IB Math Revisionibmathrevision.com
IB Mathematics: Analysis & Approaches HLOne-page formula sheet (SL + HL) · ★ = not in the booklet
★ not in the IB formula booklet — learn it. Everything else is printed in the booklet (its notation may differ slightly).
Prior learning
Areas: \(\text{parallelogram }bh,\) \(\text{triangle }\tfrac12bh,\) \(\text{trapezoid }\tfrac12(a+b)h,\) \(\text{circle }\pi r^2\)
Circumference; cylinder: \(C=2\pi r;\) \(V=\pi r^2h,\) \(\text{curved }A=2\pi rh\)
Cuboid; prism: \(V=lwh;\) \(V=Ah\)
Distance; midpoint: \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2},\) \(\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)\)
★Indices: \(a^ma^n=a^{m+n},\) \((a^m)^n=a^{mn},\) \(a^{-n}=\tfrac1{a^n},\) \(a^{\frac mn}=\sqrt[n]{a^m}\)
1 Number & algebra
Arithmetic: \(u_n=u_1+(n-1)d,\) \(S_n=\tfrac n2\big(2u_1+(n-1)d\big)\) \({}=\tfrac n2(u_1+u_n)\)
Geometric: \(u_n=u_1r^{n-1},\) \(S_n=\frac{u_1(1-r^n)}{1-r},\) \(S_\infty=\frac{u_1}{1-r}\ (|r|<1)\)
Compound interest: \(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)
Exponents & logs: \(a^x=b\iff x=\log_ab,\) \(a^x=e^{x\ln a},\) \(\log_aa^x=x=a^{\log_ax}\)
Log laws: \(\log_axy=\log_ax+\log_ay,\) \(\log_a\tfrac xy=\log_ax-\log_ay,\) \(\log_ax^m=m\log_ax\)
Change of base: \(\log_ba=\frac{\log_ca}{\log_cb}\)
Binomial, \(n\in\mathbb N\): \((a+b)^n=\sum_{r=0}^n\tbinom nra^{n-r}b^r,\) \(\tbinom nr=\frac{n!}{r!(n-r)!}\)
Permutations: \({}^nP_r=\frac{n!}{(n-r)!}\)
Binomial, \(n\in\mathbb Q\), \(|x|<1\): \((1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\cdots\)
★\((a+bx)^n=a^n\left(1+\tfrac bax\right)^n\), valid for \(|x|<\left|\tfrac ab\right|\)
★Partial fractions: \(\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}\) \({}+\frac{B}{x-b}\)
Complex numbers: \(z=a+bi=r(\cos\theta+i\sin\theta)\) \({}=re^{i\theta}=r\,\mathrm{cis}\,\theta\)
★Modulus, argument, conjugate: \(r=|z|=\sqrt{a^2+b^2},\) \(\tan\theta=\tfrac ba,\) \(z^*=a-bi,\) \(zz^*=|z|^2\)
★Products: \(|z_1z_2|=|z_1||z_2|,\) \(\arg(z_1z_2)=\arg z_1+\arg z_2,\) \(\arg\tfrac{z_1}{z_2}=\arg z_1-\arg z_2\)
De Moivre: \(\big[r(\cos\theta+i\sin\theta)\big]^n\) \({}=r^n(\cos n\theta+i\sin n\theta)\) \({}=r^ne^{in\theta}\)
★\(n\)th roots of \(re^{i\theta}\): \(r^{1/n}e^{i(\theta+2\pi k)/n},\) \(k=0,1,\dots,n-1\)
★For \(|z|=1\): \(z^n+z^{-n}=2\cos n\theta,\) \(z^n-z^{-n}=2i\sin n\theta\)
★Roots of unity sum to 0; non-real roots of real polynomials come in conjugate pairs
★Induction: prove \(P(1)\); assume \(P(k)\); prove \(P(k+1)\); conclude for all \(n\in\mathbb Z^+\)
★Contradiction: assume the negation, reach a contradiction. Counterexample disproves
2 Functions
Lines: \(m=\frac{y_2-y_1}{x_2-x_1},\) \(y=mx+c,\) \(ax+by+d=0,\) \(y-y_1=m(x-x_1)\)
★Perpendicular: \(m_1m_2=-1\)
