Formulas & Calculator — Whole-Course Summary (SL)

Module 3 of 46Lesson 1 of 2 · Formulas & Calculator — Whole-Course Summary (SL)

IB DP AA · SL · Prof. Kitso

Formulas & Calculator — your whole course on one page

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

Paper 1 has no calculator — every formula here must also work by hand. Use the TI-Nspire on Paper 2 (and Paper 3) and to CHECK your Paper 1 practice.

🗺️ How your course fits together

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Start Herehow the course works

🧮

GDC Bootcampfor Paper 2 (and Paper 3)

📌

Formulas & calculatorthis page — the whole course

🪜

Learn unitsstep-by-step lessons

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Mock Setsfull papers + mark schemes

📝 Your exam (SL)

Paper 190 min · sections A + B🚫 no calculator

Paper 290 min · sections A + B🔢 GDC

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

Topic 1🔢 Number & algebra13 formula cards · 19 h Topic 2📈 Functions9 formula cards · 21 h Topic 3📐 Geometry & trigonometry12 formula cards · 25 h Topic 4📊 Statistics & probability9 formula cards · 27 h Topic 5∫ Calculus12 formula cards · 28 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 & algebra13 cards

AA 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’.

AA 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.

AA 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))

AA 1.3Geometric sequence — nth term📖 In the booklet

u_n=u_1r^{\,n-1}

🧮 On your TI-Nspire CX II

AA 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))

AA 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.

AA 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

AA 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]

AA 1.7Laws of logarithms📖 In the booklet

\begin{aligned}\log_a xy&=\log_a x+\log_a y\\ \log_a\dfrac{x}{y}&=\log_a x-\log_a y\\ \log_a x^{m}&=m\log_a x\end{aligned}

🧮 On your TI-Nspire CX II

[ctrl]💻 Ctrl›[10ˣ]💻 type 10^

💡 Use the calculator to CHECK a log law answer, not to replace the working.

AA 1.7Change of base📖 In the booklet

\log_b a=\dfrac{\log_c a}{\log_c b}

🧮 On your TI-Nspire CX II

AA 1.7Rational exponents🧠 Learn by heart

a^{\frac{m}{n}}=\sqrt[n]{a^{m}}

a^{-n}=\dfrac{1}{a^{n}}

a^{m}a^{n}=a^{m+n}

\left(a^{m}\right)^{n}=a^{mn}

🧮 On your TI-Nspire CX II

AA 1.8Infinite geometric series📖 In the booklet

S_\infty=\dfrac{u_1}{1-r},\quad |r|<1

🧮 On your TI-Nspire CX II

💡 Always state |r| < 1 — it is a method mark.

AA 1.9Binomial theorem📖 In the booklet

(a+b)^n=a^n+{}^{n}C_1a^{n-1}b+\dots+{}^{n}C_ra^{n-r}b^{r}+\dots+b^{n}

{}^{n}C_r=\dfrac{n!}{r!\,(n-r)!}

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›5: Probability›3: Combinations

⌨️ Type: nCr(n, r)

💡 Paper 1 has no calculator: practise nCr by hand with Pascal’s triangle.

Topic 2 📈 Functions9 cards

AA 2.1Gradient of a line📖 In the booklet

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

🧮 On your TI-Nspire CX II

AA 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).

AA 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.

AA 2.5Composite & 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.

AA 2.6Quadratic: 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

AA 2.7Quadratic formula & discriminant📖 In the booklet

x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}

\Delta=b^{2}-4ac

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›3: Algebra›3: Polynomial Tools›1: Find Roots of Polynomial…

⌨️ Type: polyRoots({a, b, c})

💡 Δ > 0 two roots · Δ = 0 one repeated root · Δ < 0 no real roots. Paper 1: by hand!

AA 2.8Rational 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}

AA 2.9Exponential & log functions📖 In the booklet

a^{x}=e^{x\ln a}

\log_a a^{x}=x=a^{\log_a x}

🧮 On your TI-Nspire CX II

AA 2.11Transformations of graphs🧠 Learn by heart

\begin{aligned}&y=f(x)+b,\ \ y=f(x-a),\ \ y=pf(x),\\ &y=f(qx),\ \ y=-f(x),\ \ y=f(-x)\end{aligned}

🧮 On your TI-Nspire CX II

Graphs page›f1(x) and f2(x) together

💡 Inside the bracket = horizontal and ‘opposite’ (f(x−3) moves RIGHT 3). Outside = vertical.

Topic 3 📐 Geometry & trigonometry12 cards

AA 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

AA 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.

AA 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.

AA 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

AA 3.2Area of a triangle📖 In the booklet

A=\dfrac12ab\sin C

🧮 On your TI-Nspire CX II

AA 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

AA 3.4Arc length & sector area (radians)📖 In the booklet

l=r\theta

A=\dfrac12r^{2}\theta

\pi\ \text{rad}=180^{\circ}

🧮 On your TI-Nspire CX II

[doc]💻 File menu›Settings›Angle: Radian

AA 3.5Exact trig values🧠 Learn by heart

\begin{array}{c|ccccc}\theta&0&\frac{\pi}{6}&\frac{\pi}{4}&\frac{\pi}{3}&\frac{\pi}{2}\\\hline \sin\theta&0&\frac12&\frac{\sqrt2}{2}&\frac{\sqrt3}{2}&1\\ \cos\theta&1&\frac{\sqrt3}{2}&\frac{\sqrt2}{2}&\frac12&0\\ \tan\theta&0&\frac{\sqrt3}{3}&1&\sqrt3&\text{—}\end{array}

💡 Paper 1 essential — no calculator. Use the unit circle for other quadrants (CAST).

