IB Group 5 · Mathematics
Mathematics: Applications and Interpretation
2021 syllabus · First assessment 2021
Five topics built on the guide’s content statements, with a strong applied and statistical focus and a GDC throughout. Each note teaches the methods and worked examples the syllabus expects, with the SL core and AHL (HL-only) content clearly split.
1. Topic 1: Number and algebra
Approximation and error, sequences and series, exponents and logs, financial maths, and (HL) complex numbers, matrices, and systems.
SL1.1 — Standard form
AISL1.1.1 · Writing numbers in standard form — Expressing very large or very small numbers as $a\times 10^{k}$ with $1\le a<10$.
AISL1.1.2 · Operations in standard form — Adding, multiplying and dividing numbers of the form $a\times 10^{k}$.
SL1.2 — Arithmetic sequences and series
AISL1.2.1 · Arithmetic sequences: the nth term — Common difference $d$ and the formula $u_n=u_1+(n-1)d$.
AISL1.2.2 · Arithmetic series and sigma notation — Summing the first $n$ terms with $S_n=\tfrac{n}{2}(2u_1+(n-1)d)$.
AISL1.2.3 · Applications — Simple interest and real-life models that are approximately arithmetic.
SL1.3 — Geometric sequences and series
AISL1.3.1 · Geometric sequences: the nth term — Common ratio $r$ and the formula $u_n=u_1 r^{\,n-1}$.
AISL1.3.2 · Geometric series and sigma notation — Summing the first $n$ terms with $S_n=\dfrac{u_1(r^{\,n}-1)}{r-1}$.
AISL1.3.3 · Applications — Population growth, salary changes and the spread of disease.
SL1.4 — Financial applications: interest and depreciation
AISL1.4.1 · Compound interest — Growth by $FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}$ with different compounding periods.
AISL1.4.2 · Using the GDC finance solver — Entering values into the TVM / built-in financial package.
AISL1.4.3 · Depreciation and real value — Annual depreciation and the effect of inflation on an investment.
SL1.5 — Exponents and logarithms
AISL1.5.1 · Laws of exponents — Combining powers with integer exponents, including zero and negatives.
AISL1.5.2 · Introduction to logarithms — Base 10 and base $e$ logs, and $a^{x}=b \iff \log_a b=x$.
SL1.6 — Approximation and error
AISL1.6.1 · Rounding and estimation — Decimal places, significant figures and judging reasonableness.
AISL1.6.2 · Upper and lower bounds — The interval a rounded value could have come from.
AISL1.6.3 · Percentage error — Comparing an approximate value with the exact value.
SL1.7 — Amortization and annuities
AISL1.7.1 · Annuities — Regular equal payments valued with a GDC finance package.
AISL1.7.2 · Amortization — Paying off a loan in equal instalments using technology.
SL1.8 — Solving equations with technology
AISL1.8.1 · Systems of linear equations — Solving up to three linear equations in three unknowns with a GDC.
AISL1.8.2 · Polynomial equations — Finding zeros and roots of polynomials with technology.
AHL1.9 — Laws of logarithms
AIAHL1.9.1 · The three laws of logarithms — $\log_a xy=\log_a x+\log_a y$, and the quotient and power laws.
AIAHL1.9.2 · Using the laws: equations and log scales — Combining, splitting and solving; applications to pH and log scaling.
AHL1.10 — Rational exponents
AIAHL1.10.1 · Laws of exponents with rational and negative powers — Simplifying $x^{m}x^{n}$, $x^{m/n}=\sqrt[n]{x^{m}}$, $x^{-n}=\tfrac{1}{x^{n}}$.
AHL1.11 — Sum of infinite geometric sequences
AIAHL1.11.1 · Convergence and the sum to infinity — When $|r|<1$, $S_\infty=\dfrac{u_1}{1-r}$; applications such as a bouncing ball.
