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IB Mathematics: Analysis & Approaches SLOne-page formula sheet · booklet formulas + ★ the ones you must learn
★ not in the IB formula booklet — learn it. Everything else is printed in the booklet (its notation may differ slightly).
Prior learning
Area: parallelogram, triangle, trapezoid: \(A=bh,\) \(A=\tfrac12bh,\) \(A=\tfrac12(a+b)h\)
Circle: \(A=\pi r^2,\) \(C=2\pi r\)
Volume: cuboid, cylinder, prism: \(V=lwh,\) \(V=\pi r^2h,\) \(V=Ah\)
Curved surface of a cylinder: \(A=2\pi rh\)
Distance; midpoint: \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2},\) \(\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)\)
★Pythagoras: \(a^2+b^2=c^2\)
★Right-angled trig: \(\sin\theta=\tfrac{\text{opp}}{\text{hyp}},\) \(\cos\theta=\tfrac{\text{adj}}{\text{hyp}},\) \(\tan\theta=\tfrac{\text{opp}}{\text{adj}}\)
★Laws of indices: \(a^ma^n=a^{m+n},\) \(\tfrac{a^m}{a^n}=a^{m-n},\) \((a^m)^n=a^{mn},\) \(a^{-n}=\tfrac1{a^n},\) \(a^{\frac mn}=\sqrt[n]{a^m}\)
★Standard form: \(a\times10^k,\) \(1\le a<10,\ k\in\mathbb Z\)
1 Number & algebra
Arithmetic sequence: \(u_n=u_1+(n-1)d\)
Arithmetic series: \(S_n=\tfrac n2\big(2u_1+(n-1)d\big)\) \({}=\tfrac n2(u_1+u_n)\)
Geometric sequence: \(u_n=u_1r^{n-1}\)
Geometric series, \(r\ne1\): \(S_n=\frac{u_1(r^n-1)}{r-1}\) \({}=\frac{u_1(1-r^n)}{1-r}\)
Sum to infinity, \(|r|<1\): \(S_\infty=\frac{u_1}{1-r}\)
Compound interest (\(k\) times a year, \(n\) years, \(r\)%): \(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)
Exponents & logs: \(a^x=b\iff x=\log_ab,\) \(a^x=e^{x\ln a},\) \(\log_aa^x=x=a^{\log_ax}\)
Laws of logs: \(\log_axy=\log_ax+\log_ay,\) \(\log_a\tfrac xy=\log_ax-\log_ay,\) \(\log_ax^m=m\log_ax\)
Change of base: \(\log_ba=\frac{\log_ca}{\log_cb}\)
★Special values: \(\log_a1=0,\) \(\log_aa=1,\) \(\ln e=1,\) \(e^{\ln x}=x\)
Binomial theorem, \(n\in\mathbb N\): \((a+b)^n=a^n+\tbinom n1a^{n-1}b+\cdots\) \({}+\tbinom nra^{n-r}b^r+\cdots+b^n\)
Binomial coefficient: \(\binom nr={}^nC_r=\frac{n!}{r!\,(n-r)!}\)
★General term: \(T_{r+1}=\binom nra^{n-r}b^r\)
Deductive proof: work from one side to the other, one justified step per line.
2 Functions
Gradient; forms of a line: \(m=\frac{y_2-y_1}{x_2-x_1},\) \(y=mx+c,\) \(ax+by+d=0,\) \(y-y_1=m(x-x_1)\)
★Parallel; perpendicular lines: \(m_1=m_2;\) \(m_1m_2=-1\)
Quadratic: axis of symmetry: \(x=-\frac{b}{2a}\)
Quadratic formula; discriminant: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\) \(\Delta=b^2-4ac\)
★Roots: \(\Delta>0\) two distinct real, \(\Delta=0\) one repeated, \(\Delta<0\) no real roots
★Vertex form, vertex \((h,k)\); intercept form: \(y=a(x-h)^2+k;\) \(y=a(x-p)(x-q)\)
★Composite; inverse: \((f\circ g)(x)=f\big(g(x)\big);\) \(f\big(f^{-1}(x)\big)=x\)
★Graph of \(f^{-1}\): reflect \(y=f(x)\) in \(y=x\); domain of \(f^{-1}\) = range of \(f\)
★Translations: \(f(x)+b\) up \(b\); \(f(x-a)\) right \(a\)
★Stretches: \(pf(x)\) vertical ×\(p\); \(f(qx)\) horizontal ×\(\tfrac1q\)
★Reflections: \(-f(x)\) in the \(x\)-axis; \(f(-x)\) in the \(y\)-axis
★\(y=\frac{ax+b}{cx+d}\) asymptotes: \(x=-\frac dc,\) \(y=\frac ac\)
★Exponential ↔ log: \(y=a^x\iff x=\log_ay;\) \(e^x\text{ and }\ln x\text{ are inverses}\)
3 Geometry & trigonometry
3D distance: \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\)
3D midpoint: \(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2,\tfrac{z_1+z_2}2\right)\)
Pyramid; cone: \(V=\tfrac13Ah;\) \(V=\tfrac13\pi r^2h,\) \(\text{curved }A=\pi rl\)
Sphere: \(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
Sine rule: \(\frac{a}{\sin A}=\frac{b}{\sin B}\) \({}=\frac{c}{\sin C}\)
Cosine rule: \(c^2=a^2+b^2-2ab\cos C,\) \(\cos C=\frac{a^2+b^2-c^2}{2ab}\)
Area of a triangle: \(A=\tfrac12ab\sin C\)
