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IB Mathematics

Analysis and Approaches HLFormula Booklet

Prior learning

DescriptionFormula
Area of a parallelogram\displaystyle \text{Area of a parallelogram}A=bh, where b is the base, h is the height\displaystyle A = bh, \text{ where } b \text{ is the base, } h \text{ is the height}
Area of a triangle\displaystyle \text{Area of a triangle}A=12(bh), where b is the base, h is the height\displaystyle A = \frac{1}{2}(bh), \text{ where } b \text{ is the base, } h \text{ is the height}
Area of a trapezoid\displaystyle \text{Area of a trapezoid}A=12(a+b)h, where a and b are the parallel sides, h is the height\displaystyle A = \frac{1}{2}(a+b)h, \text{ where } a \text{ and } b \text{ are the parallel sides, } h \text{ is the height}
Area of a circle\displaystyle \text{Area of a circle}A=πr2, where r is the radius\displaystyle A = \pi r^2, \text{ where } r \text{ is the radius}
Circumference of a circle\displaystyle \text{Circumference of a circle}C=2πr, where r is the radius\displaystyle C = 2\pi r, \text{ where } r \text{ is the radius}
Volume of a cuboid\displaystyle \text{Volume of a cuboid}V=lwh, where l is the length, w is the width, h is the height\displaystyle V = lwh, \text{ where } l \text{ is the length, } w \text{ is the width, } h \text{ is the height}
Volume of a cylinder\displaystyle \text{Volume of a cylinder}V=πr2h, where r is the radius, h is the height\displaystyle V = \pi r^2h, \text{ where } r \text{ is the radius, } h \text{ is the height}
Volume of a prism\displaystyle \text{Volume of a prism}V=Ah, where A is the area of cross-section, h is the height\displaystyle V = Ah, \text{ where } A \text{ is the area of cross-section, } h \text{ is the height}
Area of the curved surface ofa cylinder\displaystyle \text{Area of the curved surface of}\\\text{a cylinder}A=2πrh, where r is the radius, h is the height\displaystyle A = 2\pi rh, \text{ where } r \text{ is the radius, } h \text{ is the height}
Distance between twopoints (x1,y1) and (x2,y2)\displaystyle \text{Distance between two}\\\text{points $(x_1, y_1)$ and $(x_2, y_2)$}d=(x1−x2)2+(y1−y2)2\displaystyle d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}
Coordinates of the midpoint ofa line segment with endpoints(x1,y1) and (x2,y2)\displaystyle \text{Coordinates of the midpoint of}\\\text{a line segment with endpoints}\\\text{$(x_1, y_1)$ and $(x_2, y_2)$}(x1+x22,y1+y22)\displaystyle \displaystyle\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Topic 1: Number and algebra

TopicDescriptionFormula
SL 1.2\displaystyle \text{SL 1.2}The nth term of anarithmetic sequence\displaystyle \text{The $n$th term of an}\\ \text{arithmetic sequence}un=u1+(n−1)d\displaystyle u_n = u_1 + (n-1)d
SL 1.2\displaystyle \text{SL 1.2}The sum of n terms of anarithmetic sequence\displaystyle \text{The sum of $n$ terms of an}\\ \text{arithmetic sequence}Sn=n2(2u1+(n−1)d);  Sn=n2(u1+un)\displaystyle \displaystyle S_n = \frac{n}{2}(2u_1 + (n-1)d); \; \displaystyle S_n = \frac{n}{2}(u_1 + u_n)
SL 1.3\displaystyle \text{SL 1.3}The nth term of ageometric sequence\displaystyle \text{The $n$th term of a}\\ \text{geometric sequence}un=u1rn−1\displaystyle u_n = u_1r^{n-1}
SL 1.3\displaystyle \text{SL 1.3}The sum of n terms of afinite geometric sequence\displaystyle \text{The sum of $n$ terms of a}\\ \text{finite geometric sequence}Sn=u1(rn−1)r−1=u1(1−rn)1−r,  r≠1\displaystyle \displaystyle S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1-r^n)}{1-r}, \; r \neq 1
