IB Math AA Formula Booklet, Annotated (SL and HL): Free PDF

Photon Resources · IB Maths AA formula booklet, annotated · photonresources.com/ib-math-aa-formula-booklet/

Prior learning

SL and HL

Area of a parallelogram

A=bhwhere b is the base, h is the height

Area of a triangle

A=\tfrac{1}{2}(bh)where b is the base, h is the height

Area of a trapezoid

A=\tfrac{1}{2}(a+b)hwhere a and b are the parallel sides, h is the height

Area of a circle

A=\pi r^2where r is the radius

Circumference of a circle

C=2\pi rwhere r is the radius

Volume of a cuboid

V=lwhwhere l is the length, w is the width, h is the height

Volume of a cylinder

V=\pi r^2hwhere r is the radius, h is the height

Volume of a prism

V=Ahwhere A is the area of cross-section, h is the height

Area of the curved surface of a cylinder

A=2\pi rhwhere r is the radius, h is the height

Photon noteTotal surface area of a closed cylinder adds the two ends: 2\pi rh+2\pi r^2.

Distance between two points (x_1,y_1) and (x_2,y_2)

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

Coordinates of the midpoint of a line segment with endpoints (x_1,y_1) and (x_2,y_2)

\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)

Not in the IB booklet · learn or derive these

Gradients of perpendicular linesm_1m_2=-1

Use it for normals and perpendicular bisectors.

Topic 1: Number and algebra

SL 1.2 SL and HL

The nth term of an arithmetic sequence

u_n=u_1+(n-1)d

Photon note(n-1)d, not nd: the first term has had no differences added yet.

The sum of n terms of an arithmetic sequence

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

SL 1.3 SL and HL

The nth term of a geometric sequence

u_n=u_1r^{n-1}

The sum of n terms of a finite geometric sequence

S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r},\quad r\neq1

SL 1.4 SL and HL

Compound interest

FV=PV\times\left(1+\frac{r}{100k}\right)^{kn}where FV is the future value, PV is the present value, n is the number of years, k is the number of compounding periods per year, r\% is the nominal annual rate of interest

Photon noteOn the calculator's finance solver, money you pay out is negative and money you receive is positive.

SL 1.5 SL and HL

Exponents and logarithms

a^x=b\iff x=\log_abwhere a>0, b>0, a\neq1

SL 1.7 SL and HL

Exponents and logarithms

\log_axy=\log_ax+\log_ay

\log_a\frac{x}{y}=\log_ax-\log_ay

\log_ax^m=m\log_ax

Photon noteThere is no rule for \log_a(x+y). Splitting it is a very common lost mark.

\log_ax=\frac{\log_bx}{\log_ba}

Photon noteChange of base: lets you evaluate any log with the calculator's \ln or \log key.

SL 1.8 SL and HL

The sum of an infinite geometric sequence

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

Photon noteState the condition |r|<1 whenever you use it: it often carries a mark.

SL 1.9 SL and HL

Binomial theorem n\in\mathbb N

(a+b)^n=a^n+{}^n\mathrm{C}_1a^{n-1}b+\dots+{}^n\mathrm{C}_ra^{n-r}b^r+\dots+b^n

Photon noteThe general term is {}^n\mathrm C_r\,a^{n-r}b^r. Set the power of x you want and solve for r.

{}^n\mathrm{C}_r=\frac{n!}{r!(n-r)!}

AHL 1.10 HL only

Combinations

{}^n\mathrm{C}_r=\frac{n!}{r!(n-r)!}

Photon noteOrder does not matter.

Permutations

{}^n\mathrm{P}_r=\frac{n!}{(n-r)!}

Photon noteOrder matters.

Extension of binomial theorem, n\in\mathbb Q

(a+b)^n=a^n\left(1+n\left(\frac{b}{a}\right)+\frac{n(n-1)}{2!}\left(\frac{b}{a}\right)^2+\dots\right)

Photon noteValid only when \left|\dfrac{b}{a}\right|<1. State this interval of validity.

