IB Math Revisionibmathrevision.com
IB Mathematics: Applications & Interpretation 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: \(a^ma^n=a^{m+n},\) \(\tfrac{a^m}{a^n}=a^{m-n},\) \((a^m)^n=a^{mn},\) \(a^{-n}=\tfrac1{a^n}\)
1 Number & algebra
★Standard form: \(a\times10^k,\) \(1\le a<10,\ k\in\mathbb Z\)
Arithmetic sequence: \(u_n=u_1+(n-1)d\)
Arithmetic series: \(S_n=\tfrac n2\big(2u_1+(n-1)d\big)\) \({}=\tfrac n2(u_1+u_n)\)
★Common difference: \(d=u_{n+1}-u_n\)
Geometric sequence: \(u_n=u_1r^{n-1}\)
Geometric series, \(r\ne1\): \(S_n=\frac{u_1(r^n-1)}{r-1}\) \({}=\frac{u_1(1-r^n)}{1-r}\)
Compound interest (\(k\) times a year, \(n\) years, \(r\)%): \(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)
★Depreciation at \(r\)% a year: \(V=V_0\left(1-\tfrac{r}{100}\right)^n\)
Loans, annuities, amortisation: use the GDC's TVM solver (N, I%, PV, PMT, FV, P/Y, C/Y); money paid out is negative.
Exponents & logs: \(a^x=b\iff x=\log_ab\)
★Log facts: \(\log_{10}10^x=x,\) \(\ln e^x=x,\) \(e^{\ln x}=x\)
Percentage error: \(\varepsilon=\left|\frac{v_A-v_E}{v_E}\right|\times100\%\)
★Bounds: a value rounded to the nearest \(u\) lies in \([x-\tfrac u2,\ x+\tfrac u2)\)
2 Functions & models
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: \(m_1=m_2;\) \(m_1m_2=-1\)
Quadratic: axis of symmetry: \(x=-\frac{b}{2a}\)
★Linear; quadratic models: \(f(x)=mx+c;\) \(f(x)=ax^2+bx+c\)
★Exponential models: \(f(x)=ka^x+c,\) \(f(x)=ka^{-x}+c,\) \(f(x)=ke^{rx}+c\)
★Direct / inverse variation: \(f(x)=ax^n,\) \(n\in\mathbb Z\)
★Cubic model: \(f(x)=ax^3+bx^2+cx+d\)
★Sinusoidal model (degrees): \(f(x)=a\sin(bx)+d:\) \(\text{amplitude }|a|,\) \(\text{period }\tfrac{360^\circ}{b}\)
★Inverse function: swap \(x\) and \(y\); graph reflects in \(y=x\); domain of \(f^{-1}\) = range of \(f\)
★Horizontal asymptote of \(ka^x+c\): \(y=c\)
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: \(V=\tfrac13Ah;\) \(V=\tfrac13\pi r^2h,\) \(\text{curved }A=\pi rl\)
Sphere: \(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
★Slant height of a cone: \(l=\sqrt{r^2+h^2}\)
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: \(A=\tfrac12ab\sin C\)
Arc; sector (\(\theta\) in degrees): \(l=\frac{\theta}{360}\times2\pi r,\) \(A=\frac{\theta}{360}\times\pi r^2\)
★Elevation / depression measured from the horizontal; bearings clockwise from north, 3 figures
★Perpendicular bisector of \(AB\): through the midpoint of \(AB\), gradient \(-\tfrac1{m_{AB}}\)
★Voronoi: cell edges lie on perpendicular bisectors of pairs of sites; the point furthest from every site ("toxic waste dump") is at a vertex or on the boundary
4 Statistics & probability
Interquartile range: \(IQR=Q_3-Q_1\)
Mean (\(n=\sum f_i\)): \(\bar x=\frac{\sum f_ix_i}{n}\)
★Outliers: \(x<Q_1-1.5\,IQR\) \(\text{or}\) \(x>Q_3+1.5\,IQR\)
★Standard deviation: \(\sigma=\sqrt{\frac{\sum f(x-\bar x)^2}{n}}\)
★Data \(\times a\) then \(+b\): \(\text{mean}\to a\bar x+b,\) \(\text{s.d.}\to|a|\sigma\)
★Regression: \(y=ax+b\) (GDC); use to predict \(y\) within the data range only
★\(|r|\) near 1: strong linear correlation; Spearman's \(r_s\) = PMCC of the ranks
Probability: \(P(A)=\frac{n(A)}{n(U)},\) \(P(A)+P(A')=1\)
★Expected number of occurrences: \(n\times P(A)\)
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: \(E(X)=np,\) \(\mathrm{Var}(X)=np(1-p)\)
★Normal: about 68%, 95%, 99.7% of values lie within 1, 2, 3 s.d. of \(\mu\)
\(\chi^2\) statistic: \(\chi^2_\text{calc}=\sum\frac{(f_o-f_e)^2}{f_e}\)
★Expected frequency (independence): \(f_e=\frac{\text{row total}\times\text{column total}}{\text{grand total}}\)
★Degrees of freedom: \(\text{independence: }(r-1)(c-1);\) \(\text{goodness of fit: classes}-1\)
★Test decision: reject \(H_0\) if \(p\)-value \(<\) significance level (or \(\chi^2_\text{calc}>\) critical value)
★\(t\)-test (GDC): compares two population means; \(H_0\!:\mu_1=\mu_2\)
5 Calculus
Derivative of \(x^n\): \(f(x)=x^n\Rightarrow f'(x)\) \({}=nx^{n-1}\)
★Constant multiples, sums: \(f(x)=ax^n+bx^m\Rightarrow f'(x)\) \({}=anx^{n-1}+bmx^{m-1}\)
★Gradient of tangent at \(x=a\): \(f'(a);\) \(\text{tangent }y-f(a)=f'(a)(x-a)\)
★Gradient of normal: \(-\frac{1}{f'(a)}\)
★Increasing / decreasing: \(f'(x)>0;\) \(f'(x)<0\)
★Stationary points; optimisation: \(f'(x)=0;\) \(\text{check the sign of }f'\text{ either side}\)
Integral of \(x^n\), \(n\ne-1\): \(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\)
Area, \(y\ge0\): \(A=\int_a^by\,dx\)
Trapezoidal rule, \(h=\tfrac{b-a}n\): \(\int_a^by\,dx\) \({}\approx\tfrac12h\big[(y_0+y_n)+2(y_1+\cdots+y_{n-1})\big]\)
★Boundary condition: find \(C\) by substituting a known point into \(\int f'(x)\,dx\)