SL 1.4\displaystyle \text{SL 1.4}Compound interest\displaystyle \text{Compound interest}FV=PV×(1+r100k)kn, where FV is the future value,where 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\displaystyle \displaystyle FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}, \text{ where }FV\text{ is the future value,}\\[1em] \text{where }PV\text{ is the present value, }n\text{ is the number of years,}\\ \text{$k$ is the number of compounding periods per year,}\\ \text{$r\%$ is the nominal annual rate of interest}
SL 1.5\displaystyle \text{SL 1.5}Exponents and logarithms\displaystyle \text{Exponents and logarithms}ax=b⇔x=log⁡ab, where a>0,b>0,a≠1\displaystyle a^x = b \Leftrightarrow x = \log_a b, \text{ where } a > 0, b > 0, a \neq 1
SL 1.7\displaystyle \text{SL 1.7}Exponents and logarithms\displaystyle \text{Exponents and logarithms}log⁡axy=log⁡ax+log⁡aylog⁡axy=log⁡ax−log⁡aylog⁡axm=mlog⁡axlog⁡ax=log⁡bxlog⁡ba\displaystyle \displaystyle \log_a xy = \log_a x + \log_a y\\[1em] \log_a \frac{x}{y} = \log_a x - \log_a y\\[1em] \log_a x^m = m\log_a x\\[1em] \log_a x = \frac{\log_b x}{\log_b a}
SL 1.8\displaystyle \text{SL 1.8}The sum of an infinitegeometric sequence\displaystyle \text{The sum of an infinite}\\ \text{geometric sequence}S∞=u11−r,  ∣r∣<1\displaystyle \displaystyle S_\infty = \frac{u_1}{1-r}, \; |r| < 1
SL 1.9\displaystyle \text{SL 1.9}Binomial theorem n∈N\displaystyle \text{Binomial theorem } n \in \mathbb{N}(a+b)n=an+n ⁣C1an−1b+...+n ⁣Cran−rbr+...+bnn ⁣Cr=n!r!(n−r)!\displaystyle \displaystyle (a + b)^n = a^n + {^n\!C_1}a^{n-1}b + ... + {^n\!C_r}a^{n-r}b^r + ... + b^n\\[1em] {^n\!C_r} = \frac{n!}{r!(n-r)!}
AHL 1.10\displaystyle \text{AHL 1.10}Combinations\displaystyle \text{Combinations}n ⁣Cr=n!r!(n−r)!\displaystyle \displaystyle {^n\!C_r} = \frac{n!}{r!(n-r)!}
AHL 1.10\displaystyle \text{AHL 1.10}Permutations\displaystyle \text{Permutations}n ⁣Pr=n!(n−r)!\displaystyle \displaystyle {^n\!P_r} = \frac{n!}{(n-r)!}
AHL 1.10\displaystyle \text{AHL 1.10}Extension of binomialtheorem, n∈Q\displaystyle \text{Extension of binomial}\\ \text{theorem, } n \in \mathbb{Q}(a+b)n=an(1+n(ba)+n(n−1)2!(ba)2+...)\displaystyle \displaystyle (a+b)^n = a^n\left(1 + n\left(\frac{b}{a}\right) + \frac{n(n-1)}{2!}\left(\frac{b}{a}\right)^2 + ...\right)
AHL 1.12\displaystyle \text{AHL 1.12}Complex numbers\displaystyle \text{Complex numbers}z=a+bi\displaystyle z = a + bi
AHL 1.13\displaystyle \text{AHL 1.13}Modulus-argument (polar)and exponential (Euler)form\displaystyle \text{Modulus-argument (polar)}\\ \text{and exponential (Euler)}\\ \text{form}z=r(cos⁡θ+isin⁡θ)=reiθ=r cis θ\displaystyle z = r(\cos \theta + i\sin \theta) = re^{i\theta} = r\text{ cis }\theta
AHL 1.14\displaystyle \text{AHL 1.14}De Moivre’s theorem\displaystyle \text{De Moivre's theorem}[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)=rneinθ=rn cis nθ\displaystyle [r(\cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta) = r^ne^{in\theta} = r^n\text{ cis }n\theta

Topic 2: Functions

TopicDescriptionFormula
SL 2.1\displaystyle \text{SL 2.1}Equations of a straight line\displaystyle \text{Equations of a straight line}y=mx+c;  ax+by+d=0;y−y1=m(x−x1)\displaystyle y = mx + c; \; ax + by + d = 0; \\ y - y_1 = m(x - x_1)
SL 2.1\displaystyle \text{SL 2.1}Gradient formula\displaystyle \text{Gradient formula}m=y2−y1x2−x1\displaystyle \displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}