AA 3.6Trig identities📖 In the booklet

\cos^{2}\theta+\sin^{2}\theta=1

\tan\theta=\dfrac{\sin\theta}{\cos\theta}

AA 3.6Double angle identities📖 In the booklet

\sin2\theta=2\sin\theta\cos\theta

\cos2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta

AA 3.7Circular functions🧠 Learn by heart

f(x)=a\sin\big(b(x+c)\big)+d:\quad \text{amplitude}=|a|,\ \ \text{period}=\dfrac{2\pi}{|b|}\ \left(\text{or }\dfrac{360^\circ}{|b|}\right)

🧮 On your TI-Nspire CX II

Graphs page›check amplitude, period and principal axis with Analyze Graph

💡 Principal axis y = d. AI SL: f(x) = a sin(bx) + d. Match the angle mode to the question!

AA 3.8Solving trig equations🧠 Learn by heart

\sin x=k\ \text{on an interval — use symmetry and period}

🧮 On your TI-Nspire CX II

Graphs page›f1(x)=sin(x), f2(x)=k›[menu]💻 click menu on keypad›Window: set the interval›Analyze Graph›Intersection

📍 Where to find it on the TI-Nspire

⌨️ sin, cos, tan: click trig on the keypad, or type sin(

💡 Count the solutions in the window — the question’s interval decides how many you need.

Topic 4 📊 Statistics & probability9 cards

AA 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.

AA 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.

AA 4.5Basic probability📖 In the booklet

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

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

\text{expected number}=np

AA 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.

AA 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.

AA 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.

AA 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.

AA 4.10Regression line x on y🧠 Learn by heart

x=cy+d

🧮 On your TI-Nspire CX II

Linear Regression›X List = y-data, Y List = x-data

💡 Use x-on-y only to predict x from a given y.

AA 4.12Standardising (z-score)📖 In the booklet

z=\dfrac{x-\mu}{\sigma}

🧮 On your TI-Nspire CX II

invNorm(area, 0, 1) gives z

📍 Where to find it on the TI-Nspire

invNorm( is in [menu]💻 click menu on keypad›5: Probability›5: Distributions›3: Inverse Normal…

💡 AA: use z to find an unknown μ or σ. Two unknowns → two equations → solve simultaneously.

Topic 5 ∫ Calculus12 cards

AA 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

AA 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).

AA 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

AA 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.

AA 5.6Standard derivatives📖 In the booklet

\begin{aligned}&\sin x\to\cos x, && \cos x\to-\sin x, && \tan x\to\dfrac1{\cos^{2}x}\\ &e^{x}\to e^{x}, && \ln x\to\dfrac1x\end{aligned}

🧮 On your TI-Nspire CX II

💡 Radian mode ON for all calculus with trig!

AA 5.6Chain, product & quotient rules📖 In the booklet

\dfrac{dy}{dx}=\dfrac{dy}{du}\times\dfrac{du}{dx}

\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}

\dfrac{d}{dx}\left(\dfrac uv\right)=\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}

🧮 On your TI-Nspire CX II

check your answer: Numerical Derivative at a Point

AA 5.7Second derivative & inflexion🧠 Learn by heart

f''(x)=\dfrac{d^{2}y}{dx^{2}}

\text{inflexion: } f''(x)=0 \text{ and changes sign}

🧮 On your TI-Nspire CX II

Graphs page›graph f′ and find its max/min›or Numerical Derivative: 2nd

AA 5.8Stationary 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.

AA 5.9Kinematics📖 In the booklet

v=\dfrac{ds}{dt},\quad a=\dfrac{dv}{dt}=\dfrac{d^{2}s}{dt^{2}}=v\dfrac{dv}{ds},\quad \text{distance}=\int_{t_1}^{t_2}|v(t)|\,dt

🧮 On your TI-Nspire CX II

[menu]💻 click menu on keypad›4: Calculus›2: Numerical Integral›abs( from [ctrl][catalog]

⌨️ Type: nInt(abs(v(t)), t, t₁, t₂)

💡 Displacement = ∫ v dt (can be negative). Distance = ∫ |v| dt.

a = v\,\dfrac{dv}{ds}

is AI HL only.

AA 5.10Standard integrals📖 In the booklet

\int\dfrac1x\,dx=\ln|x|+C,\ \ \int\sin x\,dx=-\cos x+C,\ \ \int\cos x\,dx=\sin x+C,\ \ \int e^{x}dx=e^{x}+C

🧮 On your TI-Nspire CX II

AA 5.10Integration by substitution / inspection🧠 Learn by heart

\int f\big(g(x)\big)\,g'(x)\,dx=F\big(g(x)\big)+C

🧮 On your TI-Nspire CX II

💡 Spot the ‘inside’ and its derivative. Check a definite answer with nInt.

AA 5.11Area between curves📖 In the booklet

A=\int_a^b\left|f(x)-g(x)\right|dx

🧮 On your TI-Nspire CX II

Graphs page: Intersection gives a and b›Calculator: nInt(abs(f1(x)−f2(x)),x,a,b)

📍 Where to find it on the TI-Nspire

nInt( is in [menu]💻 click menu on keypad›4: Calculus›2: Numerical Integral

⌨️ type abs( on the keyboard; the screen shows it as | |

✅ 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: Analysis and Approaches (AA) — SL and HL Formulas & Calculator — Whole-Course Summary (SL)

IB DP Mathematics: Analysis and Approaches (AA) — 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