Grade 12 math

How to add vectors end to end and by components, why the length of a sum is usually less than the sum of the lengths, how subtraction works, and what scaling does.

Tell a circle, ellipse, parabola and hyperbola apart from Ax² + Cy² + Dx + Ey + F = 0, then complete the square to find the center, radius, vertices and axes.

Raise a complex number to a power by raising its length and multiplying its angle, then reverse the rule to find all n of its nth roots, evenly spaced on a circle.

Derive sin 2θ, cos 2θ and tan 2θ from the sum formulas, reverse them into half-angle and power-reducing forms, and use them for exact values and equations.

Why an ellipse is every point whose distances to two foci add to a constant, how squaring twice turns that into x²/a² + y²/b² = 1, and how the hyperbola follows.

Why sine, cosine and tangent need a restricted domain before they can be undone, what arcsin, arccos and arctan return, and how to solve trig equations in context.

Prove a statement for every positive integer with two steps: check it for n = 1, then show that whenever it holds for k it holds for k + 1.

How matrices store data, how to add, scale and multiply them, why the order of multiplication matters, and how the determinant decides whether an inverse exists.

How a 2×2 matrix moves every vector in the plane, why its columns show where the whole plane goes, and why the determinant measures how it scales area.

Describe a curve by giving x and y as functions of a parameter t: plotting and direction, eliminating the parameter, line segments and projectile motion.

Locate points by distance and angle, convert between polar and rectangular coordinates, and graph polar equations: circles, cardioids and roses.

How to write a system of linear equations as a single matrix equation AX = B, why multiplying by the inverse solves it, and what a zero determinant says about the system.

Why cos(α − β) = cos α cos β + sin α sin β, proved by rotating a chord of the unit circle, how the other formulas follow, and how to find exact values like cos 15°.

How to plot complex numbers, why rectangular and polar form name the same point, why multiplying multiplies lengths and adds angles, and how to find distances.

Multiply two vectors into a number: u · v = u₁v₁ + u₂v₂ equals |u||v|cos θ, which measures the angle between them, tests for right angles and gives projections.

What makes a quantity a vector, how to draw and name one, why its components come from subtracting the start from the end, and how to use vectors for velocity.

Prove that two trigonometric expressions agree for every angle: work one side at a time, rewrite in sines and cosines, and trade with the Pythagorean identities.

Find the absolute maximum and minimum of a function: the closed interval method with critical points and endpoints, and what to do on open or unbounded intervals.

The alternating series test, the error bound for stopping early, and the difference between absolute and conditional convergence.

Find antiderivatives: the family F(x) + C, the basic integration rules as derivative rules run backward, ln|x|, and initial value problems.

Find the length of a curve with an integral, L = ∫√(1 + f′(x)²) dx, built from the Pythagorean theorem on tiny pieces of the curve.

Find the area between two curves: top minus bottom with vertical slices, right minus left with horizontal slices, and splitting where the curves cross.

Find areas bounded by polar curves with A = ½∫r² dθ: one curve, the region between two curves, and finding the limits where curves meet.

Find the average value of a function with an integral, see it as the height of an equal-area rectangle, and use the Mean Value Theorem for integrals.

Differentiate without limits: the constant, power, constant-multiple and sum rules, the derivatives of sin x, cos x, e^x and ln x, and why each rule holds.

Judge a series by comparing it with one you know: the direct comparison test, the limit comparison test, and how to choose a benchmark series.

Use f″ to see how a graph bends: concave up and down, inflection points where the bend changes, and the second derivative test for maxima and minima.

What it means for a function to be continuous, the kinds of discontinuity, which functions are continuous, and how the Intermediate Value Theorem proves roots exist.

Sketch a graph from its derivatives: intercepts, asymptotes, increasing and decreasing, extrema, concavity and inflection points — and read f from a graph of f′.

Find volumes of revolution with the shell method, V = 2π∫(radius)(height) dx, and choose between shells and washers for a region and an axis.

The derivative of an inverse function is the reciprocal of a slope: use it to find derivatives of ln x, logarithms in any base, arcsin, arccos and arctan.

What a differential equation is, how to check a solution, general and particular solutions, and how a slope field shows every solution at once.

Find the volume of a solid of revolution with disks and washers: V = π∫R² dx or π∫(R² − r²) dx, around the axes and around other lines.

Approximate the solution of a differential equation step by step: follow the slope for a short step, find the new slope, repeat, and judge the error.

The link between derivatives and integrals: accumulation functions, the derivative of an integral (Part 1), and evaluating integrals with antiderivatives (Part 2).