SL 2.6\displaystyle \text{SL 2.6}Axis of symmetry of thegraph of a quadraticfunction\displaystyle \text{Axis of symmetry of the}\\ \text{graph of a quadratic}\\ \text{function}f(x)=ax2+bx+c⇒ axis of symmetry is x=−b2a\displaystyle \displaystyle f(x) = ax^2 + bx + c \Rightarrow \text{ axis of symmetry is } x = -\frac{b}{2a}
SL 2.7\displaystyle \text{SL 2.7}Solutions of a quadraticequation\displaystyle \text{Solutions of a quadratic}\\ \text{equation}ax2+bx+c=0⇒x=−b±b2−4ac2a, a≠0\displaystyle ax^2 + bx + c = 0 \Rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \text{ } a \neq 0
SL 2.7\displaystyle \text{SL 2.7}Discriminant\displaystyle \text{Discriminant}Δ=b2−4ac\displaystyle \Delta = b^2 - 4ac
SL 2.9\displaystyle \text{SL 2.9}Exponential andlogarithmic functions\displaystyle \text{Exponential and}\\ \text{logarithmic functions}ax=exln⁡a;  log⁡ax=ln⁡xln⁡a, where a,x>0,  a≠1\displaystyle a^x = e^{x\ln a}; \; \log_a x = \frac{\ln x}{\ln a}, \text{ where } a, x > 0, \; a \neq 1
AHL 2.12\displaystyle \text{AHL 2.12}Sum and product of theroots of polynomialequations of the form∑r=0narxr=0\displaystyle \text{Sum and product of the}\\ \text{roots of polynomial}\\ \text{equations of the form}\\ \sum\limits_{r=0}^n a_rx^r = 0Sum is −an−1an  ; product is (−1)na0an\displaystyle \text{Sum is } \frac{-a_{n-1}}{a_n} \; \text{; product is } \frac{(-1)^n a_0}{a_n}

Topic 3: Geometry and trigonometry

TopicDescriptionFormula
SL 3.1\displaystyle \text{SL 3.1}Distance between twopoints (x1,y1,z1) and (x2,y2,z2)\displaystyle \text{Distance between two}\\\text{points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$}d=(x1−x2)2+(y1−y2)2+(z1−z2)2\displaystyle d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}
SL 3.1\displaystyle \text{SL 3.1}Coordinates of themidpoint of a line segmentwith endpoints (x1,y1,z1)and (x2,y2,z2)\displaystyle \text{Coordinates of the}\\ \text{midpoint of a line segment}\\ \text{with endpoints $(x_1, y_1, z_1)$} \\ \text{and $(x_2, y_2, z_2)$}(x1+x22,y1+y22,z1+z22)\displaystyle \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)
SL 3.1\displaystyle \text{SL 3.1}Volume of a right-pyramid\displaystyle \text{Volume of a right-pyramid}V=13Ah, where A is the area of the base, h is the height\displaystyle V = \frac{1}{3}Ah \text{, where } A \text{ is the area of the base, } h \text{ is the height}
SL 3.1\displaystyle \text{SL 3.1}Volume of a right cone\displaystyle \text{Volume of a right cone}V=13πr2h, where r is the radius, h is the height\displaystyle V = \frac{1}{3}\pi r^2h \text{, where } r \text{ is the radius, } h \text{ is the height}
SL 3.1\displaystyle \text{SL 3.1}Area of the curved surfaceof a cone\displaystyle \text{Area of the curved surface}\\\text{of a cone}A=πrl, where r is the radius, l is the slant height\displaystyle A = \pi rl \text{, where } r \text{ is the radius, } l \text{ is the slant height}
SL 3.1\displaystyle \text{SL 3.1}Volume of a sphere\displaystyle \text{Volume of a sphere}V=43πr3, where r is the radius\displaystyle V = \frac{4}{3}\pi r^3 \text{, where } r \text{ is the radius}
SL 3.1\displaystyle \text{SL 3.1}Surface area of a sphere\displaystyle \text{Surface area of a sphere}A=4πr2, where r is the radius\displaystyle A = 4\pi r^2 \text{, where } r \text{ is the radius}
SL 3.2\displaystyle \text{SL 3.2}Sine rule\displaystyle \text{Sine rule}asin⁡A=bsin⁡B=csin⁡C\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