Quadratics: \(x=-\frac b{2a},\) \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\) \(\Delta=b^2-4ac\)
★\(\Delta>0\): two real roots; \(\Delta=0\): repeated root; \(\Delta<0\): complex conjugate roots
Sum, product of roots of \(\sum_0^na_rx^r\): \(-\frac{a_{n-1}}{a_n};\) \(\frac{(-1)^na_0}{a_n}\)
★Factor & remainder theorems: \(p(a)=0\iff(x-a)\text{ is a factor};\) \(\text{remainder of }p(x)\div(x-a)\text{ is }p(a)\)
★Odd; even: \(f(-x)=-f(x);\) \(f(-x)=f(x)\)
★Composite; inverse: \((f\circ g)(x)=f(g(x));\) \(f\big(f^{-1}(x)\big)=x\)
★Transformations: \(f(x)+b\) up \(b\); \(f(x-a)\) right \(a\); \(pf(x)\) vertical ×\(p\); \(f(qx)\) horizontal ×\(\tfrac1q\); \(-f(x)\), \(f(-x)\) reflect in \(x\)-, \(y\)-axis
★Graphs: \(|f(x)|\) reflects negative parts up; \(f(|x|)\) mirrors \(x\ge0\); \(\frac1{f(x)}\) has asymptotes at zeros of \(f\)
★\(y=\frac{ax+b}{cx+d}\) asymptotes: \(x=-\frac dc,\) \(y=\frac ac\)
3 Geometry & trigonometry
3D distance (\(\Delta x=x_1-x_2\), …): \(d=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}\)
3D midpoint: \(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2,\tfrac{z_1+z_2}2\right)\)
Pyramid; cone; sphere: \(V=\tfrac13Ah;\) \(V=\tfrac13\pi r^2h,\) \(A=\pi rl;\) \(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
Sine, cosine rules; area: \(\frac{a}{\sin A}=\frac{b}{\sin B}\) \({}=\frac{c}{\sin C},\) \(c^2=a^2+b^2-2ab\cos C,\) \(A=\tfrac12ab\sin C\)
Arc; sector (radians): \(l=r\theta,\) \(A=\tfrac12r^2\theta\)
Identities: \(\tan\theta=\frac{\sin\theta}{\cos\theta},\) \(\cos^2\theta+\sin^2\theta\) \({}=1\)
Reciprocal ratios: \(\sec\theta=\frac1{\cos\theta},\) \(\cosec\theta=\frac1{\sin\theta},\) \(\cot\theta=\frac{1}{\tan\theta}\)
Pythagorean: \(1+\tan^2\theta=\sec^2\theta,\) \(1+\cot^2\theta=\cosec^2\theta\)
Compound angles: \(\sin(A\pm B)=\sin A\cos B\) \({}\pm\cos A\sin B,\) \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)
Compound tan: \(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
Double angle: \(\sin2\theta=2\sin\theta\cos\theta,\) \(\cos2\theta=2\cos^2\theta-1\) \({}=1-2\sin^2\theta,\) \(\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}\)
★Exact values \(0,\frac\pi6,\frac\pi4,\frac\pi3,\frac\pi2\): \(\sin:0,\tfrac12,\tfrac{\sqrt2}2,\tfrac{\sqrt3}2,1;\) \(\cos\text{: reverse};\) \(\tan:0,\tfrac1{\sqrt3},1,\sqrt3,\text{undef.}\)
★Ranges: \(\arcsin x\in[-\tfrac\pi2,\tfrac\pi2],\) \(\arccos x\in[0,\pi],\) \(\arctan x\in(-\tfrac\pi2,\tfrac\pi2)\)
★\(a\sin(b(x-c))+d\), \(b>0\): \(\text{amplitude }|a|,\) \(\text{period }\tfrac{2\pi}b\)
Vector magnitude: \(|\boldsymbol v|=\sqrt{v_1^2+v_2^2+v_3^2}\)
Scalar product: \(\boldsymbol v\cdot\boldsymbol w\) \({}=v_1w_1+v_2w_2+v_3w_3\) \({}=|\boldsymbol v||\boldsymbol w|\cos\theta\)
