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Cubic models of the form ax^3+bx^2+cx+d, including graphs with 0 or 2 turning points and the short flat section seen when there are no turning points.

\n","hasLesson":true},{"id":964,"name":"Exponential Models","slug":"exponential-models","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921406598-exponentialModels.png?alt=media&token=5dae0854-86d5-4130-b58a-5f24386c0d50","type":"image"},"skills":[926,901,902,1070,931],"description":"Modeling growth and decay with functions of the form Ab^{x}+c or Ae^{bx}+c, as well as the concept of half-life for exponential decay.","hasLesson":true},{"id":952,"name":"Logarithmic models, scales and linearization","slug":"logarithmic-models-scales-and-linearization","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029248261-Artboard%2042.png?alt=media&token=5339a82a-b846-4194-aeb6-38b233ba5a7d","type":"image"},"skills":[932,1087,1041,1040],"description":"Natural logarithmic models of the form \$f(x)=a+b\\ln x\$, and logarithmic scales such as \$\\mathrm{pH}\$.","hasLesson":true},{"id":962,"name":"Sinuisoidal Models","slug":"sinuisoidal-models","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921422992-sinusoidalModels.png?alt=media&token=593230ed-6278-44de-892e-00c5f328f78b","type":"image"},"skills":[928,933],"description":"Modeling with functions of the form a\\sin\\left(bx+c\\right)+d or a\\cos\\left(bx+c\\right)+d.","hasLesson":true},{"id":961,"name":"Piecewise models","slug":"piecewise-models","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921434771-piecewiseModels.png?alt=media&token=d5b1b32b-2ed0-4ac5-b3fe-dd9ef1897a3d","type":"image"},"skills":[3],"description":"

Piecewise models with non-linear sections, including defining and graphing functions given by different expressions on different intervals, such as f(x)=\\begin{cases}x & x<0\\\\ x^2 & x\\ge 0\\end{cases}.

\n","hasLesson":true},{"id":967,"name":"Logistic Models","slug":"logistic-models","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921476991-Artboard%2044.png?alt=media&token=b57a5f82-d6d4-46b3-93fc-426a89c12329","type":"image"},"skills":[934],"description":"

Logistic growth models of the form f\\left(x\\right)=\\frac{L}{1+Ce^{-kx}}, showing near-exponential growth at first and then slowing toward the carrying capacity L as a horizontal asymptote.

\n","hasLesson":true},{"id":925,"name":"Power Models & Proportionality ","slug":"power-models-proportionality-","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029212106-Artboard%2045.png?alt=media&token=ea87ff9b-8929-4296-ad61-f73b3d4c6f53","type":"image"},"skills":[1089,929,930],"description":"

Power models of the form y=ax^b or y\\propto x^b, including direct proportion y=kx^n, inverse proportion y=\\frac{k}{x^n}, and fitting a power model from data using power regression.

\n","hasLesson":true},{"id":910,"name":"Modeling foundations and linear models","slug":"modeling-foundations-and-linear-models","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921256837-linearModels.png?alt=media&token=643bb3a9-92ae-487b-aac0-a85f8888dc5f","type":"image"},"skills":[912,913,914],"description":"Mathematical modelling with linear models \$y=ax+b\$, using assumptions and parameters to represent real situations, and interpreting or extrapolating predictions beyond the given data when appropriate. ","hasLesson":true},{"id":965,"name":"Quadratic Models","slug":"quadratic-models","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921323648-quadraticModels.png?alt=media&token=392b87a5-e3aa-4eb0-a0d6-021734bf76b3","type":"image"},"skills":[922,801,802,1086],"description":"

Quadratic models in the form ax^2+bx+c, including the parabola's vertex and axis of symmetry, maximum or minimum turning point, x-intercepts or roots, and fitting a quadratic from three data points.

\n","hasLesson":true}],"description":"

Constructing, interpreting and fitting mathematical models for real-world data, including linear, quadratic, exponential, power, logarithmic, sinusoidal, logistic and piecewise forms, with emphasis on key features, assumptions and extrapolation.

\n"},{"id":865,"name":"Rounding & Error","slug":"rounding-error","topic":1,"subjects":["AISL","AASL","AIHL","AAHL"],"isVideoReady":false,"videoId":null,"concepts":[{"id":866,"name":"Rounding and estimation","slug":"rounding-and-estimation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030041198-Artboard%205.png?alt=media&token=25922b18-0ebb-4291-af0d-e8584c14c0e9","type":"image"},"skills":[869,877],"description":"

Rounding, significant figures, decimal places

","hasLesson":true},{"id":868,"name":"Scientific Notation","slug":"scientific-notation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773515918578-Artboard%206.png?alt=media&token=69a03129-81d4-4eeb-a181-bef451d1b1cd","type":"image"},"skills":[882,885,887],"description":"

Converting numbers to and from scientific notation \\left(a\\times10^{k}\\right)

","hasLesson":true},{"id":867,"name":"Measurement and error","slug":"measurement-and-error","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773516589698-error.png?alt=media&token=46ea8335-6157-4026-a7ad-19c8cb5f066e","type":"image"},"skills":[878,880,894],"description":"

Approximation, absolute and percentage error using |V_A-V_E| and \\frac{|V_A-V_E|}{V_E}\\times100\\%, together with error bounds from rounding or measurement and how they propagate through calculations.

\n","hasLesson":true}],"description":"

Rounding, significant figures, and scientific notation. Absolute and percentage error.