SL 3.2\displaystyle \text{SL 3.2}Cosine rule\displaystyle \text{Cosine rule}c2=a2+b2−2abcos⁡C;  cos⁡C=a2+b2−c22ab\displaystyle c^2 = a^2 + b^2 - 2ab\cos C ; \; \cos C = \frac{a^2 + b^2 - c^2}{2ab}
SL 3.2\displaystyle \text{SL 3.2}Area of a triangle\displaystyle \text{Area of a triangle}A=12absin⁡C\displaystyle A = \frac{1}{2}ab\sin C
SL 3.4\displaystyle \text{SL 3.4}Length of an arc\displaystyle \text{Length of an arc}l=rθwhere r is the radius, θ is the angle measured in radians\displaystyle l = r\theta\\ \text{where } r \text{ is the radius, } \theta \text{ is the angle measured in radians}
SL 3.4\displaystyle \text{SL 3.4}Area of a sector\displaystyle \text{Area of a sector}A=12r2θwhere r is the radius, θ is the angle measured in radians\displaystyle \displaystyle A = \frac{1}{2}r^2\theta\\ \text{where } r \text{ is the radius, } \theta \text{ is the angle measured in radians}
SL 3.5\displaystyle \text{SL 3.5}Identity for tan⁡θ\displaystyle \text{Identity for } \tan \thetatan⁡θ=sin⁡θcos⁡θ\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}
SL 3.6\displaystyle \text{SL 3.6}Pythagorean identity\displaystyle \text{Pythagorean identity}cos⁡2θ+sin⁡2θ=1\displaystyle \cos^2 \theta + \sin^2 \theta = 1
SL 3.6\displaystyle \text{SL 3.6}Double angle identities\displaystyle \text{Double angle identities}sin⁡2θ=2sin⁡θcos⁡θcos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\displaystyle \sin 2\theta = 2\sin \theta \cos \theta\\[1em] \cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta
AHL 3.9\displaystyle \text{AHL 3.9}Reciprocal trigonometricidentities\displaystyle \text{Reciprocal trigonometric}\\ \text{identities}sec⁡θ=1cos⁡θcosec⁡θ=1sin⁡θ\displaystyle \sec \theta = \frac{1}{\cos \theta}\\[1em] \cosec \theta = \frac{1}{\sin \theta}
AHL 3.9\displaystyle \text{AHL 3.9}Pythagorean identities\displaystyle \text{Pythagorean identities}1+tan⁡2θ=sec⁡2θ1+cot⁡2θ=cosec⁡2θ\displaystyle 1 + \tan^2 \theta = \sec^2 \theta\\[1em] 1 + \cot^2 \theta = \cosec^2 \theta
AHL 3.10\displaystyle \text{AHL 3.10}Compound angle identities\displaystyle \text{Compound angle identities}sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡Bcos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡Btan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\displaystyle \sin(A \pm B) = \sin A\cos B \pm \cos A\sin B\\[1em] \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B\\[1em] \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
AHL 3.10\displaystyle \text{AHL 3.10}Double angle identityfor tan⁡\displaystyle \text{Double angle identity}\\ \text{for }\tantan⁡2θ=2tan⁡θ1−tan⁡2θ\displaystyle \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}
AHL 3.12\displaystyle \text{AHL 3.12}Magnitude of a vector\displaystyle \text{Magnitude of a vector}∣v∣=v12+v22+v32, where v=(v1v2v3)\displaystyle |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}
AHL 3.13\displaystyle \text{AHL 3.13}Scalar product\displaystyle \text{Scalar product}v⋅w=v1w1+v2w2+v3w3, where v=(v1v2v3),w=(w1w2w3)v⋅w=∣v∣∣w∣cos⁡θ, where θ is the angle between v and w\displaystyle \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}\\[1em] \mathbf{v} \cdot \mathbf{w} = |\mathbf{v}||\mathbf{w}|\cos \theta, \text{ where } \theta \text{ is the angle between } \mathbf{v} \text{ and } \mathbf{w}
AHL 3.13\displaystyle \text{AHL 3.13}Angle between twovectors\displaystyle \text{Angle between two}\\ \text{vectors}cos⁡θ=v1w1+v2w2+v3w3∣v∣∣w∣\displaystyle \cos \theta = \frac{v_1w_1 + v_2w_2 + v_3w_3}{|\mathbf{v}||\mathbf{w}|}