AHL1.12 — Complex numbers: Cartesian form
AIAHL1.12.1 · The imaginary unit and Cartesian form — $i^2=-1$; $z=a+bi$; real/imaginary parts, conjugate, and arithmetic.
AIAHL1.12.2 · Argand diagram and complex roots of quadratics — Plotting $z$; modulus and argument; solving $ax^2+bx+c=0$ when $b^2-4ac<0$.
AHL1.13 — Complex numbers: polar and exponential form
AIAHL1.13.1 · Polar and exponential (Euler) form — $z=r\,\mathrm{cis}\,\theta=r\mathrm{e}^{i\theta}$; converting between Cartesian, polar and exponential.
AIAHL1.13.2 · Products, quotients, powers and geometry — Multiplying rotates and stretches; adding/subtracting is vector addition.
AIAHL1.13.3 · Application: AC circuits and adding sinusoids — Voltages as complex quantities; combining sinusoids of the same frequency.
AHL1.14 — Matrices
AIAHL1.14.1 · Matrices: definition and algebra — Order $m\times n$; equality, addition, subtraction, scalar multiplication.
AIAHL1.14.2 · Matrix multiplication and its properties — Row-by-column product; associative and distributive but not commutative; $I$ and $0$.
AIAHL1.14.3 · Determinant and inverse — $\det A=ad-bc$ and $A^{-1}$ for $2\times2$ by hand, larger with technology.
AIAHL1.14.4 · Solving linear systems with matrices — Writing $A\mathbf{x}=\mathbf{b}$ and solving via $\mathbf{x}=A^{-1}\mathbf{b}$; coding messages.
AHL1.15 — Eigenvalues and eigenvectors
AIAHL1.15.1 · Eigenvalues and the characteristic polynomial — $A\mathbf{v}=\lambda\mathbf{v}$; solve $\det(A-\lambda I)=0$ for the eigenvalues.
AIAHL1.15.2 · Finding eigenvectors — For each $\lambda$, solve $(A-\lambda I)\mathbf{v}=\mathbf{0}$.
AIAHL1.15.3 · Diagonalization and powers of a matrix — $A=PDP^{-1}$ and $A^{n}=PD^{n}P^{-1}$ with distinct real eigenvalues.
AIAHL1.15.4 · Applications of eigenvalues — Long-run behaviour: population movement, predator/prey, invariant states.
2. Topic 2: Functions
Linear, quadratic and other models and their graphs, and (HL) further modelling and transformations.
SL2.1 — Straight lines and gradients
AISL2.1.1 · Gradient and forms of a straight line — Gradient $m$, the $y$-intercept, and the forms $y=mx+c$, $ax+by+d=0$ and $y-y_1=m(x-x_1)$.
AISL2.1.2 · Parallel, perpendicular and inclines — $m_1=m_2$ for parallel lines, $m_1\times m_2=-1$ for perpendicular, and gradient as steepness of a ramp or road.
SL2.2 — Concept of a function
AISL2.2.1 · Functions, domain and range — A function maps each input to one output; notation $f(x)$, $v(t)$, $C(n)$, with domain (inputs) and range (outputs).
AISL2.2.2 · Inverse functions — An inverse $f^{-1}$ undoes $f$, is a reflection in $y=x$, and exists for one-to-one functions.
SL2.3 — Graphs of functions
AISL2.3.1 · The graph of a function; draw vs sketch — The graph $y=f(x)$, and turning a context or on-screen graph into a labelled sketch.
AISL2.3.2 · Graphing with technology; sums and differences — Using a GDC to graph functions and to add or subtract them, $h(x)=f(x)\pm g(x)$.
SL2.4 — Key features of graphs
AISL2.4.1 · Key features of a graph — Maxima and minima, intercepts, symmetry, vertex, and zeros or roots, read using technology.
AISL2.4.2 · Asymptotes and points of intersection — Vertical and horizontal asymptotes, and finding where two curves meet using technology.