"},{"id":30,"name":"Sequences & Series","slug":"sequences-series","topic":1,"subjects":["AAHL","AIHL","AISL","AASL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"DI2cudLIWEk","concepts":[{"id":27,"name":"Arithmetic Sequences","slug":"arithmetic-sequences","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029981178-Artboard%207.png?alt=media&token=8a340859-acc8-458a-a5be-e962cb927b61","type":"image"},"skills":[602,603],"description":"

Introduction to the definition, properties, and general term of an arithmetic sequence

","hasLesson":true},{"id":28,"name":"Geometric Sequences","slug":"geometric-sequences","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030011943-Artboard%208.png?alt=media&token=96b137e1-fd1c-43b5-9749-9222bc9e9393","type":"image"},"skills":[604,40],"description":"

Introduction to definition, identification, and general term of geometric sequences

","hasLesson":true},{"id":31,"name":"Arithmetic Series","slug":"arithmetic-series","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030004817-Artboard%209.png?alt=media&token=4f2e2c52-6ec5-4ca1-b258-67a93d257f07","type":"image"},"skills":[607,46],"description":"

Definition of series, formula for arithmetic series

","hasLesson":true},{"id":860,"name":"Summation notation","slug":"summation-notation","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029991958-Artboard%2010.png?alt=media&token=dcfdc9b6-9e57-47bc-b6bc-9b4088d76eb5","type":"image"},"skills":[47,837,861,862,863,905],"description":"

Summation notation, properties of sums

","hasLesson":true},{"id":859,"name":"Geometric Series","slug":"geometric-series","isPaywalled":true,"isPublic":true,"thumbnail":{"type":"graph","graph":{"data":"$94","mode":"geometry"}},"skills":[606,51],"description":"

Finite geometric series with sum S_{n}=\\frac{u_1(1-r^n)}{1-r}, and infinite geometric series convergence when |r|<1 so that S_\\infty=\\frac{u_1}{1-r}.

\n","hasLesson":true},{"id":906,"name":"Sequence Mode (Calculator)","slug":"sequence-mode-calculator","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029904498-Artboard%2011.png?alt=media&token=542aa577-0f10-43f9-b206-99a890877c96","type":"image"},"skills":[],"description":"

Working with sequences on a calculator, computations and applications of a calculator

","hasLesson":true},{"id":870,"name":"Compound interest, depreciation and inflation","slug":"compound-interest-depreciation-and-inflation","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029508658-Artboard%2012.png?alt=media&token=fb6f46c3-0587-4c9c-83b6-2dd08aad7ca4","type":"image"},"skills":[872,871,883,896,873],"description":"

Compound growth, depreciation, interest, inflation, real value

","hasLesson":true}],"description":"Arithmetic sequences u_{n}=u_1+\\left(n-1\\right)d and geometric sequences v_{n}=v_1r^{n-1}, as well as their series and the \\Sigma notation for summation."},{"id":864,"name":"Financial Mathematics","slug":"financial-mathematics","topic":1,"subjects":["AISL","AIHL"],"isVideoReady":false,"videoId":null,"concepts":[{"id":870,"name":"Compound interest, depreciation and inflation","slug":"compound-interest-depreciation-and-inflation","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029508658-Artboard%2012.png?alt=media&token=fb6f46c3-0587-4c9c-83b6-2dd08aad7ca4","type":"image"},"skills":[872,871,883,896,873],"description":"

Compound growth, depreciation, interest, inflation, real value

","hasLesson":true},{"id":874,"name":"Loans","slug":"loans","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029497952-Artboard%2029.png?alt=media&token=ed1c353b-8cec-4988-8123-d1aa20e48a65","type":"image"},"skills":[1081,891,896],"description":"

Loans as amortized borrowing: money is received as an initial balance and repaid by regular payments with interest until the balance is cleared, using TVM Solver variables such as PV, PMT, FV, N, I\\%, P/Y and C/Y, with correct positive and negative cash flow signs.

\n","hasLesson":true},{"id":890,"name":"Annuities","slug":"annuities-perpetuities","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029490725-Artboard%2030.png?alt=media&token=be15ceb4-2826-4e40-9fd2-3cd0aa66344a","type":"image"},"skills":[1082,898,896],"description":"

An annuity as the reverse of a loan, with regular end-of-period withdrawals or payments and TVM Solver calculations using PV, PMT, FV, N, I\\%, and correct positive/negative cash-flow signs.

\n","hasLesson":true}],"description":"

Compound interest and depreciation models, inflation and real value, and using the TVM Solver to work with loans and annuities with correct cash flow signs.

\n"},{"id":936,"name":"Matrices","slug":"matrices","topic":1,"subjects":["AIHL"],"isVideoReady":false,"videoId":null,"concepts":[{"id":938,"name":"Foundations of Matrices","slug":"properties-of-matrices","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029585755-Artboard%2024.png?alt=media&token=29ae56e8-ee63-448e-ba87-daf47252d42e","type":"image"},"skills":[958,959,960,961,962,963,1052,1053,965,1056],"description":"

Matrix notation and order m\\times n, together with addition and subtraction of matrices of the same order, scalar multiplication, matrix multiplication and its required dimensions, the zero matrix, and the properties of multiplication including non-commutativity, associativity and distributivity, with calculator use for numeric matrices.

\n","hasLesson":true},{"id":940,"name":"Identity, Inverses & Determinants","slug":"identity-inverses-determinants","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029571575-Artboard%2025.png?alt=media&token=0f7424e2-d75d-4c28-b72a-f75d451f0f91","type":"image"},"skills":[966,967,968,973,969,970,964,1055],"description":"

Identity matrices \u001f, inverses and determinants of square matrices, including \u001f\\mathbf{IM}=\\mathbf{MI}=\\mathbf{M}, the inverse condition \\det\\mathbf{M}\\neq 0\\iff\\mathbf{M}^{-1} exists, finding 2\\times 2 determinants and inverses by hand, using technology for larger matrices, and rearranging matrix equations such as \\mathbf{AB}=\\mathbf{C} to solve for unknown matrices.