AHL 3.14\displaystyle \text{AHL 3.14}Vector equation of a line\displaystyle \text{Vector equation of a line}r=a+λb\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b}
AHL 3.14\displaystyle \text{AHL 3.14}Parametric form of theequation of a line\displaystyle \text{Parametric form of the}\\ \text{equation of a line}x=x0+λl, y=y0+λm, z=z0+λn\displaystyle x = x_0 + \lambda l, \text{ } y = y_0 + \lambda m, \text{ } z = z_0 + \lambda n
AHL 3.14\displaystyle \text{AHL 3.14}Cartesian equations of aline\displaystyle \text{Cartesian equations of a}\\ \text{line}x−x0l=y−y0m=z−z0n\displaystyle \frac{x-x_0}{l} = \frac{y-y_0}{m} = \frac{z-z_0}{n}
AHL 3.16\displaystyle \text{AHL 3.16}Vector product\displaystyle \text{Vector product}v×w=(v2w3−v3w2v3w1−v1w3v1w2−v2w1), where v=(v1v2v3),w=(w1w2w3)∣v×w∣=∣v∣∣w∣sin⁡θ, where θ is the angle between v and w\displaystyle \mathbf{v} \times \mathbf{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}\\[2em] |\mathbf{v} \times \mathbf{w}| = |\mathbf{v}||\mathbf{w}|\sin \theta, \text{ where } \theta \text{ is the angle between } \mathbf{v} \text{ and } \mathbf{w}
AHL 3.16\displaystyle \text{AHL 3.16}Area of a parallelogram\displaystyle \text{Area of a parallelogram}A=∣v×w∣ where v and w form two adjacent sides of aparallelogram\displaystyle A = |\mathbf{v} \times \mathbf{w}| \text{ where } \mathbf{v} \text{ and } \mathbf{w} \text{ form two adjacent sides of a}\\ \text{parallelogram}
AHL 3.17\displaystyle \text{AHL 3.17}Vector equation of a plane\displaystyle \text{Vector equation of a plane}r=a+λb+μc\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}
AHL 3.17\displaystyle \text{AHL 3.17}Equation of a plane(using the normal vector)\displaystyle \text{Equation of a plane}\\ \text{(using the normal vector)}r⋅n=a⋅n\displaystyle \mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}
AHL 3.17\displaystyle \text{AHL 3.17}Cartesian equation of aplane\displaystyle \text{Cartesian equation of a}\\ \text{plane}ax+by+cz=d\displaystyle ax + by + cz = d

Topic 4: Statistics and probability

TopicDescriptionFormula
SL 4.2\displaystyle \text{SL 4.2}Interquartile range\displaystyle \text{Interquartile range}IQR⁡=Q3−Q1\displaystyle \operatorname{IQR} = Q_3 - Q_1
SL 4.3\displaystyle \text{SL 4.3}Mean, xˉ, of a set of data\displaystyle \text{Mean, } \bar{x} \text{, of a set of data}xˉ=∑i=1kfixin, where n=∑i=1kfi\displaystyle \bar{x} = \frac{\sum\limits_{i=1}^k f_ix_i}{n}, \text{ where } n = \sum\limits_{i=1}^k f_i
SL 4.5\displaystyle \text{SL 4.5}Probability of an event A\displaystyle \text{Probability of an event } AP⁡(A)=n(A)n(U)\displaystyle \operatorname{P}(A) = \frac{n(A)}{n(U)}
SL 4.5\displaystyle \text{SL 4.5}Complementary events\displaystyle \text{Complementary events}P⁡(A)+P⁡(A′)=1\displaystyle \operatorname{P}(A) + \operatorname{P}(A') = 1
SL 4.6\displaystyle \text{SL 4.6}Combined events\displaystyle \text{Combined events}P⁡(A∪B)=P⁡(A)+P⁡(B)−P⁡(A∩B)\displaystyle \operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B) - \operatorname{P}(A \cap B)
SL 4.6\displaystyle \text{SL 4.6}Mutually exclusive events\displaystyle \text{Mutually exclusive events}P⁡(A∪B)=P⁡(A)+P⁡(B)\displaystyle \operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B)
SL 4.6\displaystyle \text{SL 4.6}Conditional probability\displaystyle \text{Conditional probability}P⁡(A∣B)=P⁡(A∩B)P⁡(B)\displaystyle \operatorname{P}(A|B) = \frac{\operatorname{P}(A \cap B)}{\operatorname{P}(B)}