SL2.5 — Modelling with functions
AISL2.5.1 · Linear and piecewise linear models — $f(x)=mx+c$ with $m$ a constant rate of change, and piecewise linear models over intervals.
AISL2.5.2 · Quadratic models — $f(x)=ax^2+bx+c$, $a\neq 0$: axis of symmetry, vertex, zeros and intercepts.
AISL2.5.3 · Exponential and variation models — Exponential growth and decay $f(x)=ka^x+c$ and variation $f(x)=ax^n$, with their asymptotes.
AISL2.5.4 · Cubic and sinusoidal models — $f(x)=ax^3+bx^2+cx+d$ and $f(x)=a\sin(bx)+d$: amplitude, period and principal axis.
SL2.6 — The modelling process
AISL2.6.1 · Develop and fit a model — Choosing an appropriate model, setting a reasonable domain, and finding parameters by substitution or simultaneous equations.
AISL2.6.2 · Test, use and extrapolate — Judging a model's reasonableness, using it to make predictions, and the dangers of extrapolation.
AHL2.7 — Composite and inverse functions
AIAHL2.7.1 · Composite functions — $(f\circ g)(x)=f(g(x))$: apply $g$ first, then $f$; order matters and domains must be compatible.
AIAHL2.7.2 · Inverse functions and domain restriction — $f^{-1}$ reverses $f$ with $(f\circ f^{-1})(x)=(f^{-1}\circ f)(x)=x$; restrict the domain so $f$ is one-to-one.
AHL2.8 — Transformations of graphs
AIAHL2.8.1 · Translations and reflections — $y=f(x)+b$ and $y=f(x-a)$ shift the graph; $y=-f(x)$ and $y=f(-x)$ reflect it in the axes.
AIAHL2.8.2 · Stretches — $y=p\,f(x)$ is a vertical stretch (factor $p$); $y=f(qx)$ is a horizontal stretch (factor $\tfrac{1}{q}$).
AIAHL2.8.3 · Composite transformations — Combining transformations, where the order applied changes the result.
AHL2.9 — Further modelling functions
AIAHL2.9.1 · Exponential (half-life) and natural logarithmic models — Using exponential decay to find half-life, and modelling with $f(x)=a+b\ln x$.
AIAHL2.9.2 · Sinusoidal models — $f(x)=a\sin\!\big(b(x-c)\big)+d$: amplitude, period $\tfrac{2\pi}{b}$, phase shift $c$ and principal axis $d$.
AIAHL2.9.3 · Logistic models — $f(x)=\dfrac{L}{1+Ce^{-kx}}$ models restricted growth with carrying capacity $L$.
AIAHL2.9.4 · Piecewise models — Functions defined by different rules on different intervals; finding parameters for continuity.
AHL2.10 — Scaling and linearizing data with logarithms
AIAHL2.10.1 · Scaling with logarithms — Using a logarithmic scale to display data spanning a very wide range and to emphasise rate of growth.
AIAHL2.10.2 · Linearizing data; log-log and semi-log graphs — Taking logs turns exponential and power relationships into straight lines, whose slope and intercept give the parameters.
3. Topic 3: Geometry and trigonometry
3D geometry, trigonometry and applications, Voronoi diagrams, and (HL) vectors, matrices as transformations, and graph theory.
SL3.1 — 3D geometry: distance, midpoint, solids and angles
AISL3.1.1 · Distance and midpoint in 3D — Extend the distance and midpoint formulas to points in three-dimensional space.
AISL3.1.2 · Volume and surface area of solids — Prisms, cylinders, pyramids, cones, spheres and hemispheres.
AISL3.1.3 · Combinations of solids — Add or subtract standard solids to find volumes and surface areas of composite shapes.
AISL3.1.4 · Angles in 3D shapes — Identify right-angled triangles inside a solid to find the angle between two lines or between a line and a plane.