\n","hasLesson":true},{"id":956,"name":"Geometric Transformations with Matrices","slug":"geometric-transformations-with-matrices","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029556894-Artboard%2026.png?alt=media&token=ce9bb4bf-a445-4bdf-a766-1153ebe4d76d","type":"image"},"skills":[1057,1058,1059,1060,1061,1062,11,1063],"description":"

Matrix representations of geometric transformations in the plane, including images of basis vectors under a 2x2 matrix, enlargements and horizontal or vertical stretches, rotations, reflections, translations, composition of transformations using matrix products, and area scale factor |\\det\\mathbf{M}|.

\n","hasLesson":true},{"id":941,"name":"Systems of equations with Matrices","slug":"systems-of-equations-with-matrices","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029543607-Artboard%2027.png?alt=media&token=c0b50dab-5ab4-4d2c-b7d7-58de9a7fc359","type":"image"},"skills":[971,972,974],"description":"

Writing systems of linear equations in matrix form \\mathbf{Ax}=\\mathbf{b}, solving them using \\mathbf{x}=\\mathbf{A}^{-1}\\mathbf{b} when \\mathbf{A}^{-1} exists, and interpreting \\det\\mathbf{A}=0 as giving no solution or infinitely many solutions.

\n","hasLesson":true},{"id":942,"name":"Eigenvalues & Eigenvectors","slug":"eigenvalues-eigenvectors","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029529248-Artboard%2028.png?alt=media&token=dbcd4332-d10c-4a7a-b0b1-25a68a708c58","type":"image"},"skills":[975,951,952,953,976],"description":"

Eigenvalues and eigenvectors defined by \\mathbf{Av}=\\lambda\\mathbf{v}, finding eigenvalues from \\det\\left(\\mathbf{A}-\\lambda\\mathbf{I}\\right)=0 and the characteristic polynomial, finding eigenvectors by solving the resulting simultaneous equations, and diagonalizing matrices as \\mathbf{A}=\\mathbf{PDP}^{-1} for distinct real eigenvalues, including using diagonal form to exponentiate matrices.

\n","hasLesson":true}],"description":"

Matrix addition and multiplication, determinants and inverses \\mathbf{M^{-1}}=\\frac{1}{\\det\\mathbf{M}}\\begin{pmatrix}a & -c\\\\ -c & d\\end{pmatrix}, solving systems of equations with inverses, and diagonalization using eigenvectors.

Also geometric transformations with matrices (Topic 3)

\n"},{"id":108,"name":"Transformations & asymptotes","slug":"transformations-asymptotes","topic":2,"subjects":["AAHL","AASL","AIHL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"xr5EGY1f9Ps","concepts":[{"id":263,"name":"Linear Transformations","slug":"linear-transformations","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029188684-Artboard%2038.png?alt=media&token=eab72306-0d65-4d6a-882d-85b179795b8d","type":"image"},"skills":[11,769,770,771,772,773,774],"description":"

Linear transformations of graphs and points: translations by a vector \\begin{pmatrix}a\\\\ b\\end{pmatrix}, vertical and horizontal stretches y=af(x) and y=f(qx), vertical and horizontal translations y=f(x)+b and y=f(x-a), and reflections in the x- and y-axes.

\n","hasLesson":true},{"id":780,"name":"Modulus & Inequalties","slug":"modulus-inequalties","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029139699-Artboard%2041.png?alt=media&token=029ee865-3d40-4839-ad7b-49672ad900c0","type":"image"},"skills":[],"description":"

Absolute value (modulus) \\lvert x\\rvert, and solving modulus equations and inequalities such as \\lvert f(x)\\rvert=x-3 and g(x)\\ge f(x) algebraically or graphically, including changing the inequality direction when multiplying by a negative number.

\n","hasLesson":true}],"description":"

Translations, stretches, reflections and shifts of graphs using function notation, including horizontal and vertical transformations.

\n"},{"id":356,"name":"2D & 3D Geometry","slug":"2d-geometry","topic":3,"subjects":["AAHL","AASL","AIHL","AISL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"GnG2jU8VpUg","concepts":[{"id":82,"name":"Right-angled triangles","slug":"right-angled-triangles","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029099320-Artboard%2046.png?alt=media&token=803487ed-7a2d-4c58-88e5-31cba1785454","type":"image"},"skills":[87,36,56,716,83,717],"description":"

Pythagoras's theorem, SOHCAHTOA, finding side length from angles

","hasLesson":true},{"id":37,"name":"Non-right-angled triangles","slug":"non-right-angled-triangles","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029088911-Artboard%2047.png?alt=media&token=80493296-567f-4f51-b0b3-4c328ab90285","type":"image"},"skills":[88,73,74],"description":"

Sine and cosine rules, finding general area of a triangle

\n","hasLesson":true},{"id":65,"name":"Radians, arcs and sectors","slug":"radians-arcs-and-sectors","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028966537-Artboard%2049.png?alt=media&token=21e1f320-b306-4587-8651-9991150cbd34","type":"image"},"skills":[492,66,720,721,4,68,14],"description":"

Radian measure, with 2\\pi radians in a full circle and \\pi\\text{ rad}=180^\\circ, and the formulas for circle circumference C=2\\pi r, area A=\\pi r^2, arc length l=r\\theta or l=\\frac{\\theta}{360^\\circ}2\\pi r, and sector area A=\\frac12\\theta r^2 or A=\\frac{\\theta}{360^\\circ}\\pi r^2.

\n","hasLesson":true},{"id":722,"name":"3D volume and surface area","slug":"3d-volume-and-surface-area","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028955870-Artboard%2050.png?alt=media&token=3f2ceb70-0712-4f69-81cc-c257607a3013","type":"image"},"skills":[1080,1094,723,724,725,834,835],"description":"

Spheres, cylinders, prisms, right cones, right pyramids, and combinations of solids

","hasLesson":true},{"id":89,"name":"Angles of elevation and depression","slug":"angles-of-elevation-and-depression","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028981407-Artboard%2048.png?alt=media&token=5b9c00d8-8c4e-40c8-afef-26030c38db55","type":"image"},"skills":[90],"description":"Angles measured between a horizontal line and some elevated or depressed object.","hasLesson":true}],"description":"

2D and 3D geometry including circle measures and arc/sector formulas, triangle trigonometry and bearings, and coordinates, angles, surface area and volume in three-dimensional solids.

