booklet

m=\dfrac{y_2-y_1}{x_2-x_1}

🧮 On your TI-Nspire CX II

AI 2.1Equations of a straight line📖 In the booklet

y=mx+c

ax+by+d=0

y-y_1=m(x-x_1)

💡 Perpendicular lines: m₁ × m₂ = −1 (learn this).

AI 2.2Composite & inverse functions🧠 Learn by heart

(f\circ g)(x)=f\big(g(x)\big)

f\big(f^{-1}(x)\big)=x

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›1: Actions›1: Define›f(x):=… g(x):=…

📍 Where to find it on the TI-Nspire

⌨️ store a value: type := (colon, then equals) on the keyboard

💡 The graph of f⁻¹ is the reflection of f in y = x. Domain of f⁻¹ = range of f.

AI 2.4Graph features with technology🧠 Learn by heart

\text{zeros, maximum, minimum, intersection, asymptotes}

🧮 On your TI-Nspire CX II

Graphs page›type f1(x)›[menu]💻 click menu on keypad›6: Analyze Graph›1: Zero / 2: Minimum / 3: Maximum / 4: Intersection

💡 Drag the lower bound then upper bound round each root. Do one point at a time.

AI 2.5Quadratic: vertex & axis of symmetry📖 In the booklet

y=ax^{2}+bx+c\Rightarrow x=-\dfrac{b}{2a}

🧮 On your TI-Nspire CX II

Graphs page›[menu]💻 click menu on keypad›6: Analyze Graph›2: Minimum / 3: Maximum

AI 2.5Rational function asymptotes🧠 Learn by heart

y=\dfrac{ax+b}{cx+d}:\quad x=-\dfrac{d}{c}\ \ (\text{vertical})

y=\dfrac{a}{c}\ \ (\text{horizontal})

🧮 On your TI-Nspire CX II

Graphs page›[menu]💻 click menu on keypad›6: Analyze Graph›check with Trace

💡 AI SL uses only y =

\dfrac{a}{x}

type inverse variation; AA SL needs the full

\dfrac{ax+b}{cx+d}

AI 2.6Regression models (fit a function to data)🧠 Learn by heart

\text{linear, quadratic, cubic, exponential } (ab^{x}),\ \text{power } (ax^{b}),\ \text{sinusoidal}

🧮 On your TI-Nspire CX II

Lists & Spreadsheet›x in A, y in B›[menu]💻 click menu on keypad›4: Statistics›1: Stat Calculations›…Regression

💡 AI SL models: linear, quadratic, exponential, direct/inverse variation, cubic, sinusoidal. Fitting non-linear regression models with R² is AI HL 4.13.

Topic 3 📐 Geometry & trigonometry8 cards

AI 3.1Distance & midpoint (3D)📖 In the booklet

d=\sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}+(z_1-z_2)^{2}}

M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2},\dfrac{z_1+z_2}{2}\right)

🧮 On your TI-Nspire CX II

AI 3.1Volume & surface area of solids📖 In the booklet

\begin{aligned}V_{\text{pyramid}}&=\frac13Ah & V_{\text{cone}}&=\frac13\pi r^{2}h & A_{\text{cone curved}}&=\pi rl\\ V_{\text{sphere}}&=\frac43\pi r^{3} & A_{\text{sphere}}&=4\pi r^{2}\end{aligned}

🧮 On your TI-Nspire CX II

💡 l is the SLANT height. Cylinder & prism are in the ‘prior learning’ page.

AI 3.2Sine rule📖 In the booklet

\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}

🧮 On your TI-Nspire CX II

[doc]💻 File menu›Settings›Angle: Degree›[ctrl][sin] = sin⁻¹

💡 Ambiguous case: if sin B gives B, the other answer is 180° − B. Check both.