SL3.2 — Trigonometry: right triangles, sine and cosine rules
AISL3.2.1 · Right-angled trigonometry — Use sine, cosine and tangent to find sides and angles in right-angled triangles.
AISL3.2.2 · The sine rule — Relate sides to the sines of their opposite angles in any triangle.
AISL3.2.3 · The cosine rule — Link three sides and one angle in any triangle.
AISL3.2.4 · Area of a triangle — Find the area from two sides and the included angle.
SL3.3 — Applications: elevation, depression and bearings
AISL3.3.1 · Angles of elevation and depression — Measure angles above or below the horizontal and use them in right-angled triangles.
AISL3.3.2 · Bearings — Three-figure bearings measured clockwise from north for navigation problems.
AISL3.3.3 · Building diagrams and solving — Turn a written statement into a labelled diagram, then apply trigonometry.
SL3.4 — The circle: arc length and sector area
AISL3.4.1 · Length of an arc — A fraction of the circumference, set by the central angle in degrees.
AISL3.4.2 · Area of a sector — A fraction of the circle's area, set by the same central angle.
SL3.5 — Equations of perpendicular bisectors
AISL3.5.1 · The perpendicular bisector — The line of all points equidistant from two given points.
AISL3.5.2 · Finding its equation — Combine the midpoint with the negative-reciprocal gradient.
SL3.6 — Voronoi diagrams
AISL3.6.1 · Sites, edges, vertices and cells — The anatomy of a Voronoi diagram and what each cell means.
AISL3.6.2 · Working with a Voronoi diagram — Find a boundary equation, identify the nearest site, and calculate a region's area.
AISL3.6.3 · Adding a site and interpolation — Insert a new site and estimate values by nearest neighbour.
AISL3.6.4 · The toxic waste dump problem — Find the point as far as possible from every site: a Voronoi vertex.
AHL3.7 — Radian measure, arcs and sectors
AIAHL3.7.1 · Radians and degree conversion — Define the radian and convert between degrees and radians.
AIAHL3.7.2 · Arc length and sector area — Use radians to find the length of an arc and the area of a sector.
AHL3.8 — The unit circle, identities and trigonometric equations
AIAHL3.8.1 · Unit-circle definitions of sine and cosine — Define cos and sin from the unit circle and see how the graphs are built.
AIAHL3.8.2 · Pythagorean identity and tan — Use cos squared plus sin squared equals one, and tan as sin over cos.
AIAHL3.8.3 · The ambiguous case of the sine rule — Recognise when two triangles fit the given information.
AIAHL3.8.4 · Solving trigonometric equations graphically — Read solutions in a finite interval from a graph.
AHL3.9 — Matrix transformations of the plane
AIAHL3.9.1 · Transformation matrices — Reflect, stretch, enlarge and rotate points using 2x2 matrices.
AIAHL3.9.2 · Translations and compositions — Add a translation vector and combine transformations by multiplying matrices.
AIAHL3.9.3 · Determinant and area — Read the area scale factor of a transformation from its determinant.
AIAHL3.9.4 · Iterative techniques and fractals — Repeatedly apply transformations to generate fractal patterns.
AHL3.10 — Vectors: definitions and algebra
AIAHL3.10.1 · Vectors, scalars and components — Represent vectors as directed segments and in component form.
AIAHL3.10.2 · Vector algebra — Add, subtract and scale vectors algebraically and geometrically.
AIAHL3.10.3 · Magnitude, unit vectors and normalizing — Find the length of a vector and rescale it to a chosen size.
AHL3.11 — Vector equation of a line
AIAHL3.11.1 · Vector form of a line — Describe a line by a point on it and a direction vector.
AIAHL3.11.2 · Parametric form — Split the vector equation into separate coordinate equations.
AHL3.12 — Vector applications to kinematics
AIAHL3.12.1 · Motion with constant velocity — Model position over time with r equals r-nought plus v t.
AIAHL3.12.2 · Closest approach and intersections — Find where and when two moving objects meet or come nearest.