\n"},{"id":935,"name":"Voronoi Diagrams","slug":"voronoi-diagrams","topic":3,"subjects":["AIHL","AISL"],"isVideoReady":false,"videoId":null,"concepts":[{"id":909,"name":"Voronoi structure and construction","slug":"voronoi-structure-and-construction","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1758237746834-Artboard%2051.png?alt=media&token=9ba8047e-695d-4059-a7c6-2b2229990cf2","type":"image"},"skills":[917,918,923,920,921,949],"description":"

The terminology of Voronoi diagrams, including sites, cells, edges and vertices, with edges as perpendicular bisectors, vertices equidistant from three or more sites, adding a new site, nearest neighbor interpolation, and largest empty circle or toxic waste dump problems.

\n","hasLesson":true},{"id":939,"name":"Applications of Voronoi Diagrams","slug":"applications-of-voronoi-diagrams","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028926923-Artboard%2052.png?alt=media&token=6315d13b-6e14-473a-8bc7-ee067ccd5a72","type":"image"},"skills":[920,921],"description":"

Nearest neighbor interpolation from a Voronoi diagram, where each cell takes the value of its generating site and points on boundaries may use an average of the adjacent sites, and toxic waste dump problems (largest empty circle) solved by finding the Voronoi vertex farthest from its nearest sites.

\n","hasLesson":true}],"description":"

Voronoi diagram terminology and properties, including cells, edges and vertices, perpendicular bisectors, nearest-neighbor interpolation, largest empty circle problems and adding a new site.

\n"},{"id":58,"name":"Trig equations & identities","slug":"trig-equations-identities","topic":3,"subjects":["AAHL","AASL","AIHL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"M87nP0Y-Pds","concepts":[{"id":65,"name":"Radians, arcs and sectors","slug":"radians-arcs-and-sectors","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028966537-Artboard%2049.png?alt=media&token=21e1f320-b306-4587-8651-9991150cbd34","type":"image"},"skills":[492,66,720,721,4,68,14],"description":"

\n","hasLesson":true},{"id":166,"name":"The Unit Circle","slug":"the-unit-circle","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028896524-Artboard%2053.png?alt=media&token=6f0c6598-c2ab-4090-99f1-8e7ccf54e5df","type":"image"},"skills":[69,70,730,728,732,731,734,389,61],"description":"

The unit circle: using \\cos\\theta and \\sin\\theta as the x- and y-coordinates of a point on the circle, key exact values and quadrants, symmetry rules, periodicity, tangent as \\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta}, and \\sin^2\\theta+\\cos^2\\theta=1.

\n","hasLesson":true},{"id":735,"name":"Sinusoidal functions","slug":"sinusoidal-functions","isPaywalled":false,"isPublic":true,"thumbnail":{"type":"graph","graph":{"data":"$95","mode":"geometry"}},"skills":[736,76,14,75],"description":"

Radian measure, the sine and cosine functions with domain x\\in\\R and range (-1,1), the tangent function \\tan x=\\frac{\\sin x}{\\cos x} with its asymptotes and roots, and sinusoidal functions of the form a\\sin\\left(b\\left(x+c\\right)\\right)+d or a\\cos\\left(b\\left(x+c\\right)\\right)+d.

\n","hasLesson":true},{"id":396,"name":"Trigonometric equations","slug":"trigonometric-equations","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028815933-Artboard%2055.png?alt=media&token=283b491c-074b-4773-bc66-4c839b2c8cb5","type":"image"},"skills":[746],"description":"

Solving trigonometric equations with a calculator in a specified domain.

\n","hasLesson":true}],"description":"

Unit on trigonometric identities, unit circle properties and key trig values, symmetry and periodicity of sin, cos and tan, and solving trig equations in a given domain using radians, degrees, arcs and sectors.

\n"},{"id":101,"name":"Complex Numbers","slug":"complex-numbers","topic":1,"subjects":["AAHL","AIHL"],"isVideoReady":true,"videoId":"F4W0CCR6NTg","concepts":[{"id":189,"name":"Cartesian form and Argand diagrams","slug":"cartesian-form-and-argand-diagrams","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773512764902-cartesianForm.png?alt=media&token=303c4fb4-e64a-45c1-b5b8-30eb84b6de82","type":"image"},"skills":[641,642,643,211,212,226,1077],"description":"

Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).

","hasLesson":true},{"id":973,"name":"Complex conjugate","slug":"complex-conjugate","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773512727315-Artboard%2020.png?alt=media&token=9b7969e9-67b1-4087-bfe9-80c12235b6d8","type":"image"},"skills":[213,854,852],"description":"

The complex conjugate: z=a+bi\\Longrightarrow z^{*}=a-bi, its properties, and using it to simplify fractions with complex denominators.

","hasLesson":true},{"id":972,"name":"Solving complex equations","slug":"solving-complex-equations","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773512994227-solvingComplexEquations.png?alt=media&token=7354999e-a8a2-4d51-bc46-9107906bb52e","type":"image"},"skills":[214],"description":"

Equating real and imaginary parts to solve complex equations

","hasLesson":true},{"id":242,"name":"Complex Modulus","slug":"complex-modulus","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029720036-Artboard%2021.png?alt=media&token=9f467a1c-da25-49de-9b9a-8ad580991f5e","type":"image"},"skills":[431,645,646,851],"description":"

Introduction, definition, and properties of the complex modulus

","hasLesson":true},{"id":430,"name":"Complex Argument","slug":"complex-argument","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773515382533-complexArgument.png?alt=media&token=bd14a6ec-bec0-4bdf-a175-e3cb2156deac","type":"image"},"skills":[647,648,649],"description":"

r\\operatorname{\\mathrm{cis}}\\theta and e^{i\\theta}

","hasLesson":true},{"id":974,"name":"Polar and Euler forms","slug":"polar-and-euler-forms","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773515757846-Artboard%2022.png?alt=media&token=6030e122-a11f-4b80-8f84-ccd18ea688cd","type":"image"},"skills":[1078,1079,853,433,434,435],"description":"

Using modulus and argument to write complex numbers in the form r\\operatorname{\\mathrm{cis}}\\theta and re^{i\\theta}, which make multiplication and division easier.