AHL 1.12 HL only

Complex numbers

z=a+b\mathrm{i}

AHL 1.13 HL only

Modulus-argument (polar) and exponential (Euler) form

z=r(\cos\theta+\mathrm{i}\sin\theta)=r\mathrm{e}^{\mathrm{i}\theta}=r\,\mathrm{cis}\,\theta

Photon noteFind \theta from a sketch of the Argand diagram: the calculator's \arctan gives the wrong quadrant half the time.

AHL 1.14 HL only

De Moivre's theorem

\big[r(\cos\theta+\mathrm{i}\sin\theta)\big]^n=r^n(\cos n\theta+\mathrm{i}\sin n\theta)=r^n\mathrm{e}^{\mathrm{i}n\theta}=r^n\,\mathrm{cis}\,n\theta

Not in the IB booklet · learn or derive these

Laws of exponentsa^m\times a^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\frac{1}{a^n}

Modulus and argument (HL)|z|=\sqrt{a^2+b^2},\qquad \tan\theta=\frac{b}{a}

Then fix the quadrant from the signs of a and b.

Complex conjugate (HL)z^*=a-b\mathrm{i},\qquad zz^*=|z|^2

Non-real roots of a real polynomial come in conjugate pairs.

nth roots of a complex number (HL)z^{1/n}=r^{1/n}\,\mathrm{cis}\left(\frac{\theta+2\pi k}{n}\right),\ k=0,1,\dots,n-1

The n roots are equally spaced round a circle.

Topic 2: Functions

SL 2.1 SL and HL

Equations of a straight line

y=mx+c;\quad ax+by+d=0;\quad y-y_1=m(x-x_1)

Gradient formula

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

SL 2.6 SL and HL

Axis of symmetry of the graph of a quadratic function

f(x)=ax^2+bx+c\ \Rightarrow\ \text{axis of symmetry is } x=-\frac{b}{2a}

Photon noteThe vertex is at x=-\dfrac{b}{2a}; substitute back to get its y-coordinate.

SL 2.7 SL and HL

Solutions of a quadratic equation

ax^2+bx+c=0\ \Rightarrow\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\quad a\neq0

Discriminant

\Delta=b^2-4ac

Photon note\Delta>0: two real roots; \Delta=0: one repeated root (the graph touches the axis); \Delta<0: no real roots.

SL 2.9 SL and HL

Exponential and logarithmic functions

a^x=\mathrm{e}^{x\ln a};\quad \log_aa^x=x=a^{\log_ax}where a,x>0, a\neq1

AHL 2.12 HL only

Sum and product of the roots of polynomial equations of the form \sum_{r=0}^{n}a_rx^r=0

\text{Sum is } \frac{-a_{n-1}}{a_n};\quad \text{product is } \frac{(-1)^na_0}{a_n}

Photon notea_n is the leading coefficient. Watch the (-1)^n sign for odd-degree polynomials.

Not in the IB booklet · learn or derive these

Vertex form of a quadraticf(x)=a(x-h)^2+k

Vertex at (h,k). Found by completing the square.

Transformationsy=af\big(b(x-c)\big)+d

Vertical stretch a, horizontal stretch \tfrac1b, shift c right and d up. Inside the bracket acts on x the opposite way.

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

Swap x and y, then make y the subject. The graph reflects in y=x.

Topic 3: Geometry and trigonometry

SL 3.1 SL and HL

Distance between two points (x_1,y_1,z_1) and (x_2,y_2,z_2)

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

Coordinates of the midpoint of a line segment with endpoints (x_1,y_1,z_1) and (x_2,y_2,z_2)

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

Volume of a right-pyramid

V=\tfrac{1}{3}Ahwhere A is the area of the base, h is the height

Volume of a right cone

V=\tfrac{1}{3}\pi r^2hwhere r is the radius, h is the height

Area of the curved surface of a cone

A=\pi rlwhere r is the radius, l is the slant height

Photon notel is the slant height, not the vertical height. Find it with Pythagoras: l=\sqrt{r^2+h^2}.