★Perpendicular; unit vector; \(\overrightarrow{AB}\): \(\boldsymbol v\cdot\boldsymbol w\) \({}=0;\) \(\hat{\boldsymbol v}=\frac{\boldsymbol v}{|\boldsymbol v|};\) \(\boldsymbol b-\boldsymbol a\)
Line: \(\boldsymbol r=\boldsymbol a\) \({}+\lambda\boldsymbol b;\) \(x=x_0+\lambda l,\ \dots;\) \(\frac{x-x_0}l=\frac{y-y_0}m\) \({}=\frac{z-z_0}n\)
Vector product: \(\boldsymbol v\times\boldsymbol w\) \({}=\begin{pmatrix}v_2w_3-v_3w_2\\v_3w_1-v_1w_3\\v_1w_2-v_2w_1\end{pmatrix},\) \(|\boldsymbol v\times\boldsymbol w|\) \({}=|\boldsymbol v||\boldsymbol w|\sin\theta\)
Area of parallelogram: \(A=|\boldsymbol v\times\boldsymbol w|\)
★Area of triangle: \(\tfrac12|\boldsymbol v\times\boldsymbol w|\)
Plane: \(\boldsymbol r=\boldsymbol a\) \({}+\lambda\boldsymbol b\) \({}+\mu\boldsymbol c,\) \(\boldsymbol r\cdot\boldsymbol n\) \({}=\boldsymbol a\cdot\boldsymbol n,\) \(ax+by+cz=d\)
★Angle line–plane; plane–plane: \(\sin\theta=\frac{|\boldsymbol b\cdot\boldsymbol n|}{|\boldsymbol b||\boldsymbol n|};\) \(\cos\theta=\frac{|\boldsymbol n_1\cdot\boldsymbol n_2|}{|\boldsymbol n_1||\boldsymbol n_2|}\)
★Normal to plane through \(\boldsymbol a\) with directions \(\boldsymbol b,\boldsymbol c\): \(\boldsymbol n=\boldsymbol b\times\boldsymbol c\)
★Line of intersection of two planes: direction \(\boldsymbol n_1\times\boldsymbol n_2\)
4 Statistics & probability
IQR; mean: \(IQR=Q_3-Q_1,\) \(\bar x=\frac{\sum f_ix_i}{n}\)
★Outliers: \(<Q_1-1.5\,IQR\) \(\text{or}\) \({}>Q_3+1.5\,IQR\)
★Variance: \(\sigma^2=\frac{\sum f(x-\mu)^2}{n}\) \({}=\frac{\sum fx^2}{n}-\mu^2\)
Probability: \(P(A)=\frac{n(A)}{n(U)},\) \(P(A)+P(A')=1\)
Combined events: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Independent: \(P(A\cap B)=P(A)P(B)\)
Conditional: \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)
Bayes: \(P(B\mid A)=\frac{P(B)P(A\mid B)}{P(B)P(A\mid B)+P(B')P(A\mid B')}\)
Bayes, partition \(B_1,B_2,B_3\): \(P(B_i\mid A)=\frac{P(B_i)P(A\mid B_i)}{\sum_jP(B_j)P(A\mid B_j)}\)
Discrete \(X\): \(E(X)=\sum xP(X=x),\) \(\mathrm{Var}(X)=E(X^2)-[E(X)]^2\)
★Binomial pmf: \(P(X=r)=\tbinom nrp^r(1-p)^{n-r}\)
Binomial mean, variance: \(np,\) \(np(1-p)\)
Standardised normal: \(z=\frac{x-\mu}\sigma\)
Continuous \(X\): \(E(X)=\int x\,f(x)\,dx,\) \(\mathrm{Var}(X)=\int x^2f(x)\,dx-\mu^2\)
★pdf; median \(m\): \(\int f(x)\,dx=1;\) \(\int_{-\infty}^mf(x)\,dx=\tfrac12;\) \(\text{mode: max of }f\)
Linear transformation: \(E(aX+b)=aE(X)+b,\) \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\)
★Regression: \(y\) on \(x\) to predict \(y\), \(x\) on \(y\) for \(x\); interpolate only
5 Calculus
First principles: \(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\)
Basic derivatives: \((x^n)'=nx^{n-1},\) \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((e^x)'=e^x,\) \((\ln x)'=\tfrac1x\)