★Ambiguous case: given \(a,b,A\) with \(A\) acute, two triangles if \(b\sin A<a<b\)
Arc; sector (\(\theta\) in radians): \(l=r\theta,\) \(A=\tfrac12r^2\theta\)
★Radians: \(\pi\text{ rad}=180^\circ\)
Identities: \(\tan\theta=\frac{\sin\theta}{\cos\theta},\) \(\cos^2\theta+\sin^2\theta\) \({}=1\)
Double angle: \(\sin2\theta=2\sin\theta\cos\theta,\) \(\cos2\theta=\cos^2\theta-\sin^2\theta\) \({}=2\cos^2\theta-1=1-2\sin^2\theta\)
★Exact values at \(0,\frac\pi6,\frac\pi4,\frac\pi3,\frac\pi2\): \(\sin:\ 0,\tfrac12,\tfrac{\sqrt2}2,\tfrac{\sqrt3}2,1;\) \(\cos\text{: reverse order};\) \(\tan:\ 0,\tfrac1{\sqrt3},1,\sqrt3,\text{undefined}\)
★Symmetry: \(\sin(\pi-\theta)=\sin\theta,\) \(\cos(\pi-\theta)=-\cos\theta,\) \(\cos(-\theta)=\cos\theta,\) \(\sin(-\theta)=-\sin\theta\)
★\(y=a\sin\big(b(x-c)\big)+d\), \(b>0\): \(\text{amplitude }|a|,\) \(\text{period }\tfrac{2\pi}b,\) \(\text{principal axis }y=d\)
★\(\sin x=k\): \(x=\arcsin k\) or \(\pi-\arcsin k\), then add \(2\pi n\); keep solutions in the interval
4 Statistics & probability
Interquartile range: \(IQR=Q_3-Q_1\)
Mean (\(n=\sum f_i\)): \(\bar x=\frac{\sum f_ix_i}{n}\)
★Outliers: \(x<Q_1-1.5\,IQR\) \(\text{or}\) \(x>Q_3+1.5\,IQR\)
★Variance (s.d. \(\sigma\)): \(\sigma^2=\frac{\sum f(x-\mu)^2}{n}\) \({}=\frac{\sum fx^2}{n}-\mu^2\)
★Data \(\times a\) then \(+b\): \(\text{mean}\to a\bar x+b,\) \(\text{s.d.}\to|a|\sigma\)
★Regression: use \(y\) on \(x\) to predict \(y\), \(x\) on \(y\) to predict \(x\); interpolate, don't extrapolate
Probability: \(P(A)=\frac{n(A)}{n(U)},\) \(P(A)+P(A')=1\)
Combined events: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Mutually exclusive: \(P(A\cup B)=P(A)+P(B)\)
Independent: \(P(A\cap B)=P(A)P(B)\)
Conditional: \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)
Expected value: \(E(X)=\sum x\,P(X=x)\)
★Valid distribution; fair game: \(\sum P(X=x)=1;\) \(E(\text{gain})=0\)
★Binomial \(X\sim B(n,p)\): \(P(X=r)=\binom nrp^r(1-p)^{n-r}\)
Binomial mean, variance: \(E(X)=np,\) \(\mathrm{Var}(X)=np(1-p)\)
Standardised normal: \(z=\frac{x-\mu}{\sigma}\)
★Normal: about 68%, 95%, 99.7% of values lie within 1, 2, 3 s.d. of \(\mu\)
5 Calculus
Power rule: \(f(x)=x^n\Rightarrow f'(x)\) \({}=nx^{n-1}\)
Standard derivatives: \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((e^x)'=e^x,\) \((\ln x)'=\tfrac1x\)
Chain rule: \(y=g(u),\ u=f(x)\Rightarrow\frac{dy}{dx}\) \({}=\frac{dy}{du}\cdot\frac{du}{dx}\)
Product rule: \(\frac{d}{dx}(uv)=u\frac{dv}{dx}\) \({}+v\frac{du}{dx}\)
Quotient rule: \(\frac{d}{dx}\Big(\frac uv\Big)\) \({}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\)
★Linear inside: \(\tfrac{d}{dx}e^{kx}=ke^{kx},\) \(\tfrac{d}{dx}\sin kx=k\cos kx,\) \(\tfrac{d}{dx}\ln(kx)=\tfrac1x\)
★Tangent; normal gradient at \(x=a\): \(y-f(a)=f'(a)(x-a);\) \(m_\text{normal}=-\frac{1}{f'(a)}\)
★Increasing / decreasing: \(f'(x)>0;\) \(f'(x)<0\)
★Stationary points: \(f'(x)=0:\) \(f''<0\text{ max},\) \(f''>0\text{ min}\)
★Concave up \(f''>0\); point of inflexion: \(f''=0\) and \(f''\) changes sign
Integral of \(x^n\), \(n\ne-1\): \(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\)
Standard integrals: \(\int\tfrac1x\,dx=\ln|x|+C,\) \(\int\sin x\,dx=-\cos x+C,\) \(\int\cos x\,dx=\sin x+C,\) \(\int e^x\,dx=e^x+C\)
★Linear inside: \(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+C\)
★Reverse chain rule: \(\int f'\big(g(x)\big)g'(x)\,dx\) \({}=f\big(g(x)\big)+C\)
★Definite integral: \(\int_a^bf'(x)\,dx=f(b)-f(a)\)
Area, curve and \(x\)-axis: \(A=\int_a^b|y|\,dx\)
★Area between two curves: \(A=\int_a^b|f(x)-g(x)|\,dx\)
Kinematics: \(v=\frac{ds}{dt},\) \(a=\frac{dv}{dt}=\frac{d^2s}{dt^2}\)
Distance travelled \(t_1\to t_2\): \(\int_{t_1}^{t_2}|v(t)|\,dt\)
★Displacement \(t_1\to t_2\): \(\int_{t_1}^{t_2}v(t)\,dt\)