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Precalculus

Functions in depth, complex numbers, sequences, conics and vectors.

经验教训

Core Complex numbers i² = −1, arithmetic in the complex plane, modulus and conjugates. (1 + 2i)*(3 - i) Core Polynomial functions Degree, end behaviour, zeros, turning points, and what the graph must look like. y = x^3 - 3x Core Rational functions Asymptotes, holes and domain: where a fraction of polynomials blows up. y = 1/(x^2 - 1) Core Sequences and series Arithmetic and geometric sequences, partial sums, and the geometric series formula. sum of 2^k for k = 0 to 10 Core The binomial theorem Expanding (a + b)ⁿ with Pascal's triangle and binomial coefficients. expand (a + b)^5 Advanced Conic sections Circles, ellipses, parabolas and hyperbolas as equations and as surfaces sliced. x^2 + y^2 - 25 Core Vectors Magnitude, direction, components, and adding vectors head to tail. [[3],[4]] Core Exponential and logarithmic functions Growth and decay, the number e, log rules, and change of base. y = e^x Core Polynomial division and the remainder theorem Long division, synthetic division, and why f(a) is the remainder on dividing by x − a. simplify (x^3 - 1)/(x - 1) Core Parametric equations and polar coordinates Curves traced by a parameter, and the (r, θ) description of the plane. polar form of 1 + i

The bridge between algebra and calculus: functions studied for their own sake (polynomial, rational, exponential, logarithmic) plus the complex numbers, sequences and series, conic sections and vectors that calculus will assume you know.

更多经验教训

Functions

Introduction to FunctionsFunctions and Function NotationDomain and RangeRates of Change and Behavior of GraphsComposition of FunctionsTransformation of FunctionsAbsolute Value FunctionsInverse FunctionsFunction (mathematics)PolynomialRational functionExponential functionLogarithmInverse functionComposite functionAsymptoteComplex numberSequenceArithmetic progressionGeometric progressionMathematical inductionVector (mathematics and physics)Matrix (mathematics)Conic sectionParametric equationPolar coordinate system

Linear Functions

Introduction to Linear FunctionsLinear FunctionsGraphs of Linear FunctionsModeling with Linear FunctionsFitting Linear Models to Data

Polynomial and Rational Functions

Introduction to Polynomial and Rational FunctionsQuadratic FunctionsPower Functions and Polynomial FunctionsGraphs of Polynomial FunctionsDividing PolynomialsZeros of Polynomial FunctionsInverses and Radical FunctionsModeling Using Variation

Exponential and Logarithmic Functions

Introduction to Exponential and Logarithmic FunctionsExponential FunctionsGraphs of Exponential FunctionsLogarithmic FunctionsGraphs of Logarithmic FunctionsLogarithmic PropertiesExponential and Logarithmic EquationsExponential and Logarithmic ModelsFitting Exponential Models to DataFinding Composite and Inverse FunctionsEvaluate and Graph Exponential FunctionsEvaluate and Graph Logarithmic FunctionsUse the Properties of LogarithmsSolve Exponential and Logarithmic Equations

Analytic Geometry

Introduction to Analytic GeometryThe EllipseThe HyperbolaThe ParabolaRotation of AxesConic Sections in Polar Coordinates

Sequences, Probability and Counting Theory

Introduction to Sequences, Probability and Counting TheorySequences and Their NotationsArithmetic SequencesGeometric SequencesSeries and Their NotationsCounting PrinciplesProbability

Conics

Distance and Midpoint Formulas; CirclesParabolasEllipsesHyperbolasSolve Systems of Nonlinear Equations

Sequences, Series and Binomial Theorem

SequencesGeometric Sequences and Series

其中一些经验教训从开放文本中改编: OpenStax Intermediate Algebra 2e, OpenStax Precalculus 2e, Wikipedia. 每页课目都列出其来源和执照。

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人们对人们提出的问题

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

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