Volume of a sphere

V=\tfrac{4}{3}\pi r^3where r is the radius

Surface area of a sphere

A=4\pi r^2where r is the radius

Photon noteA hemisphere's curved surface is 2\pi r^2; add \pi r^2 if it has a flat base.

SL 3.2 SL and HL

Sine rule

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

Photon noteGiven two sides and a non-included angle, check for a second (obtuse) solution: the ambiguous case.

Cosine rule

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

Photon noteUse it for three sides, or two sides and the angle between them.

Area of a triangle

A=\tfrac{1}{2}ab\sin C

Photon noteC is the angle between sides a and b.

SL 3.4 SL and HL

Length of an arc

l=r\thetawhere r is the radius, \theta is the angle measured in radians

Area of a sector

A=\tfrac{1}{2}r^2\thetawhere r is the radius, \theta is the angle measured in radians

Photon noteRadians only. Convert with 180^\circ=\pi rad.

SL 3.5 SL and HL

Identity for \tan\theta

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

SL 3.6 SL and HL

Pythagorean identity

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

Double angle identities

\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

Photon notePick the version that leaves only one trig function in your equation.

AHL 3.9 HL only

Reciprocal trigonometric identities

\sec\theta=\frac{1}{\cos\theta}

\mathrm{cosec}\,\theta=\frac{1}{\sin\theta}

Pythagorean identities

1+\tan^2\theta=\sec^2\theta

1+\cot^2\theta=\mathrm{cosec}^2\theta

AHL 3.10 HL only

Compound angle identities

\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B

\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B

Photon noteNote the sign flips in the cosine version.

\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}

Double angle identity for tan

\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}

AHL 3.12 HL only

Magnitude of a vector

|\boldsymbol{v}|=\sqrt{v_1^2+v_2^2+v_3^2}where \boldsymbol v=\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}

AHL 3.13 HL only

Scalar product

\boldsymbol{v}\cdot\boldsymbol{w}=v_1w_1+v_2w_2+v_3w_3where \boldsymbol v=\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}, \boldsymbol w=\begin{pmatrix}w_1\\w_2\\w_3\end{pmatrix}

Photon notePerpendicular vectors have \boldsymbol v\cdot\boldsymbol w=0.

\boldsymbol{v}\cdot\boldsymbol{w}=|\boldsymbol{v}||\boldsymbol{w}|\cos\thetawhere \theta is the angle between \boldsymbol v and \boldsymbol w

Angle between two vectors

\cos\theta=\frac{v_1w_1+v_2w_2+v_3w_3}{|\boldsymbol{v}||\boldsymbol{w}|}

Photon noteFor the angle between two lines, use their direction vectors. Take the acute angle unless told otherwise.

AHL 3.14 HL only

Vector equation of a line

\boldsymbol{r}=\boldsymbol{a}+\lambda\boldsymbol{b}

Photon note\boldsymbol a is a point on the line, \boldsymbol b its direction.

Parametric form of the equation of a line

x=x_0+\lambda l,\quad y=y_0+\lambda m,\quad z=z_0+\lambda n

Cartesian equations of a line

\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}

AHL 3.16 HL only

Vector product

\boldsymbol{v}\times\boldsymbol{w}=\begin{pmatrix}v_2w_3-v_3w_2\\v_3w_1-v_1w_3\\v_1w_2-v_2w_1\end{pmatrix}where \boldsymbol v=\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}, \boldsymbol w=\begin{pmatrix}w_1\\w_2\\w_3\end{pmatrix}

Photon noteThe result is perpendicular to both \boldsymbol v and \boldsymbol w: the quickest way to find a normal.

|\boldsymbol{v}\times\boldsymbol{w}|=|\boldsymbol{v}||\boldsymbol{w}|\sin\thetawhere \theta is the angle between \boldsymbol v and \boldsymbol w

Area of a parallelogram

A=|\boldsymbol{v}\times\boldsymbol{w}|where \boldsymbol v and \boldsymbol w form two adjacent sides of a parallelogram

Photon noteA triangle with these two sides has half this area.

