IB Mathematics Analysis and Approaches
156 of 156 topics written so far. Greyed-out topics are coming soon.
Units marked HL are for Higher Level only. Everything else is for both SL and HL.
Unit 1: Number and Algebra
- 1Scientific Notation
Patterns in powers of 10, how the sign and size of an exponent change a power, and writing, converting and calculating with very large and very small numbers.
Draft - 2Arithmetic Sequences
Sequences that add the same amount each time — the common difference, the general term t_n = a + (n − 1)d, and problems.
Draft - 3Arithmetic Series
Adding the terms of an arithmetic sequence quickly with S_n = n/2 (2a + (n − 1)d), and why the formula works.
Draft - 4Sigma Notation
Reading and writing sums with Σ, counting terms, evaluating arithmetic and geometric sums written in sigma form, changing the index, and checking with a GDC.
Draft - 5Geometric Sequences
Sequences that multiply by the same amount each time — the common ratio, the general term t_n = ar^(n − 1), and problems.
Draft - 6Geometric Series
Adding the terms of a geometric sequence with S_n = a(r^n − 1)/(r − 1), and why the formula works.
Draft - 7Compound Interest
Interest on interest — A = P(1 + i)^n, compounding periods, present value, and finding the rate or time.
Draft - 8Appreciation and Depreciation
How things gain or lose value over time — straight-line and percent depreciation, percent appreciation, year-by-year tables, reading value graphs, and buying new vs. used.
Draft - 9Exponent Laws
The product, quotient and power laws, what zero and negative exponents mean, and how the graph of y = 2^x compares with y = x².
Draft - 10Introduction to Logarithms
A logarithm is an exponent — evaluating logs, switching between exponential and logarithmic form, and estimating logs between whole numbers.
Draft - 11Deductive Proof
Proving numerical and algebraic results step by step — laying out an LHS-to-RHS proof, the difference between = and ≡, proofs about odd, even and consecutive integers, and why checking examples isn't a proof.
Draft - 12Rational Exponents
What fractional exponents like 8^(2/3) mean, how to evaluate and simplify them, and how to rewrite powers in a different base.
Draft - 13Laws of Logarithms
The product, quotient, and power laws, the change of base formula, and using them to simplify and evaluate expressions.
Draft - 14Solving Exponential Equations
Solving equations with the unknown in the exponent — by finding a common base or by taking logarithms — including factoring and quadratic-type equations.
Draft - 15Infinite Geometric Series
When an infinite geometric series converges, its sum a/(1 − r), and how to use it for repeating decimals, shifted indices, and bouncing-ball problems.
Draft - 16Pascal's Triangle
How Pascal's triangle is built, the patterns hiding in it, and how it counts paths and combinations.
Draft - 17Binomial Expansion
Expanding powers like (x + y)^n and (2x − 3)^4 quickly using the rows of Pascal's triangle.
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Unit 2: Number and Algebra HL
- 1Counting Principles
The additive and multiplicative counting principles, when to use a list or tree diagram instead, and factorial notation.
Draft - 2Permutations
Counting arrangements where order matters, the formula P(n, r) = n!/(n − r)!, and how to handle restrictions like fixed positions, items kept together, and items kept apart.
Draft - 3Combinations
Counting selections where order doesn't matter, the formula for n choose r, deciding between permutations and combinations, and committees with conditions.
Draft - 4The Binomial Theorem for Rational Powers
Extending the binomial theorem to negative and fractional powers — infinite series, the validity condition |x| < 1, factoring (a + bx)^n, approximations like √1.02, and finding particular terms.
Draft - 5Integration Using Partial Fractions
Splitting a rational function with distinct linear factors into simple fractions that integrate to logarithms, including when to use long division first.
Draft - 6Introduction to Complex Numbers
The number i, Cartesian form a + bi, real and imaginary parts, conjugate, modulus and argument, the Argand diagram, complex arithmetic, and quadratics with a negative discriminant.
Draft - 7Polar and Euler Form of Complex Numbers
Writing complex numbers as r cis θ and re^(iθ), converting between forms, multiplying and dividing as rotations and stretches, and adding sinusoidal waves with complex numbers.
Draft - 8Complex Conjugate Roots
Why non-real roots of polynomials with real coefficients come in conjugate pairs, and how to use one known complex root to factor cubics and quartics or to build a polynomial from its roots.
Draft - 9De Moivre's Theorem
Powers of complex numbers in polar form, the proof by induction, nth roots and roots of unity as regular polygons, and multiple-angle identities like cos 3θ.