Find dy/dx when y is not solved for: differentiate both sides, attach dy/dx to each y term by the chain rule, solve, and find tangent lines and second derivatives.

Integrals over infinite intervals and of unbounded functions, defined as limits: when they converge, when they diverge, and the p-integrals.

Use the sign of f′ to find where a function rises and falls, locate critical points, and classify local maxima and minima with the first derivative test.

Decide whether a series converges by comparing it with an integral: the integral test, the harmonic series, p-series, and bounding the remainder.

Integrate products with ∫u dv = uv − ∫v du, the product rule run backward: choosing u and dv, using parts twice, and definite integrals.

Evaluate limits of the forms 0/0 and ∞/∞ by differentiating top and bottom, and rewrite the forms 0·∞, ∞ − ∞, 1^∞, 0⁰ and ∞⁰ so the rule applies.

Evaluate limits exactly: the limit laws, direct substitution, factoring and conjugates for 0/0 forms, the squeeze theorem, and why sin x over x approaches 1.

Find limits as x grows without bound and limits that blow up: horizontal asymptotes from leading terms, vertical asymptotes from infinite limits, and growth rates.

Estimate values with the tangent line: local linearity, the linearization L(x), whether an estimate is too high or too low, and differentials for small changes.

The logistic equation dP/dt = kP(1 − P/L): growth that levels off at a carrying capacity, its S-shaped solution, and where growth is fastest.

Rewrite an integrand before integrating: divide when a fraction's numerator is too big, and complete the square to reach the arctangent.

Rolle's theorem, the Mean Value Theorem and the Extreme Value Theorem: what each guarantees, why the hypotheses matter, and what follows from them.

Integrate rates in context: position from velocity, displacement versus total distance, and amounts that change with a rate in and a rate out.

Read derivatives as rates with units, and use them for motion: velocity and acceleration from position, when an object speeds up or slows down, and total distance.

Solve optimization problems: model the quantity to maximize or minimize, use the constraint to reduce it to one variable, find critical points, and justify the answer.

Slopes, tangent lines, second derivatives and arc length for curves given by x = f(t) and y = g(t), where dy/dx is dy/dt divided by dx/dt.

Integrate rational functions by splitting them into simpler fractions: distinct linear factors, finding the constants, repeated factors, and dividing first.

Find slopes of polar curves r = f(θ) by treating them as parametric curves: dy/dx in polar form, tangent lines, and what dr/dθ tells you.

Series with a variable, the sum of cₙ(x − a)ⁿ: where they converge, finding the radius with the ratio test, and testing the two endpoints.

Differentiate products and quotients: the product rule, the quotient rule, why a product's derivative has two terms, and the derivatives of tan, cot, sec and csc.

Test series with factorials and powers by comparing each term with the one before: the ratio test, the root test, and what an inconclusive result means.

Find how fast one quantity changes from how fast another does: link them with an equation, differentiate with respect to time, and substitute values only at the end.

Approximate accumulated change and area: left, right and midpoint Riemann sums, trapezoidal sums, sums from tables, sigma notation, and over- and underestimates.

Solve dy/dx = g(x)h(y) by separating variables, find particular solutions, and model exponential growth, decay, half-life and Newton's law of cooling.

What it means for an infinite sum to converge: sequences and their limits, partial sums, geometric series, and the nth-term test for divergence.

Represent functions as power series: the Taylor series formula, the standard Maclaurin series, and new series by substitution, differentiation and integration.

Approximate a function near a point by a polynomial that matches its derivatives there, and bound the error with the Lagrange error bound.

Differentiate a function inside a function: the chain rule, why rates of change multiply, common patterns like powers and exponentials of an inner function, and a^x.

The definite integral as the limit of Riemann sums: signed area, the properties of integrals, evaluating integrals with geometry, and the difference from total area.

The derivative as a limit of difference quotients: from average to instantaneous rate of change, tangent lines, the derivative as a function, and where it fails.

Undo the chain rule: choose u, rewrite the integral in u and du, integrate, and substitute back, with constant adjustments and new limits for definite integrals.

Describe motion in the plane with a position vector: velocity, speed and acceleration as derivatives, and displacement and distance as integrals.

Find volumes by slicing: integrate the area of a cross section, for solids whose slices are squares, triangles, rectangles or semicircles on a base region.

The idea of a limit: what value f(x) approaches as x approaches a, estimated from tables and read from graphs, with one-sided limits and the ways a limit fails to exist.