AHL 3.17 HL only

Vector equation of a plane

\boldsymbol{r}=\boldsymbol{a}+\lambda\boldsymbol{b}+\mu\boldsymbol{c}

Equation of a plane (using the normal vector)

\boldsymbol{r}\cdot\boldsymbol{n}=\boldsymbol{a}\cdot\boldsymbol{n}

Cartesian equation of a plane

ax+by+cz=d

Photon noteThe coefficients (a,b,c) are a normal vector to the plane.

Not in the IB booklet · learn or derive these

Exact values\sin30^\circ=\tfrac12,\ \cos30^\circ=\tfrac{\sqrt3}{2},\ \sin45^\circ=\cos45^\circ=\tfrac{\sqrt2}{2},\ \tan60^\circ=\sqrt3

Paper 1 has no calculator: learn the 30°, 45°, 60° values and their radian forms.

Power-reducing forms\cos^2\theta=\frac{1+\cos2\theta}{2},\qquad \sin^2\theta=\frac{1-\cos2\theta}{2}

Rearranged from the booklet's \cos2\theta. Needed to integrate \sin^2x and \cos^2x (HL).

Angle between a line and a plane (HL)\sin\theta=\frac{|\boldsymbol{b}\cdot\boldsymbol{n}|}{|\boldsymbol{b}||\boldsymbol{n}|}

Sine, not cosine, because \boldsymbol n is perpendicular to the plane.

Topic 4: Statistics and probability

SL 4.2 SL and HL

Interquartile range

\text{IQR}=Q_3-Q_1

Photon noteAn outlier is more than 1.5\times\text{IQR} below Q_1 or above Q_3.

SL 4.3 SL and HL

Mean, \bar x, of a set of data

\bar x=\frac{\sum_{i=1}^{k}f_ix_i}{n}where n=\sum_{i=1}^{k}f_i

SL 4.5 SL and HL

Probability of an event A

\mathrm{P}(A)=\frac{n(A)}{n(U)}

Complementary events

\mathrm{P}(A)+\mathrm{P}(A')=1

Photon note“At least one” questions: 1-\mathrm P(\text{none}).

SL 4.6 SL and HL

Combined events

\mathrm{P}(A\cup B)=\mathrm{P}(A)+\mathrm{P}(B)-\mathrm{P}(A\cap B)

Mutually exclusive events

\mathrm{P}(A\cup B)=\mathrm{P}(A)+\mathrm{P}(B)

Photon noteMutually exclusive means \mathrm P(A\cap B)=0. That is not the same as independent.

Conditional probability

\mathrm{P}(A|B)=\frac{\mathrm{P}(A\cap B)}{\mathrm{P}(B)}

Independent events

\mathrm{P}(A\cap B)=\mathrm{P}(A)\,\mathrm{P}(B)

Photon noteTo test for independence, check whether this holds with the given numbers.

SL 4.7 SL and HL

Expected value of a discrete random variable X

\mathrm{E}(X)=\sum x\,\mathrm{P}(X=x)

Photon noteA fair game has \mathrm E(\text{gain})=0.

SL 4.8 SL and HL

Binomial distribution X\sim\mathrm B(n,p)

Photon noteFour conditions: fixed n, two outcomes, constant p, independent trials. State them when asked to justify the model.

Mean

\mathrm{E}(X)=np

Variance

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

Photon noteStandard deviation is the square root of this.

SL 4.12 SL and HL

Standardized normal variable

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

Photon noteUsed to find an unknown \mu or \sigma: get z from the inverse normal on the calculator.

AHL 4.13 HL only

Bayes' theorem

\mathrm{P}(B|A)=\frac{\mathrm{P}(B)\,\mathrm{P}(A|B)}{\mathrm{P}(B)\,\mathrm{P}(A|B)+\mathrm{P}(B')\,\mathrm{P}(A|B')}

Photon noteA tree diagram usually gets you there faster.