Draft - 10Proof by Mathematical Induction
Proving a statement for every positive integer n — the base case, the inductive step and the conclusion — with model proofs for sums, divisibility and repeated derivatives.
Draft - 11Proof by Contradiction and Counterexample
Proving a statement by showing its opposite leads to something impossible — √2 and log₂3 are irrational, there are infinitely many primes — and disproving a claim with a counterexample.
Draft - 12Systems of Linear Equations (Three Unknowns)
Solving up to three linear equations in three unknowns by row reduction and with a GDC — unique solution, infinitely many (a general solution with a parameter λ), or none — and finding parameter values that change the outcome.
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Unit 3: Functions
- 1Equations of Lines
The slope formula, zero and undefined slopes, finding a line's equation from a graph, a table, a point and slope, or two points, and switching between y = mx + b, Ax + By + C = 0 and Ax + By = D.
Draft - 2Parallel and Perpendicular Lines
Parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes. Testing pairs of lines and finding equations of parallel and perpendicular lines through a point.
Draft - 3Function Notation
Writing f(x), evaluating functions, and solving f(x) = a value — from equations, tables, and graphs.
Draft - 4Domain and Range
Finding the set of possible inputs and outputs of a function from graphs, equations, and real-world situations.
Draft - 5Key Features of Graphs
Sketching graphs from information or a GDC screen, finding intercepts, maximums and minimums, symmetry and asymptotes with technology, intersecting curves, and graphing sums and differences of functions.
Draft - 6Composition of Functions
Putting one function inside another — finding f(g(x)) from tables, graphs, and equations, its domain and range, decomposing functions, and real-world chains.
Draft - 7Inverse Functions
Undoing a function — finding, graphing, and checking inverses.
Draft - 8Vertex Form and Transformations
Graphing y = a(x − h)² + k as a transformation of y = x², sketching by hand with the step pattern, and finding the equation of a parabola from its graph.
Draft - 9Factored Form of a Quadratic
Using y = a(x − r)(x − s) to find zeros, the axis of symmetry and the vertex, sketching from factored form, and finding an equation from its zeros.
Draft - 10The Quadratic Formula
Where the quadratic formula comes from, solving any quadratic equation with it (exact and decimal answers), choosing a solving method, and what a negative b² − 4ac means.
Draft - 11Zeros of Quadratics and the Discriminant
Finding the zeros of a quadratic function by factoring or the quadratic formula, and using the discriminant to count them.
Draft - 12Graphs of Rational Functions
Sketching y = (ax + b)/(cx + d) from its asymptotes and intercepts, spotting holes, and a first look at oblique asymptotes.
Draft - 13Exponential Functions
Graphs and key properties of y = a^x, and how to tell exponential, linear, and quadratic patterns apart.
Draft - 14Logarithmic Functions
Graphing y = log_b x as the inverse of y = b^x — its asymptote, domain, range, intercept, and how the base changes the graph.
Draft - 15Solving Equations and Inequalities Graphically
Solving equations like 2^x = x + 3 and cos x = x, and inequalities like 2x² < 2^x, by graphing and by narrowing in on zeros to two decimal places.
Draft - 16Combining Transformations
Graphing y = af(k(x − d)) + c by applying stretches, reflections, and translations in the right order.
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Unit 4: Functions HL
- 1The Factor Theorem
Using P(a) = 0 to find factors, testing possible integer and rational zeros, and factoring cubics and quartics.
Draft - 2The Remainder Theorem
Finding the remainder of a polynomial division without dividing, and using remainders to find unknown coefficients.
Draft - 3Solving Polynomial Equations
Solving polynomial equations up to degree 4 by factoring, connecting real roots to x-intercepts, and solving problems such as box volumes.
Draft - 4Sum and Product of Roots
Finding the sum and product of the roots of a polynomial equation straight from its coefficients, for quadratics, cubics and quartics (including complex roots), and forming equations with related roots.
Draft - 5Rational Functions with Quadratics
Graphing f(x) = (ax + b)/(cx² + dx + e) and f(x) = (ax² + bx + c)/(dx + e) — vertical, horizontal and oblique asymptotes, intercepts, and behaviour near the asymptotes.
Draft - 6Even and Odd Functions
Testing whether a function is even, odd, or neither using f(−x), the symmetry of their graphs, and what happens when you add or multiply them.
Draft - 7Polynomial Inequalities
Solving linear and factorable polynomial inequalities using graphs, intervals and test points, and showing the solution on a number line.