AIAHL3.12.3 · Motion with variable velocity — Handle velocity that changes with time in two dimensions.
AHL3.13 — The scalar and vector products
AIAHL3.13.1 · Scalar (dot) product and angle — Multiply vectors to a number and read off the angle between them.
AIAHL3.13.2 · Vector (cross) product and area — Build a perpendicular vector whose length gives areas.
AIAHL3.13.3 · Resolving a vector along another — Split a vector into parts along and perpendicular to a direction.
AHL3.14 — Graph theory: definitions
AIAHL3.14.1 · Graphs, vertices, edges and degree — Model structures as vertices joined by edges and count degrees.
AIAHL3.14.2 · Types of graph — Distinguish simple, complete, weighted and connected graphs.
AIAHL3.14.3 · Directed graphs, subgraphs and trees — Give edges direction, and pick out subgraphs and trees.
AHL3.15 — Adjacency matrices and walks
AIAHL3.15.1 · Adjacency matrices — Record which vertices are joined in a square matrix.
AIAHL3.15.2 · Counting walks with matrix powers — Read the number of k-length walks from powers of the matrix.
AIAHL3.15.3 · Weighted tables and transition matrices — Store weights in a table and build transition matrices.
AHL3.16 — Tree, cycle and route algorithms
AIAHL3.16.1 · Walks, trails, paths and cycles — Name the ways of moving through a graph, and Eulerian and Hamiltonian routes.
AIAHL3.16.2 · Minimum spanning trees: Kruskal and Prim — Connect all vertices at least total cost with no cycle.
AIAHL3.16.3 · The Chinese postman problem — Find the shortest closed route using every edge at least once.
AIAHL3.16.4 · The travelling salesman problem — Find a least-weight cycle visiting every vertex, with bounds.
4. Topic 4: Statistics and probability
Data, correlation and regression, probability and distributions, statistical tests, and (HL) further tests, distributions and Markov chains.
SL4.1 — Sampling and data
AISL4.1.1 · Populations, samples and types of data — Population vs sample, random samples, and discrete vs continuous data.
AISL4.1.2 · Reliability, bias, missing data and outliers — Judging data sources, dealing with errors, and identifying outliers with the 1.5 x IQR rule.
AISL4.1.3 · Sampling techniques — Simple random, convenience, systematic, quota and stratified sampling and their effectiveness.
SL4.2 — Presentation of data
AISL4.2.1 · Frequency distributions and histograms — Frequency tables with class intervals as inequalities, and frequency histograms.
AISL4.2.2 · Cumulative frequency graphs — Building the cumulative frequency curve and reading off median, quartiles, percentiles and IQR.
AISL4.2.3 · Box and whisker diagrams — The five-number summary, outliers as crosses, and comparing two distributions.
SL4.3 — Measures of central tendency and dispersion
AISL4.3.1 · Mean, median and mode — Central tendency for raw and grouped data, using mid-interval values and technology.
AISL4.3.2 · Dispersion: range, IQR, variance and standard deviation — Measuring spread, with standard deviation and variance from the GDC.
AISL4.3.3 · Effect of constant changes on data — How adding or multiplying by a constant changes the mean and standard deviation.
SL4.4 — Correlation and linear regression
AISL4.4.1 · Scatter diagrams and correlation — Describing bivariate data and Pearson's product-moment correlation coefficient r.
AISL4.4.2 · The regression line of y on x — Finding y = ax + b with technology and interpreting the parameters a and b.
AISL4.4.3 · Prediction, extrapolation and causation — Using the line to predict, and the correlation-does-not-imply-causation principle.
SL4.5 — Introduction to probability
AISL4.5.1 · Sample spaces and theoretical probability — Trials, outcomes, events, and P(A) = n(A) / n(U).
AISL4.5.2 · Complementary events and expected occurrences — The complement A', experimental vs theoretical probability, and expected number of occurrences.