AI 3.2Cosine rule📖 In the booklet

c^{2}=a^{2}+b^{2}-2ab\cos C

\cos C=\dfrac{a^{2}+b^{2}-c^{2}}{2ab}

🧮 On your TI-Nspire CX II

AI 3.2Area of a triangle📖 In the booklet

A=\dfrac12ab\sin C

🧮 On your TI-Nspire CX II

AI 3.2Right-angled triangles🧠 Learn by heart

\sin\theta=\dfrac{\text{opp}}{\text{hyp}},\quad \cos\theta=\dfrac{\text{adj}}{\text{hyp}},\quad \tan\theta=\dfrac{\text{opp}}{\text{adj}},\quad a^{2}+b^{2}=c^{2}

🧮 On your TI-Nspire CX II

AI 3.4Arc length & sector area (degrees)📖 In the booklet

l=\dfrac{\theta}{360}\times2\pi r

A=\dfrac{\theta}{360}\times\pi r^{2}

🧮 On your TI-Nspire CX II

AI 3.5Perpendicular bisectors & Voronoi diagrams🧠 Learn by heart

\text{bisector of }AB:\ \text{through } M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right),\ \ m=-\dfrac{1}{m_{AB}}

💡 Voronoi edges are perpendicular bisectors between neighbouring sites. Toxic-waste-dump problems: the best point is a vertex.

Topic 4 📊 Statistics & probability10 cards

AI 4.3Mean, quartiles, standard deviation📖 In the booklet

\bar x=\dfrac{\sum_{i=1}^{k}f_ix_i}{n}

\text{IQR}=Q_3-Q_1

🧮 On your TI-Nspire CX II

Lists & Spreadsheet›data in A (freq in B)›[menu]💻 click menu on keypad›4: Statistics›1: Stat Calculations›1: One-Variable Statistics…

⌨️ Type: OneVar {2,4,4,5,7,9}

💡 Outlier: more than 1.5 × IQR below Q₁ or above Q₃. σx = population SD, sx = sample SD.

AI 4.4Correlation & regression line (y on x)🧠 Learn by heart

y=ax+b

r\ \text{(Pearson’s product-moment correlation coefficient)}

🧮 On your TI-Nspire CX II

Lists & Spreadsheet›[menu]💻 click menu on keypad›4: Statistics›1: Stat Calculations›3: Linear Regression (mx+b)… (mx+b)

💡 Only predict y from x inside the data range (interpolation). r tells strength, NOT cause.

AI 4.5Basic probability📖 In the booklet

P(A)=\dfrac{n(A)}{n(U)}

P(A')=1-P(A)

\text{expected number}=np

AI 4.6Combined & conditional events📖 In the booklet

\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\ P(A\mid B)&=\dfrac{P(A\cap B)}{P(B)}\\ \text{independent: }P(A\cap B)&=P(A)P(B)\end{aligned}

💡 Mutually exclusive: P(A ∩ B) = 0. Draw a Venn or tree diagram first.

AI 4.7Discrete random variable📖 In the booklet

E(X)=\sum x\,P(X=x)

\sum P(X=x)=1

🧮 On your TI-Nspire CX II

⌨️ Type: sum(xlist · plist)

💡 Fair game: E(gain) = 0.

AI 4.8Binomial distribution📖 In the booklet

X\sim B(n,p):\quad E(X)=np

\operatorname{Var}(X)=np(1-p)

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›6: Statistics›5: Distributions›A: Binomial Pdf… (=) / B: Binomial Cdf… (≤, ≥, between)

⌨️ Type: binomPdf(n,p,x) · binomCdf(n,p,lower,upper)

💡 P(X > 3) means lower bound 4 — integers! Write B(10, 0.3) in your working.

AI 4.9Normal distribution🧠 Learn by heart

X\sim N(\mu,\sigma^{2})

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›6: Statistics›5: Distributions›2: Normal Cdf… / 3: Inverse Normal…

⌨️ Type: normCdf(lower,upper,μ,σ) · invNorm(area,μ,σ)

💡 Enter σ (standard deviation), NOT σ². ‘−∞’ = −9E999. invNorm uses the area to the LEFT.