Draft - 8Rational Inequalities
Solving inequalities with variables in the denominator, using sign charts built from zeros and asymptotes, and checking with graphs.
Draft - 9Modulus, Reciprocal and Squared Graphs
Sketching y = |f(x)|, y = f(|x|), y = 1/f(x), y = f(ax + b) and y = [f(x)]² from the graph of y = f(x), with the key points and asymptotes of each.
Draft - 10Modulus Equations and Inequalities
Solving equations and inequalities with absolute values by cases, by squaring and graphically, and solving g(x) ≥ f(x) both graphically and analytically.
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Unit 5: Geometry and Trigonometry
- 1Vectors in Three Dimensions
The right-handed 3-D coordinate system, plotting points, vectors [x, y, z] and the unit vectors i, j, k, magnitude, distance between points, operations in component form, and collinearity.
Draft - 2Spheres, Cones, Pyramids and Composite Solids
Volume and surface area of spheres, hemispheres, right pyramids and right cones, and of solids built from them, with answers to 3 significant figures.
Draft - 3Angles in 3-D Solids
Finding the angle between two lines and between a line and a plane in cuboids, pyramids and cones, by spotting the right triangle, with 3-D distance and midpoint along the way.
Draft - 4The Primary Trigonometric Ratios
Sine, cosine and tangent in right triangles — naming the sides, SOH CAH TOA, and finding missing sides and angles.
Draft - 5The Sine Law
Solving any triangle when you know two angles and a side, using a/sin A = b/sin B = c/sin C.
Draft - 6The Cosine Law
Solving triangles from two sides and the angle between them, or from three sides, using c² = a² + b² − 2ab cos C.
Draft - 7Area of a Triangle Using Sine
The formula Area = ½ab sin C, where it comes from, using it with the sine and cosine rules, and finding an angle from a known area.
Draft - 8Right Triangle Problems
Angles of elevation and depression, clinometers, ladders, ramps, navigation, and problems with two right triangles.
Draft - 9Trig Problems in Three Dimensions
Solving 3-D problems — towers, cliffs, boxes, and pyramids — by splitting them into 2-D right and oblique triangles.
Draft - 10Radian Measure
What a radian is, converting between degrees and radians, arc length a = rθ, and angular velocity.
Draft - 11Arc Length and Sector Area
Arc length, sector area and sector perimeter with the angle in degrees or radians, and the area of a segment as a sector minus a triangle.
Draft - 12Trig Ratios in Radians
Primary and reciprocal trig ratios of angles in radians, with a calculator and exactly for the special angles, using the CAST rule and related acute angles.
Draft - 13The Ambiguous Case
When two sides and a non-included angle (SSA) give no triangle, one triangle, or two — and how to find them all.
Draft - 14Proving Trig Identities
The Pythagorean, quotient, and reciprocal identities, and how to prove simple identities step by step.
Draft - 15Double Angle Formulas
Formulas for sin 2x, cos 2x (in three forms), and tan 2x, derived from the compound angle formulas, with exact values and finding sin 2x from one ratio and a quadrant.
Draft - 16Sinusoidal Functions in Radians
The graphs of y = sin x and y = cos x in radians, and y = a sin(k(x − d)) + c — amplitude, period 2π/k, phase shift, sketching, and writing equations.
Draft - 17The Tangent Function
Graphing y = tan x in radians — asymptotes, period π, zeros, domain and range — and its link to sin x / cos x.
Draft - 18Sinusoidal Applications in Radians
Modelling Ferris wheels, tides, springs, daylight, and predator–prey cycles with sinusoidal functions in radians, and answering questions with the model and its graph.
Draft - 19Solving Linear Trig Equations
Solving equations like 2 sin x + 1 = 0 for 0 ≤ x ≤ 2π in radians, with exact answers from special angles, decimal answers from related angles, and equations with kx.
Draft - 20Solving Quadratic Trig Equations
Solving trig equations that are quadratic, like 2 cos²x − cos x − 1 = 0, by factoring, common factoring, and using identities first, for 0 ≤ x ≤ 2π in radians.
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Unit 6: Geometry and Trigonometry HL
- 1Reciprocal Trig Functions
Graphing y = csc x, y = sec x, and y = cot x in radians from sin, cos, and tan — domain, range, period, asymptotes, and notation.
Draft - 2Inverse Trig Functions
The functions arcsin x, arccos x and arctan x — restricted domains, their domains, ranges and graphs, exact values, and compositions like sin(arccos x).