\mathrm{P}(B_i|A)=\frac{\mathrm{P}(B_i)\,\mathrm{P}(A|B_i)}{\mathrm{P}(B_1)\mathrm{P}(A|B_1)+\mathrm{P}(B_2)\mathrm{P}(A|B_2)+\mathrm{P}(B_3)\mathrm{P}(A|B_3)}

AHL 4.14 HL only

Variance \sigma^2

\sigma^2=\frac{\sum_{i=1}^{k}f_i(x_i-\mu)^2}{n}=\frac{\sum_{i=1}^{k}f_ix_i^2}{n}-\mu^2

Standard deviation \sigma

\sigma=\sqrt{\frac{\sum_{i=1}^{k}f_i(x_i-\mu)^2}{n}}

Linear transformation of a single random variable

\mathrm{E}(aX+b)=a\mathrm{E}(X)+b

\mathrm{Var}(aX+b)=a^2\,\mathrm{Var}(X)

Photon noteAdding b shifts the mean but not the spread.

Expected value of a continuous random variable X

\mathrm{E}(X)=\mu=\int_{-\infty}^{\infty}x\,f(x)\,\mathrm{d}x

Variance

\mathrm{Var}(X)=\mathrm{E}\big[(X-\mu)^2\big]=\mathrm{E}(X^2)-\big[\mathrm{E}(X)\big]^2

Variance of a discrete random variable X

\mathrm{Var}(X)=\sum(x-\mu)^2\,\mathrm{P}(X=x)=\sum x^2\,\mathrm{P}(X=x)-\mu^2

Variance of a continuous random variable X

\mathrm{Var}(X)=\int_{-\infty}^{\infty}(x-\mu)^2f(x)\,\mathrm{d}x=\int_{-\infty}^{\infty}x^2f(x)\,\mathrm{d}x-\mu^2

Not in the IB booklet · learn or derive these

Probabilities sum to 1\sum\mathrm{P}(X=x)=1

The first step for any unknown in a discrete probability table.

Probability density function (HL)\int_{-\infty}^{\infty}f(x)\,\mathrm{d}x=1,\qquad \mathrm{P}(a<X<b)=\int_a^bf(x)\,\mathrm{d}x

The median m satisfies \int_{-\infty}^{m}f(x)\,\mathrm dx=\tfrac12.

Topic 5: Calculus

SL 5.3 SL and HL

Derivative of x^n

f(x)=x^n\Rightarrow f'(x)=nx^{n-1}

Photon noteRewrite roots and fractions as powers first: \sqrt x=x^{1/2}, \dfrac1{x^2}=x^{-2}.

SL 5.5 SL and HL

Integral of x^n

\int x^n\,\mathrm{d}x=\frac{x^{n+1}}{n+1}+C,\quad n\neq-1

Photon noteNever drop the +C on an indefinite integral.

Area between a curve y=f(x) and the x-axis, where f(x)>0

A=\int_a^by\,\mathrm{d}x

SL 5.6 SL and HL

Derivative of \sin x

f(x)=\sin x\Rightarrow f'(x)=\cos x

Photon noteRadians only.

Derivative of \cos x

f(x)=\cos x\Rightarrow f'(x)=-\sin x

Derivative of \mathrm e^x

f(x)=\mathrm{e}^x\Rightarrow f'(x)=\mathrm{e}^x

Derivative of \ln x

f(x)=\ln x\Rightarrow f'(x)=\frac{1}{x}

Chain rule

y=g(u),\ \text{where } u=f(x)\ \Rightarrow\ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\times\frac{\mathrm{d}u}{\mathrm{d}x}

Product rule

y=uv\Rightarrow\frac{\mathrm{d}y}{\mathrm{d}x}=u\frac{\mathrm{d}v}{\mathrm{d}x}+v\frac{\mathrm{d}u}{\mathrm{d}x}

Quotient rule

y=\frac{u}{v}\Rightarrow\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{v\frac{\mathrm{d}u}{\mathrm{d}x}-u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}

Photon noteThe minus sign makes order matter: v\,u' comes first.