SL4.6 — Combined events and conditional probability
AISL4.6.1 · Diagrams for probability — Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes.
AISL4.6.2 · The addition rule and mutually exclusive events — P(A or B) = P(A) + P(B) - P(A and B) and the special case P(A and B) = 0.
AISL4.6.3 · Conditional probability — P(A | B) = P(A and B) / P(B) and problems with and without replacement.
AISL4.6.4 · Independent events — P(A and B) = P(A) P(B) and how independence differs from mutual exclusivity.
SL4.7 — Discrete random variables
AISL4.7.1 · Probability distributions — Discrete random variables, their distribution tables, and probabilities summing to 1.
AISL4.7.2 · Expected value E(X) — Computing the mean of a discrete random variable and interpreting a fair game.
SL4.8 — The binomial distribution
AISL4.8.1 · The binomial model and probabilities — When B(n, p) applies and finding probabilities with technology.
AISL4.8.2 · Mean and variance of the binomial distribution — E(X) = np and Var(X) = np(1 - p).
SL4.9 — The normal distribution
AISL4.9.1 · The normal distribution and its properties — The bell curve, notation X ~ N(mu, sigma^2), and the 68-95-99.7 rule.
AISL4.9.2 · Normal probability calculations — Finding probabilities and proportions using technology.
AISL4.9.3 · Inverse normal calculations — Finding a data value from a given probability, with mean and standard deviation given.
SL4.10 — Spearman's rank correlation coefficient
AISL4.10.1 · Spearman's rank correlation coefficient — Ranking data and computing rs with technology, averaging tied ranks.
AISL4.10.2 · Choosing between Pearson and Spearman — Linearity vs monotonicity and the effect of outliers on each coefficient.
SL4.11 — Hypothesis testing
AISL4.11.1 · Hypotheses, significance and p-values — Setting up H0 and H1, significance levels, and the p-value decision rule.
AISL4.11.2 · The chi-squared test for independence — Contingency tables, expected frequencies, degrees of freedom and the test statistic.
AISL4.11.3 · The chi-squared goodness of fit test — Testing whether data fits a given distribution with df = n - 1.
AISL4.11.4 · The two-sample t-test — Comparing two population means using the p-value with one- and two-tailed tests.
AHL4.12 — Data collection, reliability and validity
AIAHL4.12.1 · Designing valid data collection — Surveys and questionnaires; biased vs unbiased and structured questioning; selecting relevant variables.
AIAHL4.12.2 · Categorising data for the $\chi^2$ test — Choosing categories with expected frequencies above $5$ and the correct degrees of freedom.
AIAHL4.12.3 · Reliability and validity — Test-retest and parallel forms; content and criterion-related validity.
AHL4.13 — Non-linear regression and $R^2$
AIAHL4.13.1 · Least-squares regression with non-linear models — Fitting quadratic, cubic, exponential, power and sine models on the GDC.
AIAHL4.13.2 · Residuals and the coefficient of determination — $\mathrm{SS_{res}}$ as a measure of fit and $R^2$ as the proportion of variability explained.
AIAHL4.13.3 · Interpreting and comparing models — Limitations of $R^2$ and its link to Pearson's $r$ for linear models.
AHL4.14 — Transformations of random variables and unbiased estimators
AIAHL4.14.1 · Linear transformation of a random variable — $E(aX+b)=aE(X)+b$ and $\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)$.
AIAHL4.14.2 · Combinations of several random variables — Expectation of any linear combination; variance for independent variables.
AIAHL4.14.3 · Unbiased estimators of $\mu$ and $\sigma^2$ — $\bar{x}$ estimates $\mu$ and $s_{n-1}^2$ estimates $\sigma^2$.
AHL4.15 — Combining normal variables and the central limit theorem
AIAHL4.15.1 · Linear combinations of normal variables — A combination of independent normals is normal; the sample mean $\bar{X}\sim N(\mu,\sigma^2/n)$.