Draft - 3Compound Angle Formulas
The sine, cosine, and tangent of a sum or difference of two angles, where the formulas come from, and how to use them for exact values and simplifying.
Draft - 4Equivalent Trig Expressions
Cofunction, even–odd, and related-angle identities in radians, from right triangles, the unit circle, and transformations, and how to check equivalence with a graph.
Draft - 5Introduction to Vectors
Scalars vs vectors, magnitude and direction, vector notation, equal and opposite vectors, and describing directions with bearings and angles.
Draft - 6Adding and Subtracting Vectors
Adding vectors tip to tail and with the parallelogram law, subtracting by adding the opposite, the zero vector, properties of addition, and finding a resultant with the cosine and sine laws.
Draft - 7Scalar Multiplication of Vectors
Multiplying a vector by a number, collinear vectors, unit vectors, the distributive properties, linear combinations, and simple geometric proofs with vectors.
Draft - 8Cartesian Vectors
Writing 2-D vectors in component form [x, y], position vectors, converting between components and magnitude-direction, the vector between two points, unit vectors i and j, and operations in component form.
Draft - 9The Dot Product
Multiplying two vectors to get a number — the geometric and component formulas, the angle between vectors, perpendicular vectors, properties, and work.
Draft - 10Equations of Lines in 3-Space
Vector, parametric, and symmetric equations of a line in 3-space, why a line has no single scalar equation, and writing a line as the intersection of two planes.
Draft - 11Vector Kinematics
Position, velocity, and speed for objects moving in a straight line with constant velocity, when and where two objects meet, closest approach, and (AI HL) motion with variable velocity in 2-D.
Draft - 12Intersections of Lines and Planes
How two lines in 3-space can meet (intersecting, parallel, coincident, or skew), how a line meets a plane, and how to find the point of intersection.
Draft - 13The Cross Product
Multiplying two vectors in 3-space to get a perpendicular vector — the component formula, the determinant shortcut, the right-hand rule, the magnitude formula, and properties.
Draft - 14Applications of the Cross Product
Using the cross product to find areas of parallelograms and triangles, volumes with the scalar triple product, torque on a wrench, and vectors normal to two given vectors.
Draft - 15Equations of Planes
Normal vectors, scalar, vector, and parametric equations of a plane, converting between them with the cross product, the plane through three points, and parallel and perpendicular planes.
Draft - 16Intersections of Planes
How two or three planes can meet — in a point, a line, a plane, or not at all — solving systems of three equations by elimination, and using normals to predict the result.
Draft - 17Angles Between Lines and Planes
The acute angle between two lines, between a line and a plane, and between two planes, all from direction vectors, normal vectors, and the scalar product.
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Unit 7: Statistics and Probability
- 1Types of Data
Why statistical studies collect data, why data varies, and how to classify variables and data sets.
Draft - 2Sampling Methods
Populations and samples, random and non-random sampling methods, and organizing data in a spreadsheet.
Draft - 3Bias in Sampling
How sampling bias, non-response bias, response bias, and measurement bias distort results, and how to reduce them.
Draft - 4Displaying One-Variable Data
Choosing the right graph for the data — bar and circle graphs, histograms, stem-and-leaf plots, and boxplots — and spotting graphs that mislead.
Draft - 5Cumulative Frequency
Cumulative frequency tables and graphs for grouped data, reading off the median, quartiles, percentiles and IQR, and turning the results into a box-and-whisker diagram.
Draft - 6Quartiles and Percentiles
Quartiles, the interquartile range, the five-number summary, the 1.5 × IQR rule for outliers, boxplots, and percentiles.
Draft - 7Measures of Central Tendency
Mean, median, and mode; weighted means; estimating the mean from grouped data; the effect of outliers; and choosing the best measure.
Draft - 8Standard Deviation
Measuring spread with the range, variance, and standard deviation — population vs sample formulas, by hand and with technology.
Draft - 9Scatter Plots and Correlation
Independent and dependent variables, scatter plots, describing a relationship, the correlation coefficient r, and side-by-side boxplots.
Draft - 10Linear Regression
The line of best fit by least squares, interpreting slope and intercept, interpolation and extrapolation, residuals, and the effect of outliers.
Draft - 11Regression Line of x on y
When to use the regression line of x on y instead of y on x, why the two lines differ but both pass through the mean point, finding it with technology, and making reliable predictions.
Draft - 12Probability and Sample Spaces
Outcomes, sample spaces, and events; theoretical probability for equally likely outcomes; and probability distributions.