SL 4.6\displaystyle \text{SL 4.6}Independent events\displaystyle \text{Independent events}P⁡(A∩B)=P⁡(A)P⁡(B)\displaystyle \operatorname{P}(A \cap B) = \operatorname{P}(A)\operatorname{P}(B)
SL 4.7\displaystyle \text{SL 4.7}Expected value of adiscrete random variable X\displaystyle \text{Expected value of a}\\ \text{discrete random variable } XE⁡(X)=∑xP⁡(X=x)\displaystyle \operatorname{E}(X) = \sum x\operatorname{P}(X = x)
SL 4.8\displaystyle \text{SL 4.8}Binomial distributionX∼B(n,p)\displaystyle \text{Binomial distribution}\\ X \sim \text{B}(n, p)E⁡(X)=np\displaystyle \operatorname{E}(X) = np
SL 4.8\displaystyle \text{SL 4.8}Mean\displaystyle \text{Mean}E⁡(X)=np\displaystyle \operatorname{E}(X) = np
SL 4.8\displaystyle \text{SL 4.8}Variance\displaystyle \text{Variance}Var⁡(X)=np(1−p)\displaystyle \operatorname{Var}(X) = np(1-p)
SL 4.12\displaystyle \text{SL 4.12}Standardized normalvariable\displaystyle \text{Standardized normal}\\ \text{variable}z=x−μσ\displaystyle z = \frac{x - \mu}{\sigma}
AHL 4.13\displaystyle \text{AHL 4.13}Bayes’ theorem\displaystyle \text{Bayes' theorem}P⁡(B∣A)=P⁡(B)P⁡(A∣B)P⁡(B)P⁡(A∣B)+P⁡(B′)P⁡(A∣B′)P⁡(Bi∣A)=P⁡(Bi)P⁡(A∣Bi)P⁡(B1)P⁡(A∣B1)+P⁡(B2)P⁡(A∣B2)+P⁡(B3)P⁡(A∣B3)\displaystyle \operatorname{P}(B|A) = \frac{\operatorname{P}(B)\operatorname{P}(A|B)}{\operatorname{P}(B)\operatorname{P}(A|B) + \operatorname{P}(B')\operatorname{P}(A|B')}\\[2em] \operatorname{P}(B_i|A) = \frac{\operatorname{P}(B_i)\operatorname{P}(A|B_i)}{\operatorname{P}(B_1)\operatorname{P}(A|B_1) + \operatorname{P}(B_2)\operatorname{P}(A|B_2) + \operatorname{P}(B_3)\operatorname{P}(A|B_3)}
AHL 4.14\displaystyle \text{AHL 4.14}Variance σ2\displaystyle \text{Variance } \sigma^2σ2=∑i=1kfi(xi−μ)2n=∑i=1kfixi2n−μ2\displaystyle \sigma^2 = \frac{\sum\limits_{i=1}^k f_i(x_i - \mu)^2}{n} = \frac{\sum\limits_{i=1}^k f_ix_i^2}{n} - \mu^2
AHL 4.14\displaystyle \text{AHL 4.14}Standard deviation σ\displaystyle \text{Standard deviation } \sigmaσ=∑i=1kfi(xi−μ)2n\displaystyle \sigma = \sqrt{\frac{\sum\limits_{i=1}^k f_i(x_i - \mu)^2}{n}}
AHL 4.14\displaystyle \text{AHL 4.14}Linear transformation of asingle random variable\displaystyle \text{Linear transformation of a}\\ \text{single random variable}E⁡(aX+b)=aE⁡(X)+bVar⁡(aX+b)=a2Var⁡(X)\displaystyle \operatorname{E}(aX + b) = a\operatorname{E}(X) + b\\[1em] \operatorname{Var}(aX + b) = a^2\operatorname{Var}(X)
AHL 4.14\displaystyle \text{AHL 4.14}Expected value of acontinuous randomvariable X\displaystyle \text{Expected value of a}\\ \text{continuous random}\\ \text{variable } XE⁡(X)=μ=∫−∞∞xf(x) dx\displaystyle \operatorname{E}(X) = \mu = \int_{-\infty}^{\infty} xf(x)\,dx
AHL 4.14\displaystyle \text{AHL 4.14}Variance\displaystyle \text{Variance}Var⁡(X)=E⁡[(X−μ)2]=E⁡(X2)−[E⁡(X)]2\displaystyle \operatorname{Var}(X) = \operatorname{E}[(X-\mu)^2] = \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
AHL 4.14\displaystyle \text{AHL 4.14}Variance of a discreterandom variable X\displaystyle \text{Variance of a discrete}\\ \text{random variable } XVar⁡(X)=∑(x−μ)2P⁡(X=x)=∑x2P⁡(X=x)−μ2\displaystyle \operatorname{Var}(X) = \sum(x-\mu)^2\operatorname{P}(X=x) = \sum x^2\operatorname{P}(X=x) - \mu^2
AHL 4.14\displaystyle \text{AHL 4.14}Variance of a continuousrandom variable X\displaystyle \text{Variance of a continuous}\\ \text{random variable } XVar⁡(X)=∫−∞∞(x−μ)2f(x) dx=∫−∞∞x2f(x) dx−μ2\displaystyle \operatorname{Var}(X) = \int_{-\infty}^{\infty}(x-\mu)^2f(x)\,dx = \int_{-\infty}^{\infty}x^2f(x)\,dx - \mu^2

Topic 5: Calculus

TopicDescriptionFormula
