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IB Mathematics: Applications & Interpretation 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: \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2},\) \(\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)\)
★Pythagoras; SOHCAHTOA; indices: \(a^2+b^2=c^2;\) \(\tan\theta=\tfrac{\text{opp}}{\text{adj}};\) \(a^ma^n=a^{m+n}\)
1 Number & algebra
★Standard form: \(a\times10^k,\) \(1\le a<10\)
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: \(u_n=u_1r^{n-1},\) \(S_n=\frac{u_1(r^n-1)}{r-1}\) \({}=\frac{u_1(1-r^n)}{1-r}\)
Compound interest: \(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)
Annuities, loans, amortisation: GDC TVM solver; money paid out is negative.
Percentage error: \(\varepsilon=\left|\frac{v_A-v_E}{v_E}\right|\times100\%\)
Exponents & logs: \(a^x=b\iff x=\log_ab\)
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: \(\log_ba=\frac{\log_ca}{\log_cb}\)
Complex numbers: \(z=a+bi=r(\cos\theta+i\sin\theta)\) \({}=re^{i\theta}=r\,\cis\theta\)
★Modulus, argument: \(|z|=\sqrt{a^2+b^2},\) \(\arg z=\theta,\) \(\tan\theta=\tfrac ba\ \text{(check quadrant)}\)
★Products: \(|z_1z_2|=|z_1||z_2|,\) \(\arg(z_1z_2)=\arg z_1+\arg z_2\)
★Sinusoids: \(A\cos(\omega t)+B\cos(\omega t+\phi)\) = real part of \((A+Be^{i\phi})e^{i\omega t}\), so amplitudes and phases add as complex numbers
2×2 determinant; inverse: \(\det A=ad-bc,\) \(A^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)
★3×3 determinant: \(\det A=a\begin{vmatrix}e&f\\h&i\end{vmatrix}\) \({}-b\begin{vmatrix}d&f\\g&i\end{vmatrix}\) \({}+c\begin{vmatrix}d&e\\g&h\end{vmatrix}\)
★Matrix rules: \(AB\ne BA\text{ in general},\) \((AB)^{-1}=B^{-1}A^{-1},\) \(AX=B\Rightarrow X=A^{-1}B\)
★Eigenvalues; eigenvectors: \(\det(M-\lambda I)=0;\) \(M\boldsymbol p=\lambda\boldsymbol p\)
Matrix powers: \(M^n=PD^nP^{-1}\)
★\(P\) = eigenvectors as columns, \(D\) = diagonal matrix of the matching eigenvalues
2 Functions & models
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: \(m_1m_2=-1\)
Quadratic: axis of symmetry: \(x=-\frac b{2a}\)
★Models: \(f(x)=ka^x+c,\) \(ke^{rx}+c;\) \(ax^n;\) \(a\sin(bx)+d\ (\text{period }\tfrac{360^\circ}b)\)
★Logistic model: \(f(x)=\frac{L}{1+Ce^{-kx}}\) \((L,C,k>0)\)
★Linearising: \(y=ax^b\Rightarrow\ln y\) \({}=\ln a+b\ln x;\) \(y=ka^x\Rightarrow\ln y\) \({}=\ln k+x\ln a\)
★Composite; inverse: \((f\circ g)(x)=f(g(x));\) \(f\big(f^{-1}(x)\big)=x\)
★Transformations: \(f(x)+b\) up; \(f(x-a)\) right; \(pf(x)\) vertical ×\(p\); \(f(qx)\) horizontal ×\(\tfrac1q\)
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, degrees: \(l=\tfrac{\theta}{360}2\pi r,\) \(A=\tfrac{\theta}{360}\pi r^2\)
Arc; sector, radians: \(l=r\theta,\) \(A=\tfrac12r^2\theta\)
Identities: \(\cos^2\theta+\sin^2\theta\) \({}=1,\) \(\tan\theta=\frac{\sin\theta}{\cos\theta}\)