Trig derivatives: \((\tan x)'=\sec^2x,\) \((\sec x)'=\sec x\tan x,\) \((\cosec x)'=-\cosec x\cot x,\) \((\cot x)'=-\cosec^2x\)
More derivatives: \((a^x)'=a^x\ln a,\) \((\log_ax)'=\tfrac1{x\ln a}\)
Inverse trig: \((\arcsin x)'=\tfrac1{\sqrt{1-x^2}},\) \((\arccos x)'=-\tfrac1{\sqrt{1-x^2}},\) \((\arctan x)'=\tfrac1{1+x^2}\)
Chain, product, quotient: \(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx},\) \((uv)'=uv'+vu',\) \(\Big(\frac uv\Big)'=\frac{vu'-uv'}{v^2}\)
★Implicit: \(\tfrac{d}{dx}\big(y^n\big)\) \({}=ny^{n-1}\tfrac{dy}{dx},\) \(\tfrac{d}{dx}(xy)=x\tfrac{dy}{dx}+y\)
★Related rates: \(\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}\)
★Tangent; normal: \(y-f(a)=f'(a)(x-a);\) \(m_\text{n}=-\tfrac1{f'(a)}\)
★Max / min; inflexion: \(f'=0\text{ and }f''<0\ /\ f''>0;\) \(f''=0\text{ with sign change}\)
★L'Hôpital, for \(\frac00\) or \(\frac\infty\infty\): \(\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}\)
Standard integrals: \(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C,\) \(\int\tfrac1x\,dx=\ln|x|+C,\) \(\int\sin x\,dx=-\cos x+C,\) \(\int\cos x\,dx=\sin x+C,\) \(\int e^x\,dx=e^x+C\)
More integrals: \(\int a^x\,dx=\tfrac{a^x}{\ln a}+C,\) \(\int\tfrac{dx}{a^2+x^2}=\tfrac1a\arctan\tfrac xa+C,\) \(\int\tfrac{dx}{\sqrt{a^2-x^2}}\) \({}=\arcsin\tfrac xa+C\)
★Learn: \(\int\sec^2x\,dx=\tan x+C,\) \(\int\tfrac{f'(x)}{f(x)}\,dx\) \({}=\ln|f(x)|+C,\) \(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+C\)
★\(\sin^2x\), \(\cos^2x\): \(\sin^2x=\tfrac12(1-\cos2x),\) \(\cos^2x=\tfrac12(1+\cos2x)\)
By parts: \(\int u\frac{dv}{dx}\,dx=uv\) \({}-\int v\frac{du}{dx}\,dx\)
★Substitution: \(\int f(g(x))g'(x)\,dx=\int f(u)\,du;\) \(\text{change the limits too}\)
Areas: \(\int_a^b|y|\,dx,\) \(\int_c^d|x|\,dy\)
Volumes of revolution: \(V=\int_a^b\pi y^2\,dx,\) \(V=\int_c^d\pi x^2\,dy\)
Kinematics: \(v=\tfrac{ds}{dt},\) \(a=\tfrac{dv}{dt}=\tfrac{d^2s}{dt^2};\) \(\text{distance }\int|v|\,dt\)
Euler's method: \(y_{n+1}=y_n+h\,f(x_n,y_n),\) \(x_{n+1}=x_n+h\)
★Separable: \(\frac{dy}{dx}=f(x)g(y)\) \({}\Rightarrow\int\frac{dy}{g(y)}\) \({}=\int f(x)\,dx\)
★Homogeneous \(\frac{dy}{dx}=f\big(\frac yx\big)\): \(y=vx,\) \(\frac{dy}{dx}=v+x\frac{dv}{dx}\)
Integrating factor for \(y'+P(x)y=Q(x)\): \(e^{\int P(x)\,dx}\)
★Then: \(y\,e^{\int P\,dx}=\int Q\,e^{\int P\,dx}\,dx\)
Maclaurin: \(f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots\)
Series: \(e^x=1+x+\tfrac{x^2}{2!}+\cdots,\) \(\ln(1+x)=x-\tfrac{x^2}2\) \({}+\tfrac{x^3}3-\cdots\)
sin, cos: \(\sin x=x-\tfrac{x^3}{3!}\) \({}+\tfrac{x^5}{5!}-\cdots,\) \(\cos x=1-\tfrac{x^2}{2!}\) \({}+\tfrac{x^4}{4!}-\cdots\)
arctan: \(\arctan x=x-\tfrac{x^3}3\) \({}+\tfrac{x^5}5-\cdots\)