SL 5.3\displaystyle \text{SL 5.3}Derivative of xn\displaystyle \text{Derivative of } x^nf(x)=xn⇒f′(x)=nxn−1\displaystyle f(x) = x^n \Rightarrow f'(x) = nx^{n-1}
SL 5.5\displaystyle \text{SL 5.5}Integral of xn\displaystyle \text{Integral of } x^n∫xn dx=xn+1n+1+C, n≠−1\displaystyle \int x^n \,dx = \frac{x^{n+1}}{n+1} + C, \text{ } n \neq -1
SL 5.5\displaystyle \text{SL 5.5}Area between a curvey=f(x) and the x-axis,where f(x)>0\displaystyle \text{Area between a curve}\\ \text{$y = f(x)$ and the $x$-axis,}\\ \text{where } f(x) > 0A=∫aby dx\displaystyle A = \int_a^b y\,dx
SL 5.6\displaystyle \text{SL 5.6}Derivative of sin⁡x\displaystyle \text{Derivative of } \sin xf(x)=sin⁡x⇒f′(x)=cos⁡x\displaystyle f(x) = \sin x \Rightarrow f'(x) = \cos x
SL 5.6\displaystyle \text{SL 5.6}Derivative of cos⁡x\displaystyle \text{Derivative of } \cos xf(x)=cos⁡x⇒f′(x)=−sin⁡x\displaystyle f(x) = \cos x \Rightarrow f'(x) = -\sin x
SL 5.6\displaystyle \text{SL 5.6}Derivative of ex\displaystyle \text{Derivative of } e^xf(x)=ex⇒f′(x)=ex\displaystyle f(x) = e^x \Rightarrow f'(x) = e^x
SL 5.6\displaystyle \text{SL 5.6}Derivative of ln⁡x\displaystyle \text{Derivative of } \ln xf(x)=ln⁡x⇒f′(x)=1x\displaystyle f(x) = \ln x \Rightarrow f'(x) = \frac{1}{x}
SL 5.6\displaystyle \text{SL 5.6}Chain rule\displaystyle \text{Chain rule}y=g(u), where u=f(x)⇒dydx=dydu×dudx\displaystyle y = g(u), \text{ where } u = f(x) \Rightarrow \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}
SL 5.6\displaystyle \text{SL 5.6}Product rule\displaystyle \text{Product rule}y=uv⇒dydx=udvdx+vdudx\displaystyle y = uv \Rightarrow \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}
SL 5.6\displaystyle \text{SL 5.6}Quotient rule\displaystyle \text{Quotient rule}y=uv⇒dydx=vdudx−udvdxv2\displaystyle y = \frac{u}{v} \Rightarrow \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}
SL 5.9\displaystyle \text{SL 5.9}Acceleration\displaystyle \text{Acceleration}a=dvdt=d2sdt2\displaystyle a = \frac{dv}{dt} = \frac{d^2s}{dt^2}
SL 5.9\displaystyle \text{SL 5.9}Distance travelled fromt1 to t2\displaystyle \text{Distance travelled from}\\ \text{$t_1$ to $t_2$}distance=∫t1t2∣v(t)∣ dt\displaystyle \text{distance} = \int_{t_1}^{t_2} |v(t)|\,dt
SL 5.9\displaystyle \text{SL 5.9}Displacement fromt1 to t2\displaystyle \text{Displacement from}\\ \text{$t_1$ to $t_2$}displacement=∫t1t2v(t) dt\displaystyle \text{displacement} = \int_{t_1}^{t_2} v(t)\,dt
SL 5.10\displaystyle \text{SL 5.10}Standard integrals\displaystyle \text{Standard integrals}∫1x dx=ln⁡∣x∣+C∫sin⁡x dx=−cos⁡x+C∫cos⁡x dx=sin⁡x+C∫ex dx=ex+C\displaystyle \int \frac{1}{x}\,dx = \ln|x| + C\\[1em] \int \sin x\,dx = -\cos x + C\\[1em] \int \cos x\,dx = \sin x + C\\[1em] \int e^x\,dx = e^x + C
SL 5.11\displaystyle \text{SL 5.11}Area of region enclosedby a curve and x-axis\displaystyle \text{Area of region enclosed}\\ \text{by a curve and $x$-axis}A=∫ab∣y∣ dx\displaystyle A = \int_a^b |y|\,dx
AHL 5.12\displaystyle \text{AHL 5.12}Derivative of f(x) fromfirst principles\displaystyle \text{Derivative of } f(x) \text{ from}\\ \text{first principles}y=f(x)⇒dydx=f′(x)=lim⁡h→0(f(x+h)−f(x)h)\displaystyle y = f(x) \Rightarrow \frac{dy}{dx} = f'(x) = \lim_{h \to 0}\left(\frac{f(x+h) - f(x)}{h}\right)
AHL 5.15\displaystyle \text{AHL 5.15}Standard derivativestan x\displaystyle \text{Standard derivatives}\\ \text{tan }xf(x)=tan⁡x⇒f′(x)=sec⁡2x\displaystyle f(x) = \tan x \Rightarrow f'(x) = \sec^2 x
AHL 5.15\displaystyle \text{AHL 5.15}sec⁡x\displaystyle \sec xf(x)=sec⁡x⇒f′(x)=sec⁡xtan⁡x\displaystyle f(x) = \sec x \Rightarrow f'(x) = \sec x\tan x