AIAHL4.15.2 · The central limit theorem — For large $n$ the sample mean is approximately normal whatever the population.
AHL4.16 — Confidence intervals for a mean
AIAHL4.16.1 · Confidence intervals: normal vs $t$ — Use the normal distribution when $\sigma$ is known and the $t$-distribution when it is unknown.
AIAHL4.16.2 · Finding and interpreting a confidence interval — Reading a GDC interval and stating what the confidence level means.
AHL4.17 — The Poisson distribution
AIAHL4.17.1 · The Poisson distribution, mean and variance — $P(X=x)=\dfrac{\lambda^{x}e^{-\lambda}}{x!}$ with mean and variance both $\lambda$.
AIAHL4.17.2 · Sums of Poisson variables and calculations — Independent Poissons add; finding probabilities with technology.
AIAHL4.17.3 · Choosing normal, binomial or Poisson — Recognising from context which distribution is appropriate.
AHL4.18 — Hypothesis testing and Type I and II errors
AIAHL4.18.1 · Critical regions and testing a population mean — Critical values, one- and two-tailed tests, and the normal or $t$ test for a mean.
AIAHL4.18.2 · Tests for a proportion, a Poisson mean and correlation — Binomial and Poisson one-tailed tests and testing $\rho=0$ with technology.
AIAHL4.18.3 · Type I and Type II errors — Defining both errors and calculating their probabilities.
AHL4.19 — Transition matrices and Markov chains
AIAHL4.19.1 · Transition matrices, diagrams and powers — The state after $n$ steps is $s_n=T^n s_0$; reading and building a transition diagram.
AIAHL4.19.2 · Regular Markov chains and the steady state — Long-term probabilities from repeated multiplication or by solving $\pi T=\pi$.
5. Topic 5: Calculus
Differentiation and integration and their applications, and (HL) further calculus, differential equations and numerical methods.
SL5.1 — Limits and the derivative
AISL5.1.1 · The concept of a limit — Introduction to the idea of a limit; estimating a limit from a table or graph.
AISL5.1.2 · Derivative as gradient function and rate of change — The derivative interpreted as the gradient function and as a rate of change; notation.
AISL5.1.3 · Gradient of a curve as a limit — Informal understanding of the gradient of a curve as a limit of gradients of chords.
SL5.2 — Increasing and decreasing functions
AISL5.2.1 · Increasing and decreasing functions — Using the sign of the derivative to identify where a function increases or decreases.
AISL5.2.2 · Graphical interpretation of the derivative's sign — Reading f'(x)>0, f'(x)=0 and f'(x)<0 from the shape of a graph.
SL5.3 — Differentiating polynomials (the power rule)
AISL5.3.1 · The power rule — Derivative of f(x)=ax^n is f'(x)=anx^(n-1) for integer n.
AISL5.3.2 · Differentiating polynomials term by term — Differentiating sums such as ax^n + bx^(n-1) + ... with integer exponents.
SL5.4 — Tangents and normals
AISL5.4.1 · Equation of a tangent — Finding the equation of the tangent to a curve at a given point.
AISL5.4.2 · Equation of a normal — Finding the equation of the normal (perpendicular to the tangent) at a given point.
SL5.5 — Integration and area
AISL5.5.1 · Integration as anti-differentiation — Introduction to integration as the reverse of differentiation for polynomial terms.
AISL5.5.2 · Boundary condition to find C — Using a known point (boundary condition) to determine the constant of integration.
AISL5.5.3 · Definite integrals and area under a curve — Definite integrals with technology; area between a positive curve and the x-axis.
SL5.6 — Stationary points
AISL5.6.1 · Stationary points: f'(x)=0 — Values of x where the gradient is zero; solving f'(x)=0.
AISL5.6.2 · Local maximum and minimum points — Classifying stationary points as local maxima or minima, and the greatest/least value caveat.