AI 4.10Spearman’s rank correlation🧠 Learn by heart

r_s=\text{Pearson’s } r \text{ applied to the ranks}

🧮 On your TI-Nspire CX II

rank each list by hand (ties: average rank)›Linear Regression on the ranks›read r

💡 Spearman measures a monotonic (always up / always down) relationship.

AI 4.11χ² test for independence🧠 Learn by heart

\chi^{2}_{calc}=\sum\dfrac{(f_o-f_e)^{2}}{f_e},\quad f_e=\dfrac{\text{row total}\times\text{column total}}{\text{grand total}},\quad \nu=(r-1)(c-1)

🧮 On your TI-Nspire CX II

observed table as a matrix [⊞]›[menu]💻 click menu on keypad›6: Statistics›7: Stat Tests›8: χ² 2-way Test…

⌨️ Type: χ²2way [20,30;30,20]

💡 If any expected value < 5, combine rows/columns first. Reject H₀ if p < significance level.

AI 4.11χ² goodness of fit & t-test🧠 Learn by heart

\nu=n-1-(\text{parameters estimated})

t\text{-test: compare two means}

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›6: Statistics›7: Stat Tests›7: χ² GOF… / 4: 2-Sample t Test… (Pooled: Yes)

💡 H₀ for t-test: μ₁ = μ₂. State hypotheses in words AND symbols.

Topic 5 ∫ Calculus6 cards

AI 5.2Increasing & decreasing🧠 Learn by heart

f'(x)>0\Rightarrow f\ \text{increasing}

f'(x)<0\Rightarrow f\ \text{decreasing}

🧮 On your TI-Nspire CX II

Graphs page›graph f′ and look above/below the x-axis

AI 5.3Derivative of axⁿ📖 In the booklet

f(x)=ax^{n}\Rightarrow f'(x)=anx^{\,n-1}

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›4: Calculus›1: Numerical Derivative at a Point…

⌨️ Type: d/dx(f(x))|x=a

💡 The calculator gives a NUMBER (gradient at a point), not the formula f′(x).

AI 5.4Tangents & normals🧠 Learn by heart

y-y_1=m(x-x_1)

m_{\text{normal}}=-\dfrac{1}{m_{\text{tangent}}}

🧮 On your TI-Nspire CX II

Graphs page›[menu]💻 click menu on keypad›6: Analyze Graph›5: dy/dx (gradient)›or Geometry → Points & Lines → Tangent

AI 5.5Integral of axⁿ & area📖 In the booklet

\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C,\ n\ne-1

A=\int_a^b y\,dx

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›4: Calculus›2: Numerical Integral

⌨️ Type: nInt(f(x), x, a, b)

💡 Area below the x-axis is negative — use |y| or split the integral.

AI 5.6Stationary points & optimisation🧠 Learn by heart

f'(x)=0;\quad f''(x)<0\Rightarrow\max,\ \ f''(x)>0\Rightarrow\min

🧮 On your TI-Nspire CX II

Graphs page›[menu]💻 click menu on keypad›6: Analyze Graph›3: Maximum / 2: Minimum

💡 Optimisation: write the function, differentiate, set f′ = 0, CHECK it is a max/min, answer in context.

AI 5.8Trapezoidal rule📖 In the booklet

\int_a^b y\,dx\approx\dfrac{h}{2}\Big[(y_0+y_n)+2(y_1+y_2+\dots+y_{n-1})\Big],\quad h=\dfrac{b-a}{n}

🧮 On your TI-Nspire CX II

Lists & Spreadsheet›x-values in A›B = f(a[])›or type the formula

💡 n = number of STRIPS, not number of points. Over-estimate if the curve is concave up.

✅ Next: open the Learn units for full step-by-step lessons on each topic. HL students: the HL version of this page has every SL and HL formula.
Booklet = the IB formula booklet for this course (first exams 2021, used until the new guide). TI-Nspire CX II (non-CAS) key paths may differ slightly by OS version.

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL Formulas & Calculator — Whole-Course Summary (SL)

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

Formulas & Calculator — your whole course on one page

🗺️ How your course fits together

📝 Your exam (SL)

🧭 The five topics at a glance — tap one to open it