","hasLesson":true},{"id":958,"name":"Complex Phasors","slug":"complex-phasors","isPaywalled":false,"isPublic":true,"thumbnail":null,"skills":[13],"description":"

Adding same-frequency complex waves using phasors and Euler's form to rewrite r_1\\sin\\left(ax+\\alpha_1\\right)+r_2\\sin\\left(ax+\\alpha_2\\right) as a single sine r\\sin\\left(ax+\\alpha\\right), with resultant amplitude r=\\left|r_1e^{i\\alpha_1}+r_2e^{i\\alpha_2}\\right| and phase \\alpha=\\arg\\left(r_1e^{i\\alpha_1}+r_2e^{i\\alpha_2}\\right).

\n","hasLesson":true},{"id":243,"name":"Powers of Complex Numbers","slug":"powers-of-complex-numbers","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029793266-Artboard%2023.png?alt=media&token=f8552fe8-946e-4330-8fd0-26ad39aab281","type":"image"},"skills":[12,436],"description":"

Using De Moivre's Theorem to find powers of complex numbers in polar form, z=r e^{i\\theta}, so that z^n=r^n e^{in\\theta}=r^n\\mathrm{cis}(n\\theta), and applying the argument rule \\arg(zw)=\\arg(z)+\\arg(w).

\n","hasLesson":true}],"description":"

The imaginary number i=\\sqrt{-1}, complex numbers in the form a+bi and their representation in the complex plane. Polar and Euler form of complex numbers, as well as complex phasors.

"},{"id":175,"name":"Vectors","slug":"vectors","topic":3,"subjects":["AAHL","AIHL"],"isVideoReady":true,"videoId":"_5Md2ZQdRo8","concepts":[{"id":186,"name":"Vector arithmetic & geometry","slug":"vector-arithmetic-geometry","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028742211-Artboard%2059.png?alt=media&token=b2257a9e-2d72-4974-88d7-651675f64f8b","type":"image"},"skills":[811,516,812,815,517,15,21,813,1095,1096,1098,816],"description":"

3D and 2D vector notation with base vectors and position vectors, including zero and negative vectors, scalar multiples, unit vectors, vector magnitude, addition and subtraction by components or triangle law, displacement vectors, and parallel vectors.

\n","hasLesson":true},{"id":180,"name":"Scalar product","slug":"scalar-product","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775711238503-scalarProduct.png?alt=media&token=f5a47138-79e7-4141-a420-c068e9a45b58","type":"image"},"skills":[783,181,221,818,833],"description":"

The scalar (aka dot) product, and its connection to angles between vectors.

","hasLesson":true},{"id":977,"name":"Vector Product","slug":"vector-product","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775711427076-vectorProduct.png?alt=media&token=1592590a-c509-45ec-bd5f-3c80e2a669f7","type":"image"},"skills":[817,184,820,523,17],"description":"

The vector (aka cross) product \\mathbf{u}\\times\\mathbf{v} and its connection to areas in 3D space.

","hasLesson":true},{"id":190,"name":"Vector equations of lines","slug":"vector-equations-of-lines","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028714655-Artboard%2061.png?alt=media&token=9b843942-74f7-46d4-b21d-859eb6009b32","type":"image"},"skills":[785,786,220,819,1028],"description":"

Equations of a vector line in 3D using a point and direction vector, written in vector form \\mathbf{r}=\\mathbf{a}+\\lambda\\mathbf{b} or parametric form x=x_0+\\lambda l,\\ y=y_0+\\lambda m,\\ z=z_0+\\lambda n, with applications to motion and angles between lines.

\n","hasLesson":true}],"description":"

2D and 3D vector representation, position and displacement vectors, vector addition and subtraction, scalar multiples and unit vectors, magnitudes and parallel/perpendicular vectors, scalar and vector products, angles and areas, and vector equations of lines and motion models.

\n"},{"id":1,"name":"Graph Theory","slug":"graph-theory","topic":3,"subjects":["AIHL"],"isVideoReady":false,"videoId":null,"concepts":[{"id":2,"name":"Foundations of Graphs","slug":"foundations-of-graphs","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775711438275-graphFoundations.png?alt=media&token=1750da65-1016-4905-9c10-66dc384e52f6","type":"image"},"skills":[977,981,982,978,979,983,986,984,985,990,1102],"description":"

Graphs as vertex-edge networks: terminology for vertices, edges and self-edges; adjacent vertices and edges; degree of a vertex; simple, complete and connected graphs; directed graphs with in-degree and out-degree; strongly connected graphs; and weighted graphs.

\n","hasLesson":true},{"id":3,"name":"Adjacency matrices & walks","slug":"adjacency-matrices-walks","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775711453903-walks.png?alt=media&token=63663974-d137-43d4-806c-d3d7a14bf08a","type":"image"},"skills":[7,991,992,994,995,996,997,1108,1109],"description":"Using matrices to represent connectivity in a graph, and counting walks using powers of those matrices.","hasLesson":true},{"id":978,"name":"Subgraphs, Trees and MSTs","slug":"subgraphs-trees-and-msts","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775711444651-subgraphs.png?alt=media&token=e7e0c6f3-13c2-400b-8870-3557800106d6","type":"image"},"skills":[987,989,988,5,6,1104,1105],"description":"

Subgraphs formed by deleting vertices or edges, trees as connected simple graphs with no cycles and n-1 edges, and minimum spanning trees found by adding the smallest edges without creating cycles using Kruskal's or Prim's algorithm, including from an adjacency matrix.

\n","hasLesson":true},{"id":979,"name":"Eulerian trails, circuits and graphs","slug":"eulerian-trails-circuits-and-graphs","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775950980022-eulerian.png?alt=media&token=baea4a1c-0094-4356-bbda-0f9493e438ec","type":"image"},"skills":[1000,999,1001,1003,1110],"description":"

Eulerian graphs and semi-Eulerian graphs, including Eulerian circuits and trails that use every edge exactly once, and the degree conditions: all vertices even for an Eulerian graph and exactly two odd vertices for a semi-Eulerian graph.

\n","hasLesson":true},{"id":5,"name":"Chinese Postman Problem","slug":"chinese-postman-problem","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775937600457-chinesePostman.png?alt=media&token=0066abfc-e314-4585-a7ed-f827222c9c1f","type":"image"},"skills":[1005,1006,1007,1111],"description":"

Introduction to the Chinese Postman Problem of finding a closed walk that visits every edge at least once. Algorithms for finding solutions in the case of an Eulerian graph, a semi-Eulerian graph, and a graph with 4 odd vertices.

\n","hasLesson":true},{"id":6,"name":"Traveling Salesman Problem","slug":"traveling-salesman-problem","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775711473466-TSP.png?alt=media&token=b16fc4f6-2370-4c8f-b31f-0bfc69b1ad24","type":"image"},"skills":[1008,1010,1011,1004,1112,1113,1114,1115],"description":"

Finding the shortest Hamiltonian cycle that starts and ends at the same vertex and visits every other vertex once, including the classical case on complete graphs, the practical case using least distances between vertices, and estimating bounds with the nearest neighbor and deleted vertex algorithms.

\n","hasLesson":true}],"description":"

Graph terminology and structure, including directed and weighted graphs, connectivity, walks/trails/paths/cycles, trees and spanning trees, adjacency matrices, Eulerian and Hamiltonian graphs, and optimization algorithms such as Prim's, Kruskal's, the Chinese postman problem and the traveling salesman problem.

\n"},{"id":346,"name":"Differentiation","slug":"differentiation","topic":5,"subjects":["AAHL","AASL","AIHL","AISL"],"isVideoReady":true,"videoId":"7GKpUTrLk_o","concepts":[{"id":111,"name":"Limits and Derivatives","slug":"limits-and-derivatives","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027621748-Artboard%2086.png?alt=media&token=f862e464-90a6-482b-9b52-1be5899b056c","type":"image"},"skills":[565,566,567,570,569,571,535,534,19],"description":"

The basic idea of a limit \\lim_{x\\to a}f(x) from tables and graphs, slope as a limit, rate of change and gradient, derivatives of x^n, and linearity for sums and scalar multiples.

\n","hasLesson":true},{"id":112,"name":"Differentiation rules","slug":"differentiation-rules","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027576694-Artboard%2088.png?alt=media&token=b370fb29-0b88-471e-867c-06e32f483988","type":"image"},"skills":[541,539,538,336,533,335,537,535,534],"description":"

Derivatives of x^{r},e^{x},\\ln x,\\sin x and \\cos x. The chain, product and quotient rules, as well as the derivatives of of \\tan x.

","hasLesson":true},{"id":326,"name":"Tangents and normals","slug":"tangents-and-normals","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027557648-Artboard%2088_1.png?alt=media&token=ba008df4-2d3c-426c-bf8f-0809dadd8f51","type":"image"},"skills":[575,576],"description":"

Definitions of tangent and normal, finding tangent and normal lines to a function

","hasLesson":true},{"id":331,"name":"Applications of the First Derivative","slug":"applications-of-the-first-derivative","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027540062-Artboard%2089.png?alt=media&token=8765b377-ec1c-498c-8770-e7f8904405d4","type":"image"},"skills":[573,16,531],"description":"

Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation

","hasLesson":true},{"id":7,"name":"Second Derivatives and Applications","slug":"second-derivatives-and-applications","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027522497-Artboard%2090.png?alt=media&token=08f7442f-e526-4cd9-af09-980e1d6aa69c","type":"image"},"skills":[348,330,574,18],"description":"

Second derivatives f'', using the sign of f'' to determine concavity, locate inflexion points where f''(x)=0 and changes sign, and classify stationary points with the second derivative test.

\n","hasLesson":true},{"id":323,"name":"Related Rates","slug":"related-rates","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027477811-Artboard%2091.png?alt=media&token=c21ece92-2ee3-49de-ace8-c64be842afc3","type":"image"},"skills":[20,577],"description":"

Using the chain rule to relate changing quantities, including \\frac{\\differentialD y}{\\differentialD t}=\\frac{\\differentialD y}{\\differentialD x}\\cdot\\frac{\\differentialD x}{\\differentialD t}, and applying this to volume, distance, and angle rates.

\n","hasLesson":true}],"description":"

Differentiation, including derivative rules for common functions, tangents and normals, rates of change, stationary points and concavity, and applications such as optimisation and related rates.

\n"},{"id":114,"name":"Integration","slug":"integration","topic":5,"subjects":["AAHL","AASL","AIHL","AISL"],"isVideoReady":true,"videoId":"Ex7Qii1AXn8","concepts":[{"id":450,"name":"Definite Integrals and Basic Anti-Derivatives","slug":"definite-integrals-and-basic-anti-derivatives","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027038579-Artboard%2093.png?alt=media&token=bb418ca4-71b9-42ce-b97d-090a1379edea","type":"image"},"skills":[418,293,294,295,587,581,583,545,584,585,41,1020],"description":"

Concept of an integral, areas with definite integrals, and basic anti-derivative solving skills

\n","hasLesson":true},{"id":425,"name":"Anti-Derivative Rules","slug":"anti-derivative-rules","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027022392-Artboard%2094.png?alt=media&token=67fe5d1c-0469-4131-b445-d19e642ef49d","type":"image"},"skills":[546,422,426,420,1022],"description":"

Basic anti-derivative rules for common functions: \\int x^n\\,dx=\\frac{x^{n+1}}{n+1}+C for n\\ne -1, \\int \\sin x\\,dx=-\\cos x+C, \\int \\cos x\\,dx=\\sin x+C, \\int e^x\\,dx=e^x+C, \\int \\frac{1}{x}\\,dx=\\ln\\left|x\\right|+C, and \\int \\sec^2 x\\,dx=\\tan x+C.