SL 5.9 SL and HL

Acceleration

a=\frac{\mathrm{d}v}{\mathrm{d}t}=\frac{\mathrm{d}^2s}{\mathrm{d}t^2}

Distance travelled from t_1 to t_2

\text{distance}=\int_{t_1}^{t_2}|v(t)|\,\mathrm{d}t

Photon noteDistance uses |v|; displacement does not. Questions ask for both.

Displacement from t_1 to t_2

\text{displacement}=\int_{t_1}^{t_2}v(t)\,\mathrm{d}t

SL 5.10 SL and HL

Standard integrals

\int\frac{1}{x}\,\mathrm{d}x=\ln|x|+C

\int\sin x\,\mathrm{d}x=-\cos x+C

\int\cos x\,\mathrm{d}x=\sin x+C

\int\mathrm{e}^x\,\mathrm{d}x=\mathrm{e}^x+C

SL 5.11 SL and HL

Area of region enclosed by a curve and x-axis

A=\int_a^b|y|\,\mathrm{d}x

Photon noteIf the curve crosses the axis, split the integral at the root or use the modulus on the calculator.

AHL 5.12 HL only

Derivative of f(x) from first principles

y=f(x)\Rightarrow\frac{\mathrm{d}y}{\mathrm{d}x}=f'(x)=\lim_{h\to0}\left(\frac{f(x+h)-f(x)}{h}\right)

Photon noteWrite “lim” on every line until you let h\to0.

AHL 5.15 HL only

Standard derivatives: \tan x

f(x)=\tan x\Rightarrow f'(x)=\sec^2x

\sec x

f(x)=\sec x\Rightarrow f'(x)=\sec x\tan x

\mathrm{cosec}\,x

f(x)=\mathrm{cosec}\,x\Rightarrow f'(x)=-\mathrm{cosec}\,x\cot x

\cot x

f(x)=\cot x\Rightarrow f'(x)=-\mathrm{cosec}^2x

a^x

f(x)=a^x\Rightarrow f'(x)=a^x(\ln a)

\log_ax

f(x)=\log_ax\Rightarrow f'(x)=\frac{1}{x\ln a}

\arcsin x

f(x)=\arcsin x\Rightarrow f'(x)=\frac{1}{\sqrt{1-x^2}}

\arccos x

f(x)=\arccos x\Rightarrow f'(x)=-\frac{1}{\sqrt{1-x^2}}

\arctan x

f(x)=\arctan x\Rightarrow f'(x)=\frac{1}{1+x^2}

Standard integrals

\int a^x\,\mathrm{d}x=\frac{1}{\ln a}a^x+C

\int\frac{1}{a^2+x^2}\,\mathrm{d}x=\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C

\int\frac{1}{\sqrt{a^2-x^2}}\,\mathrm{d}x=\arcsin\left(\frac{x}{a}\right)+C,\quad |x|<a

AHL 5.16 HL only

Integration by parts

\int u\frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x=uv-\int v\frac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x\quad\text{or}\quad\int u\,\mathrm{d}v=uv-\int v\,\mathrm{d}u

Photon noteChoose u as the part that gets simpler when differentiated (often a power of x, or \ln x).

AHL 5.17 HL only

Area of region enclosed by a curve and y-axis

A=\int_a^b|x|\,\mathrm{d}y

Volume of revolution about the x or y-axes

V=\int_a^b\pi y^2\,\mathrm{d}x\quad\text{or}\quad V=\int_a^b\pi x^2\,\mathrm{d}y

Photon noteSquare first, then integrate. Limits must match the variable.

AHL 5.18 HL only

Euler's method

y_{n+1}=y_n+h\times f(x_n,y_n);\quad x_{n+1}=x_n+hwhere h is a constant (step length)

Photon noteSet it up as a calculator table. Smaller h gives a better approximation.