SL5.7 — Optimization
AISL5.7.1 · Setting up an optimization problem — Building a model to maximize or minimize a quantity in context.
AISL5.7.2 · Solving optimization problems — Worked optimization examples using the derivative and a GDC.
SL5.8 — The trapezoidal rule
AISL5.8.1 · The trapezoidal rule formula — Approximating the area under a curve using strips of equal width.
AISL5.8.2 · Applying the trapezoidal rule — Estimating an area from a function or a table of data, and judging over/under-estimates.
AHL5.9 — Further differentiation: rules and further functions
AIAHL5.9.1 · Derivatives of further functions — Derivatives of $\sin x$, $\cos x$, $\tan x$, $e^x$, $\ln x$ and $x^n$ for $n\in\mathbb{Q}$.
AIAHL5.9.2 · Chain, product and quotient rules — Differentiating composite functions, products and quotients.
AIAHL5.9.3 · Related rates of change — Linking rates through a chain of derivatives with respect to time.
AHL5.10 — The second derivative
AIAHL5.10.1 · Second derivative and concavity — Notation $f''(x)$ and $\tfrac{d^2y}{dx^2}$; concave-up and concave-down.
AIAHL5.10.2 · Second derivative test and inflexion — Classifying stationary points and locating points of inflexion.
AHL5.11 — Further integration
AIAHL5.11.1 · Integrals of further functions — Integrating $x^n$ (including $n=-1$), $\sin x$, $\cos x$, $\tfrac{1}{\cos^2 x}$ and $e^x$.
AIAHL5.11.2 · Integration by inspection or substitution — Reversing the chain rule: $\int f(g(x))g'(x)\,dx$.
AHL5.12 — Areas and volumes of revolution
AIAHL5.12.1 · Area enclosed by a curve and an axis — Area with respect to the $x$- or $y$-axis, including negative integrals.
AIAHL5.12.2 · Volumes of revolution — Rotating a region about the $x$- or $y$-axis to form a solid.
AHL5.13 — Kinematics
AIAHL5.13.1 · Displacement, velocity and acceleration — Differentiating to move from $s$ to $v$ to $a$.
AIAHL5.13.2 · Distance, displacement and speed by integration — Integrating velocity, and using $|v|$ for total distance.
AHL5.14 — Differential equations: separation of variables
AIAHL5.14.1 · Setting up a differential equation — Translating a rate description in context into an equation.
AIAHL5.14.2 · Separation of variables and the general solution — Separating variables and integrating both sides.
AHL5.15 — Slope fields
AIAHL5.15.1 · Slope fields and their diagrams — Reading, drawing and interpreting slope fields for $\tfrac{dy}{dx}=f(x,y)$.
AHL5.16 — Euler's method and coupled systems
AIAHL5.16.1 · Euler's method for first order equations — Stepping along tangents to approximate $\tfrac{dy}{dx}=f(x,y)$.
AIAHL5.16.2 · Numerical solution of coupled systems — Two linked rates $\tfrac{dx}{dt}$ and $\tfrac{dy}{dt}$, e.g. predator-prey.
AHL5.17 — Phase portraits and eigenvalues
AIAHL5.17.1 · Exact solutions via eigenvalues — Solving $\tfrac{d}{dt}\mathbf{x}=M\mathbf{x}$ for real distinct eigenvalues.
AIAHL5.17.2 · Phase portraits and classifying behaviour — Trajectories, equilibrium and stability from the eigenvalues.
AHL5.18 — Second order differential equations
AIAHL5.18.1 · Second order equations as coupled systems — Rewriting $\tfrac{d^2x}{dt^2}=f(x,\tfrac{dx}{dt},t)$ and using Euler's method.
AIAHL5.18.2 · Exact solutions and the phase portrait method — Damped equations $\tfrac{d^2x}{dt^2}+a\tfrac{dx}{dt}+bx=0$ via eigenvalues.
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