AHL 5.15\displaystyle \text{AHL 5.15}cosec⁡x\displaystyle \cosec xf(x)=cosec⁡x⇒f′(x)=−cosec⁡xcot⁡x\displaystyle f(x) = \cosec x \Rightarrow f'(x) = -\cosec x\cot x
AHL 5.15\displaystyle \text{AHL 5.15}cot⁡x\displaystyle \cot xf(x)=cot⁡x⇒f′(x)=−cosec⁡2x\displaystyle f(x) = \cot x \Rightarrow f'(x) = -\cosec^2 x
AHL 5.15\displaystyle \text{AHL 5.15}ax\displaystyle a^xf(x)=ax⇒f′(x)=ax(ln⁡a)\displaystyle f(x) = a^x \Rightarrow f'(x) = a^x(\ln a)
AHL 5.15\displaystyle \text{AHL 5.15}log⁡ax\displaystyle \log_a xf(x)=log⁡ax⇒f′(x)=1xln⁡a\displaystyle f(x) = \log_a x \Rightarrow f'(x) = \frac{1}{x\ln a}
AHL 5.15\displaystyle \text{AHL 5.15}arcsin⁡x\displaystyle \arcsin xf(x)=arcsin⁡x⇒f′(x)=11−x2\displaystyle f(x) = \arcsin x \Rightarrow f'(x) = \frac{1}{\sqrt{1-x^2}}
AHL 5.15\displaystyle \text{AHL 5.15}arccos⁡x\displaystyle \arccos xf(x)=arccos⁡x⇒f′(x)=−11−x2\displaystyle f(x) = \arccos x \Rightarrow f'(x) = -\frac{1}{\sqrt{1-x^2}}
AHL 5.15\displaystyle \text{AHL 5.15}arctan⁡x\displaystyle \arctan xf(x)=arctan⁡x⇒f′(x)=11+x2\displaystyle f(x) = \arctan x \Rightarrow f'(x) = \frac{1}{1+x^2}
AHL 5.15\displaystyle \text{AHL 5.15}Standard integrals\displaystyle \text{Standard integrals}∫ax dx=1ln⁡aax+C∫1a2+x2 dx=1aarctan⁡(xa)+C∫1a2−x2 dx=arcsin⁡(xa)+C, ∣x∣<a\displaystyle \int a^x\,dx = \frac{1}{\ln a}a^x + C\\[1em] \int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C\\[1em] \int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\left(\frac{x}{a}\right) + C, \text{ }|x| < a
AHL 5.16\displaystyle \text{AHL 5.16}Integration by parts\displaystyle \text{Integration by parts}∫udvdx dx=uv−∫vdudx dx or ∫u dv=uv−∫v du\displaystyle \int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx \text{ or } \int u\,dv = uv - \int v\,du
AHL 5.17\displaystyle \text{AHL 5.17}Area of region enclosedby a curve and y-axis\displaystyle \text{Area of region enclosed}\\ \text{by a curve and $y$-axis}A=∫cd∣x∣ dy\displaystyle A = \int_c^d |x|\,dy
AHL 5.17\displaystyle \text{AHL 5.17}Volume of revolutionabout the x or y-axes\displaystyle \text{Volume of revolution}\\ \text{about the $x$ or $y$-axes}V=∫abπy2 dx or V=∫cdπx2 dy\displaystyle V = \int_a^b \pi y^2\,dx \text{ or } V = \int_c^d \pi x^2\,dy
AHL 5.18\displaystyle \text{AHL 5.18}Euler’s method\displaystyle \text{Euler's method}yn+1=yn+h×f(xn,yn);  xn+1=xn+h, where h is a constant(step length)\displaystyle y_{n+1} = y_n + h \times f(x_n, y_n); \; x_{n+1} = x_n + h, \text{ where $h$ is a constant}\\ \text{(step length)}
AHL 5.18\displaystyle \text{AHL 5.18}Integrating factor fory′+P(x)y=Q(x)\displaystyle \text{Integrating factor for}\\ y' + P(x)y = Q(x)e∫P(x) dx\displaystyle e^{\int P(x)\,dx}
AHL 5.19\displaystyle \text{AHL 5.19}Maclaurin series\displaystyle \text{Maclaurin series}f(x)=f(0)+xf′(0)+x22!f′′(0)+...\displaystyle f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + ...
AHL 5.19\displaystyle \text{AHL 5.19}Maclaurin series forspecial functions\displaystyle \text{Maclaurin series for}\\ \text{special functions}ex=1+x+x22!+...ln⁡(1+x)=x−x22+x33−...sin⁡x=x−x33!+x55!−...cos⁡x=1−x22!+x44!−...arctan⁡x=x−x33+x55−...\displaystyle e^x = 1 + x + \frac{x^2}{2!} + ...\\[1em] \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...\\[1em] \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...\\[1em] \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...\\[1em] \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - ...