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Composing functions like f\\circ g

","hasLesson":true},{"id":969,"name":"Even and odd functions","slug":"even-and-odd-functions","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770854405577-even%20and%20odd%20functions.png?alt=media&token=54d61f6a-8238-47e0-aef7-83975be2a030","type":"image"},"skills":[756,757],"description":"

Even functions satisfy f\\left(-x\\right)=f\\left(x\\right) and have symmetry in the y-axis, while odd functions satisfy f\\left(-x\\right)=-f\\left(x\\right) and have origin symmetry.

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The concept of an inverse function, its graph as a reflection in the line y=x, finding the inverse of a specific value, and domain and range of inverse functions. When inverse functions exist, and how to find them.

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

Function notation, domain and range, graphing and interpreting functions with technology, composite and inverse functions, and key graph features such as intercepts, asymptotes, maxima and minima, symmetry and function tests.

\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)

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Rounding, significant figures, and scientific notation.

"},{"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

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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":"$7f","mode":"geometry"}},"skills":[606,608,51],"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

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Compound growth, depreciation, interest, inflation, real value

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Pascal's triangle and binomial coefficients, using factorials and ^{n}C_{r}=\\frac{n!}{r!(n-r)!} to find combinations and the coefficients in binomial expansions.

\n","hasLesson":true},{"id":248,"name":"Binomial Theorem","slug":"binomial-theorem","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029881955-Artboard%2014.png?alt=media&token=1442315c-0b38-4a37-a959-022b43646b65","type":"image"},"skills":[249,632,250],"description":"

Expansion of \\left(a+b\\right)^n using binomial coefficients, including the general term \\binom{n}{r}a^{n-r}b^r and finding the coefficient of a specified term.

\n","hasLesson":true},{"id":247,"name":"Counting","slug":"counting","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029871896-Artboard%2015.png?alt=media&token=28368164-f4e4-4694-ba5c-02e0bdebbad5","type":"image"},"skills":[635,634,245,1038,843],"description":"Counting with combinations and permutations.","hasLesson":true},{"id":441,"name":"Counting with Restrictions","slug":"counting-with-restrictions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029862325-Artboard%2016.png?alt=media&token=2127db25-0b2c-4fbf-ac94-c5138be93db1","type":"image"},"skills":[1039,638,639,640],"description":"

Counting arrangements by splitting into subcases and using complementary counts, including treating items that must stay together as one block and placing items in separate slots to keep them apart.

\n","hasLesson":true}],"description":"Pascal's triangle and the coefficients of binomial expansions like \\left(x+a\\right)^5, as well as combinatorics."},{"id":102,"name":"Proof and Reasoning","slug":"proof-and-reasoning","topic":1,"subjects":["AAHL","AASL"],"isVideoReady":true,"videoId":"syZ6AKBabRA","concepts":[{"id":240,"name":"Proof by deduction","slug":"proof-by-deduction","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029849287-Artboard%2017.png?alt=media&token=312ca3d8-5599-4f64-8439-464901c0de57","type":"image"},"skills":[652,650],"description":"

Strict logical proofs by deduction, including parity arguments with even and odd integers written as n=2k and n=2k+1, and direct proofs that two expressions are equivalent by showing \\text{LHS}\\equiv\\text{RHS} for all variables.

\n","hasLesson":true},{"id":239,"name":"Proof by induction","slug":"proof-by-induction","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029838490-Artboard%2018.png?alt=media&token=627b4f1e-74e8-4962-b371-a4b91d559a96","type":"image"},"skills":[651,656,659,660,661,662],"description":"

Proof by induction for statements about integers, divisibility, inequalities, summations and derivatives: establish a base case, assume the result for n=k, then use the inductive hypothesis to prove n=k+1, often by comparing successive cases, splitting a sum, or using the product rule.

\n","hasLesson":true},{"id":188,"name":"Contradiction and Counterexamples","slug":"contradiction-and-counterexamples","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029822219-Artboard%2019.png?alt=media&token=86054dfb-bff6-443b-a067-2d6f1d88d305","type":"image"},"skills":[653,655,654,657],"description":"

Proof by contradiction, where assuming the negation of a statement leads to an inconsistency, and counterexamples used to disprove claims, often with examples involving prime numbers and rational numbers.

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

Simple deductive proofs to show LHS\\equiv RHS, as well as proofs by contradiction and proofs by induction.

"},{"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":[11,769,770,771,772,773,774,775],"description":"

Linear transformations of graphs and points, including translations by vectors, vertical and horizontal stretches, vertical and horizontal translations, reflections in the axes, and combined transformations such as y=f(ax+b).

\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":[781,782,1088],"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},{"id":255,"name":"Rational functions","slug":"rational-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029160193-Artboard%2039.png?alt=media&token=cced4769-0d16-429a-8ae4-bd3a62d893ca","type":"image"},"skills":[808,364,381,809],"description":"

Graphs and asymptotes of rational functions, including the reciprocal function f\\left(x\\right)=\\frac{1}{x}, linear rational functions \\frac{ax+b}{cx+d}, forms with oblique asymptotes from polynomial division, and rationals with quadratic denominators.

\n","hasLesson":true},{"id":265,"name":"Further Transformations","slug":"further-transformations","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029151885-Artboard%2040.png?alt=media&token=aadaf839-7cb4-4ded-a078-385bf09c5195","type":"image"},"skills":[776,777,779,778,775],"description":"Graph transformations of functions including combined horizontal translations and stretches \$f(ax+b)\$, and the effects of modulus, squaring and reciprocals: \$f(|x|)\$, \$|f(x)|\$, \$[f(x)]^2\$, and \$\\frac{1}{f(x)}\$.","hasLesson":true}],"description":"

Translations, reflections and stretches of graphs, including modulus and reciprocal transformations, and identifying vertical, horizontal and oblique asymptotes of rational functions.

\n"},{"id":107,"name":"Polynomials","slug":"polynomials","topic":2,"subjects":["AAHL"],"isVideoReady":true,"videoId":"4XB0k4z9IDs","concepts":[{"id":204,"name":"Polynomial Structure","slug":"polynomial-structure","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921504562-polynomialBasics.png?alt=media&token=0b8e8316-ac3a-4357-9190-4eb4d3b38a70","type":"image"},"skills":[205,206,208,791,210],"description":"

Polynomial definition in x with coefficients and degree, identifying terms like a_kx^k, finding degrees of polynomial products and specific coefficients in products, and matching coefficients in polynomial identities that hold for all x\\in\\mathbb{R}.

\n","hasLesson":true},{"id":200,"name":"Polynomial division, factors and remainders","slug":"polynomial-division-factors-and-remainders","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921518202-polynomialDivision.png?alt=media&token=7f7f60a3-b928-427f-9168-4cf324b30129","type":"image"},"skills":[792,216,217,219,218],"description":"

Polynomial division with quotient and remainder, including long division by a linear divisor, the degree condition that \\deg R < \\deg D, and the remainder and factor theorems (P(a) is the remainder on division by (x-a), and (x-a) is a factor iff P(a)=0).

\n","hasLesson":true},{"id":203,"name":"Polynomial graphs","slug":"polynomial-graphs","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921525454-polynomialGraphs.png?alt=media&token=b79674ad-7928-4fda-b6b2-4fea53ccad3a","type":"image"},"skills":[795,796],"description":"

Polynomial graphs and how the sign of the leading coefficient affects the broad shape of cubic and quartic graphs, and how real roots appear as x-intercepts where the curve touches or crosses the axis depending on whether the root is repeated or unique.

\n","hasLesson":true},{"id":227,"name":"Polynomial roots","slug":"polynomial-roots","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1774921539920-polynomialRoots.png?alt=media&token=2bdf87b4-f460-4ebf-b677-904a68a61df2","type":"image"},"skills":[201,228,1067],"description":"

A degree n polynomial has n roots counting multiplicity, and its roots may be real or complex; for a polynomial with real coefficients, complex roots occur in conjugate pairs, and the sum and product of roots are given by the coefficient relationships \\frac{-a_{n-1}}{a_n} and \\frac{(-1)^n a_0}{a_n}.

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

Polynomial structure and degree, roots and factors (including remainder and factor theorems), division with linear divisors, and relationships among roots, coefficients and conjugate pairs.

\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":[88,73,74,718],"description":"

Area of non-right-angled triangles using A=\\frac12ab\\sin C, the sine rule \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}, the cosine rule c^2=a^2+b^2-2ab\\cos C, and the ambiguous case when solving triangles.

\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":[492,66,720,4,68,14],"description":"

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=r\\theta, A=\\frac12\\theta r^2, C=2\\pi r, and A=\\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 sectors, right-triangle and non-right-triangle trigonometry, bearings and angles of elevation/depression, 3D coordinates and angles in space, and surface area and volume of common solids.

\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":[492,66,720,4,68,14],"description":"

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=r\\theta, A=\\frac12\\theta r^2, C=2\\pi r, and A=\\pi r^2.

\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":[69,70,729,730,728,732,731,734,389,61],"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":"$80","mode":"geometry"}},"skills":[736,76,14,75],"description":"Amplitude, period, and principal axis if functions like \$a\\sin\\left(bx\\right)+c\$.","hasLesson":true},{"id":949,"name":"Tangent and reciprocal trig functions","slug":"tangent-and-reciprocal-trig-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028852382-Artboard%2054.png?alt=media&token=c153b834-4d14-48e7-b00c-ce7ed61b764b","type":"image"},"skills":[63,37,38,39],"description":"The tangent function \$\\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta}\$, reciprocal trig functions \$\\sec\\theta=\\frac{1}{\\cos\\theta}\$, \$\\cosec\\theta=\\frac{1}{\\sin\\theta}\$ and \$\\cot\\theta=\\frac{1}{\\tan\\theta}\$, including the graph, domain and range of \$\\sec x\$, \$\\cosec x\$ and \$\\cot x\$.","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":[746,1021,744,745,747],"description":"

Solving trigonometric equations algebraically in a specified domain, including \\sin\\theta=a, \\cos\\theta=a, \\tan\\theta=a, equations with arguments of the form ax+b, and trigonometric quadratics using identities such as \\sin^2\\theta+\\cos^2\\theta=1.

\n","hasLesson":true},{"id":950,"name":"Inverse trig functions","slug":"inverse-trig-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028802942-Artboard%2056.png?alt=media&token=813ae494-c58f-41f6-9eef-ae84736cb47f","type":"image"},"skills":[737,79,738],"description":"

The inverse trig functions \\arcsin, \\arccos and \\arctan, including their principal-value domains and ranges and the graphs as reflections of the restricted trig functions in y=x.

\n","hasLesson":true},{"id":393,"name":"Pythagorean and double-angle identities","slug":"pythagorean-and-double-angle-identities","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028794001-Artboard%2057.png?alt=media&token=6c43949e-408e-48a5-aa5d-2ee8a9e57c20","type":"image"},"skills":[61,739,740],"description":"

Trigonometric identities including \\sin^2\\theta+\\cos^2\\theta=1, \\sin2\\theta=2\\sin\\theta\\cos\\theta, and the equivalent forms of \\cos2\\theta: \\cos^2\\theta-\\sin^2\\theta, 2\\cos^2\\theta-1, and 1-2\\sin^2\\theta.

\n","hasLesson":true},{"id":951,"name":"Reciprocal and compound-angle identities","slug":"reciprocal-and-compound-angle-identities","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028783135-Artboard%2058.png?alt=media&token=dd1d6ad4-93fd-4db0-b398-0f141df1b421","type":"image"},"skills":[742,78,394,741,743],"description":"

Further trigonometric identities, including the compound angle formulas for \\sin(A\\pm B), \\cos(A\\pm B) and \\tan(A\\pm B), the double angle identity \\tan 2\\theta=\\frac{2\\tan\\theta}{1-\\tan^2\\theta}, and the identities 1+\\tan^2\\theta=\\sec^2\\theta and 1+\\cot^2\\theta=\\cosec^2\\theta.

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

Trigonometric identities and equations using the unit circle, key values, reciprocal and inverse trig functions, compound and double angle formulas, and solving trig equations in given domains.

\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":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":[12,436,437],"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, and their utility for finding powers and roots of complex numbers.

"},{"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":[785,786,784,220,819],"description":"

Equations of a vector line in 3D using a point and direction vector, in vector form \\mathbf{r}=\\mathbf{a}+\\lambda\\mathbf{b}, parametric form x=x_0+\\lambda l,\\ y=y_0+\\lambda m,\\ z=z_0+\\lambda n, Cartesian form \\frac{x-x_0}{l}=\\frac{y-y_0}{m}=\\frac{z-z_0}{n}, and the angle between lines from their direction vectors.

\n","hasLesson":true},{"id":514,"name":"Line relationships and intersections","slug":"line-relationships-and-intersections","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028703375-Artboard%2062.png?alt=media&token=042dd612-7b82-4ea2-aa06-db1077b59117","type":"image"},"skills":[822,191,823,819,821],"description":"

Classifying vector lines in 3D as coincident, parallel, intersecting or skew using direction vectors and point checks, with the angle between lines found by \\theta=\\cos^{-1}\\left(\\frac{\\mathbf{b}_1\\cdot\\mathbf{b}_2}{\\left|\\mathbf{b}_1\\right|\\left|\\mathbf{b}_2\\right|}\\right).

\n","hasLesson":true},{"id":194,"name":"Equations of a plane","slug":"equations-of-a-plane","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028689158-Artboard%2063.png?alt=media&token=ebc626e0-e14a-4009-95f9-4d94e3560064","type":"image"},"skills":[789,788,787],"description":"

Equations of a plane in 3D space using vector form \\mathbf{r}=\\mathbf{a}+\\lambda\\mathbf{b}+\\mu\\mathbf{c}, scalar product form \\mathbf{r}\\cdot\\mathbf{n}=\\mathbf{a}\\cdot\\mathbf{n}, and Cartesian form n_1x+n_2y+n_3z=d, with a point and two direction vectors or a normal vector.

\n","hasLesson":true},{"id":824,"name":"Intersections with planes","slug":"intersections-with-planes","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028677121-Artboard%2064.png?alt=media&token=882bb3ea-73e5-4c56-98aa-7ca462260ad7","type":"image"},"skills":[819,826,827,833,831,1090,825,828,830],"description":"

Finding angles between lines, between a line and a plane, and between two planes using direction vectors and normals, plus intersections of lines and planes and systems of three planes in 3D.

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

Vectors in 2D and 3D, including representation, magnitude and unit vectors, vector operations and products, and using vector equations to model and analyze lines and planes in space.

\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":[565,566,567,570,569,571,332,535,534,19],"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, using the derivative definition f'(x)=\\lim_{h\\to0}\\frac{f(x+h)-f(x)}{h}, 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":[541,539,538,336,533,335,537,535,534,540,536,855,1116],"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 reciprocal trig functions\\log_{a}x, and a^{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":[348,330,574,18,532],"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":982,"name":"Implicit differentiation","slug":"implicit-differentiation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1776137641317-implicit.png?alt=media&token=80156229-ae46-40dc-81d4-732e2f65beb4","type":"image"},"skills":[347,1076],"description":"

Differentiating both sides of an equation with respect to x to find \\frac{\\mathrm{d}y}{\\mathrm{d}x} for relations not written as y=f(x), using the product and chain rules, and finding horizontal and vertical tangents and normals from the resulting derivative.

\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":[347,20,577,578,579,580],"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},{"id":349,"name":"n^th Derivative","slug":"nth-derivative","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027507807-Artboard%2092.png?alt=media&token=36a37a08-d37d-46e2-9dd4-8a3048e88187","type":"image"},"skills":[530],"description":"

The nth derivative of a function, written f^{\\left(n\\right)}\\left(x\\right) or \\frac{d^{n}y}{dx^{n}}, found by differentiating n times.

\n","hasLesson":true},{"id":8,"name":"L'Hôpital's rule","slug":"lhopitals-rule","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027460767-hopital.png?alt=media&token=28ee0503-5f53-40af-8c03-28b0bacd0634","type":"image"},"skills":[568],"description":"

Evaluating indeterminate limits of the form \\lim_{x\\to a}\\frac{f\\left(x\\right)}{g\\left(x\\right)} when both numerator and denominator approach 0 or \\infty, by replacing them with \\lim_{x\\to a}\\frac{f'(x)}{g'(x)} and repeating if necessary.

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

Rules and applications of differentiation, including derivatives of common functions, composite and implicit differentiation, tangents and normals, stationary points, concavity, limits, related rates and optimisation.

\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":[418,293,294,295,296,587,581,583,545,584,585,41],"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":[546,422,426,420,547,424,1022,1023,540,536,855,1118],"description":"

Anti-derivatives of sums and scalar multiples, using linearity together with standard forms such as \\int x^n\\,dx=\\frac{x^{n+1}}{n+1}+C, \\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|x|+C, and the usual trig and inverse trig results.

\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":[316,317,588,237,235,590,589],"description":"

Techniques for evaluating integrals, including substitution for f(ax+b) and compositions, handling definite integrals with changed bounds, partial fractions, integration by parts, repeated parts, and cyclical parts.

\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],"description":"

Displacement and distance, velocity and acceleration, speed as |v|, average velocity and acceleration as \\Delta s/\\Delta t and \\Delta v/\\Delta t, and the relations v=\\mathrm{d}s/\\mathrm{d}t, a=\\mathrm{d}v/\\mathrm{d}t=\\mathrm{d}^2s/\\mathrm{d}t^2, \\int_{t_1}^{t_2} v(t)\\,\\mathrm{d}t for change in displacement, and \\int_{t_1}^{t_2} |v(t)|\\,\\mathrm{d}t for distance.

\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 indefinite and definite integrals, substitution and integration by parts, partial fractions, applications to area, volume and motion, and use of GDC to evaluate definite integrals.

\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":[302,236,300,238,594],"description":"

Solving differential equations by direct integration, separable variables, homogeneous equations using the substitution y=vx, and linear first-order equations with an integrating factor, then using initial conditions to find particular solutions.

\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],"description":"Euler's method as a numerical method for finding particular solutions.","hasLesson":true}],"description":"

Solving first-order differential equations using direct integration, separation of variables, integrating factors and homogeneous substitutions, with particular solutions from initial conditions and numerical approximation by Euler's method.

\n"},{"id":117,"name":"Maclaurin","slug":"maclaurin-series","topic":5,"subjects":["AAHL"],"isVideoReady":true,"videoId":"OjEhrxGxUhE","concepts":[{"id":304,"name":"Maclaurin Series Basics","slug":"maclaurin-series-basics","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775760759702-maclaurin.png?alt=media&token=14f98c81-c663-4d0c-a881-ed4a883d4fa6","type":"image"},"skills":[507,508,509,511,512,510,250],"description":"

Maclaurin series express a function as a polynomial in powers of x using derivatives at 0, with general form f\\left(x\\right)=\\sum_{n=0}^{\\infty}\\frac{f^{\\left(n\\right)}\\left(0\\right)x^{n}}{n!}, and include standard expansions for e^x, \\sin x, \\cos x, \\ln\\left(x+1\\right), \\arctan x, and the binomial extension for rational exponents.

\n","hasLesson":true},{"id":598,"name":"Operations on Maclaurin Series","slug":"operations-on-maclaurin-series","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1775760732396-maclaurinOps.png?alt=media&token=94d693ce-61d7-4106-9f3d-d982cf8cdb48","type":"image"},"skills":[599,513,308,307,600,601,856],"description":"

Operations on Maclaurin series including term-by-term differentiation and integration, adding and subtracting coefficients, multiplying and dividing series to find initial terms, using substitutions for composite functions, and applying series expansions to evaluate limits.

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

Maclaurin series for common functions, constructing and manipulating series using differentiation, integration, addition, multiplication, division and composition, and using series to approximate functions, expand rational powers and evaluate limits.

\n"},{"id":96,"name":"Probability","slug":"probability","topic":4,"subjects":["AAHL","AASL","AIHL","AISL"],"isVideoReady":true,"videoId":"nTKI6Ze6SxE","concepts":[{"id":145,"name":"Foundations of probability","slug":"foundations-of-probability","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028648442-Artboard%2065.png?alt=media&token=5706f4de-134c-4cf9-b2be-c3b79211ab31","type":"image"},"skills":[701,702,699,700,32,703],"description":"

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

","hasLesson":true},{"id":154,"name":"Combined Events","slug":"combined-events","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028633410-Artboard%2066.png?alt=media&token=875764d4-1b49-4d32-9717-25ae5901b2f5","type":"image"},"skills":[156,155,157,163,34,33,1117,704],"description":"

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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Theoretical vs experimental probability, combined events, independence vs dependence and Bayes's Theorem.

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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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Variance as \\sigma^2=\\frac{\\sum f_i(x_i-\\mu)^2}{n} and standard deviation as \\sigma=\\sqrt{\\sigma^2}, including calculator Sx vs \\sigma x and how adding a constant changes the mean but not the standard deviation, while scaling multiplies both mean and standard deviation.

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Summarizing and interpreting data using measures of centre and spread, frequency tables and graphs, sampling 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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Correlation and linear regression for paired data, including best fit lines, Pearson's r, predictions from regression models, and the limits of extrapolation.

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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, z-values z=\\frac{x-\\mu}{\\sigma} and standardization to Z\\sim\\mathrm{N}(0,1), the empirical rule, and calculator-based normal and inverse normal probability calculations.

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Continuous random variables with pdfs, probabilities as areas under f(x), normalization to total area 1, and finding E(X), median, mode, variance, standard deviation, and the effects of linear transformations aX+b.

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Discrete and continuous random variables, probability distributions, expected value and variance, binomial and normal models, z-values and linear transformations.

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Decomposing rational expressions into partial fractions, including linear factors such as \\frac{A}{x+3}+\\frac{B}{x-1} and more complex forms, by cross-multiplying and solving for the unknown constants.

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Solving systems of equations in three unknowns by substitution, elimination or calculator methods, and determining whether the system has no solution, a unique solution or infinitely many solutions.

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Systems of equations with 2 or 3 unknowns, including solution methods and the number of possible solutions, and partial fractions for linear and non-linear rational expressions.

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IB Math AA HL