★Radians; bearings: \(\pi\text{ rad}=180^\circ;\) \(\text{bearings clockwise from north}\)
★Perpendicular bisectors & Voronoi: cell edges on perpendicular bisectors of sites; largest empty circle at a vertex or on the boundary
Reflection in \(y=(\tan\theta)x\): \(\begin{pmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{pmatrix}\)
Stretch ×\(k\): horizontal, vertical; enlargement: \(\begin{pmatrix}k&0\\0&1\end{pmatrix},\) \(\begin{pmatrix}1&0\\0&k\end{pmatrix};\) \(\begin{pmatrix}k&0\\0&k\end{pmatrix}\)
Rotation about \(O\) by \(\theta\): anticlockwise; clockwise: \(\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix};\) \(\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}\)
★Area of image: \(|\det A|\times\text{area of object}\)
★Composite transformation: \(B\) then \(A\) is \(AB\)
Vector magnitude: \(|\boldsymbol v|=\sqrt{v_1^2+v_2^2+v_3^2}\)
Scalar product; angle: \(\boldsymbol v\cdot\boldsymbol w\) \({}=v_1w_1+v_2w_2+v_3w_3\) \({}=|\boldsymbol v||\boldsymbol w|\cos\theta\)
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: \(A=|\boldsymbol v\times\boldsymbol w|\)
Line: \(\boldsymbol r=\boldsymbol a\) \({}+\lambda\boldsymbol b\)
★Motion with constant velocity: \(\boldsymbol r=\boldsymbol r_0\) \({}+t\boldsymbol v,\) \(\text{speed}=|\boldsymbol v|\)
★Unit vector; perpendicular: \(\hat{\boldsymbol v}=\frac{\boldsymbol v}{|\boldsymbol v|};\) \(\boldsymbol v\cdot\boldsymbol w\) \({}=0\)
★Component of \(\boldsymbol v\) along \(\boldsymbol w\): \(\boldsymbol v\cdot\hat{\boldsymbol w}\)
★Graphs: sum of degrees = \(2\times\) edges; \(K_n\) has \(\tfrac12n(n-1)\) edges; a tree on \(n\) vertices has \(n-1\) edges
★Planar connected graph: \(v-e+f=2,\) \(e\le3v-6\ (v\ge3)\)
★Eulerian circuit: every vertex even; Eulerian trail: exactly two odd vertices
★Walks of length \(k\) from \(i\) to \(j\): \((A^k)_{ij}\)
★Chinese postman: add the cheapest pairing of odd vertices, repeated once each
★TSP: upper bound = nearest neighbour; lower bound = MST of the rest + two shortest edges from the deleted vertex
4 Statistics & probability
IQR; mean: \(IQR=Q_3-Q_1,\) \(\bar x=\frac{\sum f_ix_i}{n}\)
★Outliers: \(<Q_1-1.5\,IQR\) \(\text{or}\) \({}>Q_3+1.5\,IQR\)
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: \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)
Expected value: \(E(X)=\sum xP(X=x)\)
★Variance: \(\mathrm{Var}(X)=E(X^2)-[E(X)]^2\)
★Binomial pmf: \(P(X=r)=\tbinom nrp^r(1-p)^{n-r}\)
Binomial mean, variance: \(np,\) \(np(1-p)\)
★Poisson \(X\sim\mathrm{Po}(m)\): \(P(X=x)=\frac{m^xe^{-m}}{x!}\)
Poisson mean, variance: \(E(X)=m,\) \(\mathrm{Var}(X)=m\)
★Sum of independent Poissons: \(\mathrm{Po}(m_1)+\mathrm{Po}(m_2)\) \({}=\mathrm{Po}(m_1+m_2)\)
Linear combinations: \(E(a_1X_1\pm a_2X_2)=a_1E(X_1)\) \({}\pm a_2E(X_2)\)
… independent \(X_i\): \(\mathrm{Var}(a_1X_1\pm a_2X_2)\) \({}=a_1^2\mathrm{Var}(X_1)\) \({}+a_2^2\mathrm{Var}(X_2)\)
