Articles
Monotonic Functions and the First Derivative Test
Monotonic functions and the First Derivative Test: where a function rises and falls from the sign of f′, local extrema, sign charts and inequalities, with code.
17 min readThe Mean Value Theorem
The Mean Value Theorem and Rolle’s Theorem: statements, proofs, how to find c, speed and rates, and the corollaries about constant functions, with examples and code.
15 min readExtreme Values of Functions on Closed Intervals
Extreme values on closed intervals: absolute and local extremes, the Extreme Value Theorem, critical points and the closed interval method, with examples and code.
17 min readLinearization and Differentials
Linearization and differentials: the tangent-line approximation L(x) = f(a) + f′(a)(x − a), df = f′(x) dx, estimates, errors, percentage change and code.
16 min readRelated Rates
Related rates: differentiate an equation in time to find an unknown rate, with the four-step strategy, six worked examples, a sliding ladder and common mistakes.
13 min readInverse Trigonometric Functions
Inverse trigonometric functions: domains and ranges, the derivatives of arcsin, arctan and arcsec with proofs, cofunction identities, and the best viewing angle.
14 min readDerivatives of Inverse Functions and Logarithms
Derivatives of inverse functions and logarithms: the rule for inverses, d/dx ln x = 1/x, a^x, log base a, logarithmic differentiation, and the number e.
15 min readImplicit Differentiation
Implicit differentiation explained: find dy/dx without solving for y, tangent and normal lines, second derivatives, rational powers, pitfalls, and code.
19 min readThe Chain Rule
The chain rule explained: how to find the inside and outside of a composite function, the power chain rule, nested functions, second derivatives, and code.
15 min readDerivatives of Trigonometric Functions
Derivatives of trigonometric functions with proofs: sin x and cos x, then tan, cot, sec and csc, why radians, simple harmonic motion, higher derivatives, and code.
18 min readThe Derivative as a Rate of Change
The derivative as a rate of change: velocity, speed, acceleration and jerk, free fall, marginal cost and revenue, sensitivity to change, and rates from data in code.
15 min readDifferentiation Rules
Differentiation rules with proofs and examples: constant, power, multiple, sum, product and quotient rules, the derivative of e^x, and higher-order derivatives.
18 min readThe Derivative as a Function
The derivative as a function: f′(x) from the definition, notation, graphing f′ from f, step functions, domains, one-sided derivatives and corners, and code.
19 min readTangent Lines and the Derivative at a Point
Tangent lines and the derivative at a point: secant slopes, the definition of f′(a), tangent and normal lines, rates of change, and where no derivative exists.
18 min readThe Pythagorean Theorem: Proofs, Triples and Applications
The Pythagorean theorem explained: the statement, four proofs, the converse and the 3-4-5 rule, Pythagorean triples, the distance formula, and generalizations.
18 min readCircles: Circumference, Area, Angles and Equations
Circles explained: pi, circumference and area, arcs and sectors, angle theorems, chords and tangents, the equation of a circle, and how lines and circles meet.
20 min readTriangles: Types, Angles, Area, Centers and Congruence
Triangles explained: types by sides and angles, the 180° angle sum, the triangle inequality, area and Heron's formula, centers, congruence and the law of sines.
18 min readQuadrilaterals: Types, Properties, Perimeter and Area
Quadrilaterals explained: square, rectangle, rhombus, parallelogram, kite and trapezoid, their properties, perimeter, area and coordinates, with practice.
17 min readGeometric Sequences and Series: Ratio, Sums and Growth
Geometric sequences and series explained: common ratio, the n-th term, finite and infinite sums, geometric means, compound interest and decay, with practice.
16 min readArithmetic Sequences and Series: Formulas, Sums and Examples
Arithmetic sequences and series explained: common difference, the n-th term, the sum formula, Gauss’s trick, arithmetic means and word problems, with practice.
19 min readBinary Numbers: How Computers Count, Convert and Calculate
Binary numbers explained: bits and place value, converting to decimal and hex, binary arithmetic, two’s complement, bitwise operators and binary fractions.
19 min readIrrational Numbers: Definition, Examples, Proofs and Approximations
Irrational numbers explained: why √2 is irrational, which roots are, sums and products, π, e, transcendental numbers and fraction approximations, with practice.
17 min readRational Numbers: Fractions, Operations, Decimals and Percent
Rational numbers explained: fractions, equivalent forms, comparing, the four operations, mixed numbers, decimals and percent, with worked examples and practice.
19 min readIntegers: Number Line, Operations, Divisibility, GCD and LCM
Integers explained: the number line, adding and multiplying negatives, divisibility, primes, GCD and LCM, and remainders, with worked examples and practice.
16 min readAlgebraic Expressions: Terms, Simplifying, Expanding and Factoring
Algebraic expressions explained: terms, like terms, evaluating, expanding brackets, factoring and algebraic fractions, with worked examples and practice.
17 min readExponents and Radicals: Laws, Rules, Examples and Practice
Exponents and radicals explained: the five laws, zero, negative and fractional exponents, simplifying roots, rationalizing and scientific notation, with practice.
18 min readNumbers in Chinese: Mandarin Numerals from 0 to Billions
How to count and read numbers in Chinese: Mandarin numerals from 0 to billions, 万 and 亿, zeros, 两 vs 二, formal numerals, with a converter and quizzes.
19 min readReal Numbers: Definition, Types, Examples and Properties
Real numbers explained: definition, types (rational and irrational), examples, properties and the number line, with interactive practice, a proof and FAQ.