\n","hasLesson":true},{"id":542,"name":"Techniques of Integration","slug":"techniques-of-integration","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027007339-Artboard%2094%20copy.png?alt=media&token=a987c620-4834-46ff-afb4-3aab9c0b265e","type":"image"},"skills":[316,317,588],"description":"

Integration of composite functions using substitution, including integrals of the form f(ax+b), and handling definite integrals by changing the bounds or substituting back.

\n","hasLesson":true},{"id":231,"name":"Kinematics","slug":"kinematics","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757026968679-Artboard%2095.png?alt=media&token=8619b033-cbb4-41e2-b0f2-f691121bf19f","type":"image"},"skills":[311,592,591,500,313,1026,1027,1028],"description":"

Displacement, distance, velocity and acceleration in one and two dimensions, including v=\\frac{\\mathrm{d}s}{\\mathrm{d}t}, a=\\frac{\\mathrm{d}v}{\\mathrm{d}t}=\\frac{\\mathrm{d}^2s}{\\mathrm{d}t^2}, average velocity and acceleration, \\int v(t)\\,\\mathrm{d}t for displacement, \\int |v(t)|\\,\\mathrm{d}t for distance, speed as |v|, speeding up/slowing down, and variable velocity vectors.

\n","hasLesson":true},{"id":544,"name":"Volumes of Revolution","slug":"volumes-of-revolution","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757026951449-Artboard%2096.png?alt=media&token=106d5899-1675-42aa-839d-490ee2277217","type":"image"},"skills":[297,298],"description":"

Finding the volume of a solid revolved around a function or axis to create a 3\\mathrm{D} figure

","hasLesson":true}],"description":"

Integration as reverse differentiation, including basic antiderivatives, definite integrals, substitution, accumulation applications, and numerical/technology methods for areas, distance and volumes.

\n"},{"id":116,"name":"Differential Equations","slug":"differential-equations","topic":5,"subjects":["AAHL","AIHL"],"isVideoReady":true,"videoId":"XYGRGxmuOlg","concepts":[{"id":593,"name":"Solving Differential Equations","slug":"solving-differential-equations","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757026775513-Artboard%2097.png?alt=media&token=5dd3864b-1439-485b-922e-d3f79d8b7183","type":"image"},"skills":[236,594],"description":"

Solving differential equations by direct integration or separable variables.

\n","hasLesson":true},{"id":299,"name":"Euler's Method","slug":"eulers-method","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757026762514-Artboard%2098.png?alt=media&token=a42ed813-17b9-43bc-b71c-5d8aeb2cdde5","type":"image"},"skills":[595,596,1013],"description":"Euler's method as a numerical method for finding particular solutions.","hasLesson":true},{"id":947,"name":"Coupled systems","slug":"coupled-systems","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757026745780-Artboard%2099.png?alt=media&token=b8e564df-18e5-4075-b6fa-48f50beba64b","type":"image"},"skills":[1015,1033],"description":"Coupled systems of differential equations where \$x\$ and \$y\$ depend on each other, written as \$\\frac{\\differentialD x}{\\differentialD t}=f_1(x,y,t)\$ and \$\\frac{\\differentialD y}{\\differentialD t}=f_2(x,y,t)\$.","hasLesson":true},{"id":1003,"name":"Linear systems and phase portraits","slug":"linear-systems-and-phase-portraits-1","isPaywalled":false,"isPublic":true,"thumbnail":null,"skills":[1064,1016,1014,1065],"description":"","hasLesson":true},{"id":957,"name":"Second Order Differential Equations","slug":"second-order-differential-equations","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757026723742-Artboard%20100.png?alt=media&token=4c76dd3d-7916-4e13-b01e-8043c79f8a02","type":"image"},"skills":[1017],"description":"

Second order differential equations of the form \\frac{\\differentialD^2x}{\\differentialD t}=f\\left(x,\\frac{\\differentialD x}{\\differentialD t},t\\right), rewritten as the coupled system \\frac{\\differentialD x}{\\differentialD t}=y and \\frac{\\differentialD y}{\\differentialD t}=f\\left(x,y,t\\right) and solved using first order techniques.

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Solving and approximating differential equations, including separable equations, slope fields, Euler's method, initial conditions and particular solutions, and an introduction to coupled and second order systems with phase portraits and eigenvector methods.

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Theoretical and experimental probability, complementary events, expected number of outcomes, sample space

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Venn diagrams, intersection and union of events, conditional probability, independent events

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Tree diagrams, selection with/without replacement, dependence

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Markov chains and transition diagrams with state vectors and transition matrices, including finding future probabilities using \\mathbf{s_n}=\\mathbf{T^n s_0} and identifying steady state from \\mathbf{Ts}=\\mathbf{s}.

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Theoretical vs experimental probability, combined events, independence vs dependence and Markov Chains.

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Understand the basics of population, data collection, and sampling.

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Learn different ways to measure the \"center,\" or typical values, of a set of data: mean, median, and mode.

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Learn the concept of dispersion, range, IQR, outliers, and box and whisker plots.

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Learn about standard deviation and variance, which we use to measure how tightly or loosely clustered the data is around the mean.

