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Higher Mathematics Knowledge Framework

2026-10-29   0  Report

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Limits

Core concepts

Limit of a sequence: as n grows without bound, the sequence {aₙ} approaches a constant A, written

Limit of a function:(or ) the function values approach a constant A, written

Left-hand and right-hand limits:、 approach from the left and right respectively

Infinitesimals and infinities: a variable with limit 0 is an infinitesimal; one whose absolute value grows without bound is an infinity

Comparing infinitesimals: higher-order, same-order and equivalent infinitesimals; equivalent substitution simplifies computation

Key theorems

Necessary and sufficient condition for a limit to exist: both one-sided limits exist and are equal (functions); a monotone bounded sequence always has a limit

Squeeze theorem: if and then

Two important limits:；

Arithmetic rules: the limit of a sum, difference, product or quotient equals the sum, difference, product or quotient of the limits (denominator limit ≠ 0)

Limit rule for composite functions: substitution is valid when the inner limit exists and the outer function is continuous

Continuity and discontinuity: a function is continuous at a point iff; discontinuities split into first kind (removable, jump) and second kind (infinite, oscillatory)

Properties of continuous functions on a closed interval: boundedness, extreme value theorem, intermediate value theorem, zero-point theorem

Review points

Common methods for limits: direct substitution, equivalent infinitesimal substitution, L'Hôpital's rule, squeeze, monotone bounded, Taylor expansion

Pitfall: equivalent infinitesimals may be substituted only in products/quotients, not in sums/differences

Derivatives

Core concepts

Definition:, i.e. the instantaneous rate of change at that point

Geometric meaning: of the curve at the point — the slope of the tangent

Differentiability and continuity: differentiability implies continuity, but continuity does not imply differentiability

Higher-order derivatives: derivatives of second order and above; denotes the rate of change of the rate of change

Differentiating implicit and parametric functions: differentiate both sides w.r.t. x for implicit functions; for parametric equations

Key theorems

Basic derivative formulas:、、、 etc.

Arithmetic differentiation rules:、、

Differentiating composite functions (chain rule):

Derivative of an inverse function:

Applications

Monotonicity test: means increasing, means decreasing

Extremum test: first-derivative sign change, second-derivative test ( gives a maximum)

Finding extrema: compare function values at stationary points, non-differentiable points and endpoints

Concavity and inflection points: are determined by the sign of

Asymptotes: finding horizontal, vertical and oblique asymptotes

L'Hôpital's rule: and type indeterminate forms

Differentials

Core concepts

Definition:, then , the differential is the linear principal part of the increment

Differentiability: for a function of one variable, having a differential and having a derivative are equivalent

Geometric meaning of the differential: the increment of the tangent's ordinate

Key theorems and rules

Invariance of differential form: whether u is the independent or an intermediate variable, holds

Arithmetic rules for differentials:、、

First-order invariance of the differential form is used in approximation and integration by substitution

Applications

Approximation:

Error estimation: estimate function-value error from argument error

Relation to the derivative: the derivative is the differential quotient , while the differential focuses on linear approximation of increments

Integrals

Indefinite integrals

Concept:, then , the inverse operation of differentiation

Basic integral formulas: standard tables for power, exponential, trigonometric and inverse trigonometric functions

Integration by substitution: first kind (adjusting the differential), second kind (change of variable, incl. trigonometric)

Integration by parts:, suited to products as integrands

Integrating rational functions: partial fraction decomposition, then integrate term by term

Definite integrals

Concept: the limit of Riemann sums

Geometric meaning: the signed area of a curvilinear trapezoid

Newton–Leibniz formula:, turning definite-integral computation into evaluating an antiderivative

Properties of definite integrals: linearity, additivity over intervals, sign preservation, estimation theorem, mean value theorem for integrals

Integral with variable upper limit:，

Key theorems and methods

Mean value theorem for integrals:，

Fundamental theorem of calculus: the derivative of an integral with variable upper limit equals the integrand — the basis of the Newton–Leibniz formula

Substitution and integration by parts for definite integrals: change the limits along with the variable

Applications

Area: plane-figure area (Cartesian, polar)

Volume: solids of revolution (disk method, shell method)

Arc length and area of surfaces of revolution

Physics applications: work of a variable force, fluid pressure, centroid

Differential equations

Core concepts

Definition: an equation containing an unknown function and its derivatives; the order is that of the highest derivative

General and particular solutions: the general solution contains as many independent arbitrary constants as the order; initial conditions determine the particular solution

First-order differential equations

Separable equations:, separate and integrate both sides

Homogeneous equations:, let reduces it to a separable equation

First-order linear equations:, general solution

Bernoulli equations:, let transforms it into a first-order linear equation

Exact differential equations: and a potential function exists

Higher-order differential equations

Reducible types: integrate successively; set ； for , taking y as the independent variable

Second-order homogeneous linear equations with constant coefficients:, the characteristic equation has roots that determine the form of the general solution (two distinct real roots, repeated root, conjugate complex roots)

Second-order non-homogeneous linear equations with constant coefficients: general solution = homogeneous solution + particular solution; guess the particular form by the type of the forcing term (、 ) to set up the undetermined-coefficient form

Key theorems

Structure theorem: the general solution of a non-homogeneous equation equals the general solution of the associated homogeneous equation plus one particular solution

Superposition principle: when the forcing term is a sum of several terms, solve for each particular solution and add them

Existence and uniqueness theorem: under continuity conditions, the initial value problem has a unique solution

Series

Series with constant terms

Concept: converges if the sequence of partial sums has a limit, otherwise diverges

Necessary condition for convergence:(otherwise it diverges)

Geometric series:， converges, diverges

p-series:， converges, diverges

Key convergence tests

Comparison test for positive-term series: compare with a known convergent/divergent series (including the limit form)

Ratio test (d'Alembert):， means convergence, means divergence

Root test (Cauchy):, with the same decision rule as the ratio test

Leibniz test for alternating series: converges when it decreases monotonically to 0

Absolute and conditional convergence: absolute convergence implies convergence; conditional convergence must be tested separately

Power series

Concept:, the interval of convergence is centred on the radius of convergence R

Finding the radius of convergence:，(or obtained by the root test)

Domain of convergence: check the endpoints separately on top of the interval

Properties of power series: term-by-term differentiation and integration within the interval, with unchanged radius

Function expansions: Taylor and Maclaurin series, common expansions (、、、、）

Fourier series

Concept: expand a periodic function into a trigonometric series, with coefficients given by integral formulas

Dirichlet convergence theorem: under the conditions, the Fourier series converges to the function (at discontinuities, to the mean of the one-sided limits)

Fourier series of odd/even functions: odd functions have only sine terms, even functions only cosine terms