Draft - 13Complements and Mutually Exclusive Events
The complement rule, mutually exclusive events, the additive principle for P(A or B), and Venn diagrams.
Draft - 14Independent and Dependent Events
When one event affects another, multiplying probabilities for "and", drawing with and without replacement, and tree diagrams.
Draft - 15Conditional Probability
The probability of B given A, from formulas, two-way tables, and tree diagrams — and why the order matters.
Draft - 16Discrete Random Variables
Random variables, probability distributions in tables, probability histograms, and the uniform distribution.
Draft - 17Expected Value
The long-run average of a random variable — computing E(X), fair games, raffles, and the link to the weighted mean.
Draft - 18Binomial Distribution
Counting successes in independent trials — the binomial conditions and formula, tables and histograms, E(X) = np, and how the shape changes with n.
Draft - 19The Normal Distribution
The bell-shaped normal model, its properties, the notation X ~ N(μ, σ²), and estimating probabilities with the 68–95–99.7 rule.
Draft - 20Z-Scores and the Standard Normal Distribution
Standardizing values with z-scores, finding normal probabilities and percentiles with a table or technology, and working backwards with the inverse normal.
Draft - 21Inverse Normal with Unknown Mean or SD
Standardizing normal variables and using inverse normal z-values to find an unknown mean, an unknown standard deviation, or both from given probabilities.
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Unit 8: Statistics and Probability HL
- 1Bayes' Theorem
Reversing a conditional probability with Bayes' theorem, for two or three events, using tree diagrams and the formula — medical tests, machines in a factory, and more.
Draft - 2Variance of a Discrete Random Variable
Variance and standard deviation of a discrete random variable using Var(X) = E(X²) − [E(X)]², and how E(aX + b) and Var(aX + b) change under a linear transformation.
Draft - 3Probability Density Functions
Continuous random variables and their probability density functions — finding k, probabilities as integrals, the mode, the median, the mean and variance, and piecewise pdfs.
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Unit 9: Calculus
- 1Introduction to Limits
The idea of a limit — what a function approaches near a point, limit notation, one-sided limits, and when a limit does not exist.
Draft - 2The Definition of the Derivative
The derivative as a limit, derivative notation, tangent lines, and estimating derivatives from tables and graphs.
Draft - 3First Derivative Test
Finding where a function increases and decreases from the sign of f', building sign charts, and using the first derivative test to classify relative extrema.
Draft - 4The Power Rule and Basic Derivative Rules
Differentiating powers of x (including negative and fractional exponents), constants, sums, differences, and constant multiples.
Draft - 5Tangents and Normals
Equations of the tangent and normal to a curve at a point, finding where the tangent has a given gradient, tangents through an outside point, and using technology for gradients.
Draft - 6Antiderivatives and Indefinite Integrals
Finding antiderivatives with the basic rules (power, 1/x, exponential, trig, inverse trig), the + C, and solving initial-value problems.
Draft - 7Evaluating Definite Integrals with the FTC
The Fundamental Theorem of Calculus, part 2 — evaluating a definite integral as F(b) − F(a) using an antiderivative, and the net change theorem.
Draft - 8Derivatives of Sine, Cosine, eˣ, and ln x
The derivatives of sin x, cos x, e^x, and ln x, why radians matter, and using them with the basic derivative rules.
Draft - 9The Chain Rule
Differentiating composite functions with f'(g(x)) · g'(x) and dy/du · du/dx — powers, trig, exponential, and log compositions, tables of values, and choosing which rules to combine.
Draft - 10The Product Rule
Differentiating a product of two (or three) functions, including from tables of values and in context.
Draft - 11The Quotient Rule and Trig Derivatives
Differentiating quotients, and using the quotient rule to find the derivatives of tan x, cot x, sec x, and csc x.
Draft - 12Higher-Order Derivatives
Second, third, and higher derivatives — notation like f''(x), d²y/dx², and y'', what the second derivative means, and second derivatives of implicit relations.
Draft - 13Connecting f, f′, and f″
Reading the features of f from graphs of f' and f'', and sketching f and f' from each other — increasing, extrema, concavity, and inflection points.
Draft - 14Critical Points and Extrema
Absolute and relative maximums and minimums, the Extreme Value Theorem, and finding critical points where f'(x) = 0 or f'(x) does not exist.
Draft - 15Concavity and the Second Derivative Test
Concave up and concave down from the sign of f'', points of inflection, and the second derivative test for relative extrema — including when it's inconclusive.