\n","hasLesson":true},{"id":126,"name":"Frequency tables and histograms","slug":"frequency-tables-and-histograms","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028166495-Artboard%2072.png?alt=media&token=2d41e78e-b5a6-4558-849d-c5ba24514aef","type":"image"},"skills":[127,673,674],"description":"Visualizing frequency data using tables and histograms.","hasLesson":true},{"id":1004,"name":"Cumulative frequency","slug":"cumulative-frequency","isPaywalled":false,"isPublic":true,"thumbnail":null,"skills":[675,676],"description":"Completing tables of cumulative frequency, and reading cumulative frequency diagrams.","hasLesson":true},{"id":21,"name":"Data collection and measurement quality","slug":"data-collection-and-measurement-quality","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775531270444-dataCollection.png?alt=media&token=d7269d2e-470c-48cf-9a4d-418d4a8ac319","type":"image"},"skills":[49,50,52],"description":"

Designing surveys with clear, unbiased and appropriately structured questions, and considering the reliability and validity of the data collected.

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Summarizing and interpreting data using measures of centre and spread, frequency tables and graphs, sampling & data collection methods, quartiles, IQR, box plots, cumulative frequency, and outlier analysis.

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Scatter diagrams and lines of best fit y=ax+b, by eye or with GDC, the Pearson correlation coefficient r, and making predictions with awareness of limitations.

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Calculating Spearman's rank correlation coefficient by ranking x and y values, then using Pearson's r, and understanding when to use it.

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The sum of square residuals SS_{res} and the coefficient of determination R^2. Using the calculator to fit linear, quadratic, cubic, exponential, power or sine models.

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Pearson's and Spearman's correlation coefficients, goodness of fit (SS_{res}) for non-linear models, and predictions with attention to extrapolation and model limitations.

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Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution

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The normal (bell-shaped) distribution X\\sim\\mathrm{N}(\\mu,\\sigma^2), its symmetry about \\mu, the empirical rule, and calculator-based normal and inverse normal probability calculations.

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Discrete random variables take finitely many values, with probability distributions given in tables or expressions where the probabilities sum to 1, and their expected value is found by E(X)=\\sum x\\,P(X=x), with fair games and linear transformations E(aX+b)=aE(X)+b, \\mathrm{Var}(aX+b)=a^2\\mathrm{Var}(X).

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The Poisson distribution for count data, where independent events occur at a uniform average rate and X\\sim\\mathrm{Po}(\\lambda) with \\lambda=\\mathrm{E}(X)=\\mathrm{Var}(X), including probability calculations from technology and the fact that sums of independent Poisson variables are Poisson.

\n","hasLesson":true},{"id":15,"name":"Linear combinations and the central limit theorem","slug":"linear-combinations-and-the-central-limit-theorem","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027846602-Artboard%2079.png?alt=media&token=bf8e9345-81e6-48e1-94c1-cf5c356706c4","type":"image"},"skills":[698,65,67,84,1029,1030,1031,53,71],"description":"Linear combinations of independent random variables, using \$\\mathrm{E}(aX+b)=a\\mathrm{E}(X)+b\$, \$\\mathrm{Var}(aX+b)=a^2\\mathrm{Var}(X)\$, \$\\bar{X}=\\frac{X_1+\\cdots+X_n}{n}\$, and the central limit theorem for large samples, \$\\bar{X}\\sim\\mathrm{N}\\left(\\mu,\\frac{\\sigma^2}{n}\\right)\$.","hasLesson":true}],"description":"

Random variables and discrete distributions, including expected value and variance, binomial and Poisson models, linear transformations, sample means, the normal distribution, and related estimation and limit results.

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Formulating null and alternative hypotheses H_0 and H_1, then using the p-value from the data to judge evidence against H_0 at a chosen significance level.

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Expected and observed frequencies in \\chi^2 goodness of fit and independence tests, including degrees of freedom, the test statistic, and comparing \\chi^2 to a critical value to decide whether to reject H_0.

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Unbiased estimation of \\mu by \\bar{x} and of \\sigma^2 by S_x^2, and chi-squared tests with grouped categories so all expected frequencies exceed 5 and degrees of freedom are reduced by 1 for each estimated parameter, using \\mathrm{df}=n-1-k.

\n","hasLesson":true},{"id":953,"name":"Student's t-test","slug":"students-t-test","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027745203-Artboard%2084.png?alt=media&token=4e653334-7eea-41d7-b305-8b628dfef074","type":"image"},"skills":[80,82,1048,1049],"description":"

Student's t-tests for comparing means in the 1-sample, 2-sample and paired cases, as well as the assumptions made by the test.

\n","hasLesson":true},{"id":14,"name":"Confidence intervals for a population mean","slug":"confidence-intervals-for-a-population-mean","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027723432-Artboard%2084_1.png?alt=media&token=6bac3568-0413-4e64-b506-b70f77627704","type":"image"},"skills":[94,92,93,95,955,1107],"description":"Using a sample to estimate an interval containing the true po","hasLesson":true},{"id":981,"name":"Testing for population correlation","slug":"testing-for-population-correlation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775854742070-correlationTest.png?alt=media&token=149ced5a-8dc2-430e-8940-36f877dbe33f","type":"image"},"skills":[1106],"description":"

Bivariate normal data and testing whether the population correlation coefficient \\rho is zero using the sample correlation.

\n","hasLesson":true},{"id":12,"name":"Binomial and Poisson Tests","slug":"binomial-and-poisson-tests","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027670256-Artboard%2085.png?alt=media&token=1391c479-b4df-4527-a13d-a6de4e964035","type":"image"},"skills":[85,89,59],"description":"

One-tailed binomial tests for a population proportion and Poisson tests for a mean rate, using the observed count and the relevant cumulative distribution to find the p-value and compare against a hypothesized value such as p or \\lambda.

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Hypothesis testing and confidence intervals for means, proportions, rates and categorical data using p-values, significance levels, expected frequencies, chi-squared tests, t-tests and z-tests.

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