Unbiased variance estimate: \(s^2_{n-1}=\frac{n}{n-1}s^2_n\)
★Sample mean (CLT, \(n\ge30\) or \(X\) normal): \(\bar X\sim N\!\left(\mu,\tfrac{\sigma^2}{n}\right)\)
★Linear combinations of independent normals are normal; for \(n\) independent \(X_i\sim N(\mu,\sigma^2)\): \(\sum X_i\sim N(n\mu,n\sigma^2)\)
\(\chi^2\) statistic: \(\chi^2_\text{calc}=\sum\frac{(f_o-f_e)^2}{f_e}\)
★Expected frequency; df: \(\frac{\text{row}\times\text{column}}{\text{total}};\) \((r-1)(c-1)\)
★Goodness of fit df: classes − 1 − parameters estimated
★Reject \(H_0\) if \(p\)-value \(<\) significance level. Type I: reject a true \(H_0\); Type II: accept a false \(H_0\)
★Transition matrix: \(T^n\boldsymbol s_0=\boldsymbol s_n;\) \(\text{steady state }T\boldsymbol s\) \({}=\boldsymbol s,\ \sum s_i\) \({}=1\)
5 Calculus
Derivatives: \((x^n)'=nx^{n-1},\) \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((\tan x)'=\tfrac1{\cos^2x},\) \((e^x)'=e^x,\) \((\ln x)'=\tfrac1x\)
Chain, product, quotient: \(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx},\) \((uv)'=uv'+vu',\) \(\Big(\frac uv\Big)'=\frac{vu'-uv'}{v^2}\)
★Tangent; normal: \(y-f(a)=f'(a)(x-a);\) \(m_\text{n}=-\tfrac1{f'(a)}\)
★Max / min; inflexion: \(f'=0,\ f''<0\ /\ f''>0;\) \(f''=0\text{ with sign change}\)
★Related rates: \(\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}\)
Integrals: \(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C,\) \(\int\tfrac1x\,dx=\ln|x|+C,\) \(\int\sin x\,dx=-\cos x+C\)
More integrals: \(\int\cos x\,dx=\sin x+C,\) \(\int\tfrac1{\cos^2x}\,dx=\tan x+C,\) \(\int e^x\,dx=e^x+C\)
★Linear inside; by inspection: \(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+C;\) \(\int g'(x)f'(g(x))\,dx=f(g(x))+C\)
Areas: \(\int_a^b|y|\,dx,\) \(\int_c^d|x|\,dy\)
Volumes of revolution: \(V=\int_a^b\pi y^2\,dx,\) \(V=\int_c^d\pi x^2\,dy\)
Trapezoidal rule, \(h=\tfrac{b-a}n\): \(\int_a^by\,dx\) \({}\approx\tfrac12h\big[(y_0+y_n)\) \({}+2(y_1+\cdots+y_{n-1})\big]\)
Kinematics: \(v=\tfrac{ds}{dt},\) \(a=\tfrac{dv}{dt}=\tfrac{d^2s}{dt^2}\) \({}=v\tfrac{dv}{ds};\) \(\text{distance }\int_{t_1}^{t_2}|v|\,dt\)
★Separable: \(\frac{dy}{dx}=f(x)g(y)\) \({}\Rightarrow\int\frac{dy}{g(y)}\) \({}=\int f(x)\,dx\)
Euler's method: \(y_{n+1}=y_n+h\,f(x_n,y_n),\) \(x_{n+1}=x_n+h\)
Euler, coupled systems: \(x_{n+1}=x_n+h\,f_1(x_n,y_n,t_n),\) \(y_{n+1}=y_n+h\,f_2(x_n,y_n,t_n),\) \(t_{n+1}=t_n+h\)
Coupled linear system \(\dot{\boldsymbol x}=M\boldsymbol x\): \(\boldsymbol x=Ae^{\lambda_1t}\boldsymbol p_1\) \({}+Be^{\lambda_2t}\boldsymbol p_2\)
★Phase portraits: real \(\lambda\) both \(<0\) stable node, both \(>0\) unstable, opposite signs saddle; complex: \(\mathrm{Re}\,\lambda<0\) spiral in, \(>0\) out, \(=0\) closed loops
★2nd order \(\ddot x+a\dot x+bx=0\): \(\text{put }y=\dot x:\) \(\dot x=y,\) \(\dot y=-bx-ay\)