Integrating factor for y'+P(x)y=Q(x)

\mathrm{e}^{\int P(x)\,\mathrm{d}x}

Photon noteDivide through so that y' has coefficient 1 before finding P(x).

AHL 5.19 HL only

Maclaurin series

f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\dots

Maclaurin series for special functions

\mathrm{e}^x=1+x+\frac{x^2}{2!}+\dots

\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots

\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots

\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\dots

Photon noteSubstitute into these (for example x\to2x) rather than differentiating from scratch.

Not in the IB booklet · learn or derive these

Tangent and normal at x=ay-f(a)=f'(a)(x-a),\qquad m_\text{normal}=-\frac{1}{f'(a)}

Stationary pointsf'(x)=0;\quad f''(x)>0\ \text{minimum},\ f''(x)<0\ \text{maximum}

If f''(x)=0, check the sign of f' either side. A point of inflexion needs f'' to change sign.

Linear substitution\int f(ax+b)\,\mathrm{d}x=\frac{1}{a}F(ax+b)+C

For example \int\mathrm e^{3x}\,\mathrm dx=\tfrac13\mathrm e^{3x}+C.

Implicit differentiation (HL)\frac{\mathrm{d}}{\mathrm{d}x}\big(y^n\big)=ny^{n-1}\frac{\mathrm{d}y}{\mathrm{d}x}

Differentiate every term with respect to x, then collect the \dfrac{\mathrm dy}{\mathrm dx} terms.

L'Hôpital's rule (HL)\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}

Only for the forms \tfrac00 or \tfrac{\infty}{\infty}. Say which form it is.

Separating variables (HL)\frac{\mathrm{d}y}{\mathrm{d}x}=f(x)g(y)\ \Rightarrow\ \int\frac{1}{g(y)}\,\mathrm{d}y=\int f(x)\,\mathrm{d}x

One +C is enough, on one side.

Source: IB Mathematics: analysis and approaches formula booklet, first examinations 2021, version 1.3, retyped by Photon Resources. This is an unofficial study companion, not produced or endorsed by the International Baccalaureate. In the exam you use the clean official booklet your school provides. All notes and extra formulas are ours.

IB Maths AA formula booklet, annotated

Print it: one-page versions

Prior learning

SL and HL

Topic 1: Number and algebra

SL 1.2 SL and HL

SL 1.3 SL and HL

SL 1.4 SL and HL

SL 1.5 SL and HL

SL 1.7 SL and HL

SL 1.8 SL and HL

SL 1.9 SL and HL

AHL 1.10 HL only

AHL 1.12 HL only

AHL 1.13 HL only

AHL 1.14 HL only

Topic 2: Functions

SL 2.1 SL and HL

SL 2.6 SL and HL

SL 2.7 SL and HL

SL 2.9 SL and HL

AHL 2.12 HL only

Topic 3: Geometry and trigonometry

SL 3.1 SL and HL

SL 3.2 SL and HL

SL 3.4 SL and HL

SL 3.5 SL and HL

SL 3.6 SL and HL

AHL 3.9 HL only

AHL 3.10 HL only

AHL 3.12 HL only

AHL 3.13 HL only

AHL 3.14 HL only

AHL 3.16 HL only

AHL 3.17 HL only

Topic 4: Statistics and probability

SL 4.2 SL and HL

SL 4.3 SL and HL

SL 4.5 SL and HL

SL 4.6 SL and HL

SL 4.7 SL and HL

SL 4.8 SL and HL

SL 4.12 SL and HL

AHL 4.13 HL only

AHL 4.14 HL only

Topic 5: Calculus

SL 5.3 SL and HL

SL 5.5 SL and HL

SL 5.6 SL and HL

SL 5.9 SL and HL

SL 5.10 SL and HL

SL 5.11 SL and HL

AHL 5.12 HL only

AHL 5.15 HL only

AHL 5.16 HL only

AHL 5.17 HL only

AHL 5.18 HL only

AHL 5.19 HL only

Formula booklet questions

IB Maths AA topic cheatsheets

Other annotated booklets