Draft - 16Optimization
Setting up and solving optimization problems with derivatives — variables, constraint, objective function, domain — and justifying the answer for fencing, boxes, cans, cost, and distance.
Draft - 17Straight-Line Motion with Derivatives
Position, velocity, and acceleration of a particle on a line — speed, direction, changing direction, and speeding up versus slowing down.
Draft - 18Motion Along a Line with Integrals
Going backwards from velocity to position — initial conditions, displacement versus total distance, and velocity from acceleration.
Draft - 19Integration by Substitution
Undoing the chain rule with u-substitution, for indefinite and definite integrals (including changing the limits), and choosing an integration technique.
Draft - 20Area Between Curves
Finding the area between two curves with vertical slices — top minus bottom, finding intersection points, and splitting the integral when the curves cross more than twice.
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Unit 10: Calculus HL
- 1Continuity
The three conditions for continuity at a point, continuity on an interval, and where polynomial, rational, root, exponential, logarithmic, and trig functions are continuous.
Draft - 2Differentiability and Continuity
Why differentiable functions are continuous, where derivatives fail to exist (corners, cusps, vertical tangents, discontinuities), and checking piecewise functions.
Draft - 3Limits at Infinity
End behaviour and horizontal asymptotes — limits of rational functions by comparing degrees, exponential and logarithmic functions, square roots, and relative growth rates.
Draft - 4L'Hospital's Rule
Using derivatives to evaluate limits of the indeterminate forms 0/0 and ∞/∞ — checking the form first, and knowing when not to use the rule.
Draft - 5Implicit Differentiation
Finding dy/dx for curves like circles where y isn't written as a function of x — differentiating both sides, solving for dy/dx, and finding tangent lines.
Draft - 6Related Rates Problems
A step-by-step strategy for multi-step related rates — sliding ladders, filling cones, moving shadows, and approaching cars.
Draft - 7Derivatives of Inverse Trig Functions
Derivatives of arcsin x, arccos x, and arctan x (also written sin⁻¹ x, cos⁻¹ x, tan⁻¹ x), where they come from, and using them with the chain rule.
Draft - 8Derivatives of Exponential Functions
Why the derivative of a^x is a constant times a^x, how that leads to the number e, the rule d/dx a^x = a^x ln a, the chain rule with e^(kx) and A·b^(kt), and rates of growth and decay.
Draft - 9Integrating with Long Division and Completing the Square
Integrating rational functions by first using polynomial long division, or by completing the square to get an arctangent.
Draft - 10Integration by Parts
Undoing the product rule — choosing u and dv (LIATE), repeated parts, the tabular method, definite integrals by parts, and integrals that return to themselves.
Draft - 11Area Between Curves Using Horizontal Slices
Integrating with respect to y — right minus left, rewriting curves as x in terms of y, and choosing the slice direction that makes the problem easier.
Draft - 12Volumes of Revolution — the Disc Method
Spinning a region around a line to make a solid — discs with radius R, revolving about the x-axis, the y-axis, and other horizontal or vertical lines.
Draft - 13Volumes of Revolution — the Washer Method
Solids with a hole — washers with outer radius R and inner radius r, revolving about the x-axis, the y-axis, and other horizontal or vertical lines.
Draft - 14Introduction to Differential Equations
Writing differential equations from verbal descriptions, verifying solutions by substitution, and telling general solutions from particular ones.
Draft - 15Euler's Method
Approximating a solution of a differential equation step by step with tangent lines, organizing the work in a table, and deciding whether the estimate is too high or too low.
Draft - 16Separation of Variables
Solving separable differential equations, finding particular solutions from initial conditions, and stating where a solution is defined.
Draft - 17Homogeneous Differential Equations
Solving first-order differential equations of the form dy/dx = f(y/x) with the substitution y = vx, then separating, integrating, and back-substituting.
Draft - 18The Integrating Factor
Solving linear first-order differential equations y′ + P(x)y = Q(x) by multiplying by the integrating factor e^∫P dx, with initial conditions and rearranging into standard form.
Draft - 19Taylor and Maclaurin Series
Taylor series as never-ending Taylor polynomials, the four series you're expected to know (e^x, sin x, cos x, 1/(1 − x)), writing the general term, and building a series from derivatives.
Draft - 20Representing Functions as Power Series
Building new power series from known ones — substitution, multiplying by x, and differentiating or integrating term by term (radius unchanged, endpoints may change) — including the series for ln(1 + x) and arctan x.
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