Mathematics · 8. Calculus & Differential Equations
Derivatives, integrals, ODEs, Taylor series and matrix decompositions
Comprehensive calculus: tangent limits, Riemann integrals, canonical first/second-order ODE methods, Taylor/Maclaurin series expansions, and modern matrix factorizations (LU, Cholesky, QR, SVD).
Driving question
How do calculus, Taylor series and matrix factorizations link infinitesimal changes to big data modeling and AI?
Curriculum coverage
Vietnam Mathematics 11–12; Cambridge AS/A-Level differentiation, integration, differential equations, series and matrix decompositions.
Model scope and limitations
Numerical visuals suggest limits and tangent slope fields; exact analytical solutions require correct classification, checking equilibrium solutions (y = 0) and initial values.
Learning objectives
- Interpret derivative as instantaneous rate of change, tangent gradient and Lagrange Mean Value Theorem.
- Distinguish a definite integral from non-negative geometric area.
- Expand functions into Taylor and Maclaurin series with Lagrange remainder error bounds.
- Solve the 7 first-order ODE types: separable, homogeneous, linear (integrating factor μ), Bernoulli, exact (Euler), Riccati, Lagrange and Clairaut.
- Apply reduction of order, undetermined coefficients and universal Lagrange variation via Wronskian for second-order ODEs and Euler-Cauchy.
- Master principles and applications of modern matrix factorizations: LU, Cholesky, QR and Singular Value Decomposition SVD.
Prerequisite knowledge
- Functions, gradients and basic limits.
- Antiderivatives and integration techniques (substitution, by parts).
- Matrix algebra and vector spaces.
Core knowledge
Derivative, Tangent & Lagrange MVT
f′(x) is the limit of secant slopes as h → 0. At differentiable points, f′(x₀) is tangent slope. Lagrange MVT guarantees c in (a, b) with f′(c) = (f(b) − f(a))/(b − a).
Definite integral, Riemann sums & Signed area
Definite integrals are limits of Riemann sums. Regions below the axis contribute negatively; total geometric area requires splitting at roots and taking absolute values.
Taylor series, Maclaurin expansions & Lagrange remainder
Smooth functions expand into Taylor series f(x) = ∑ [f^(k)(x₀)/k!]·(x − x₀)^k + R_n(x) with Lagrange remainder bound. At x₀ = 0, standard Maclaurin series include e^x, sin x, cos x, and 1/(1−x).
Separable & Homogeneous first-order ODEs
1. Separable: dy/h(y) = g(x)dx. 2. Homogeneous: y′ = f(y/x) uses u = y/x ⇒ y′ = u + x·du/dx to get du/(f(u) − u) = dx/x. Non-homogeneous linear fractions translate coordinates to eliminate constants.
Linear first-order ODEs & Integrating factor
Standard form: y′ + P(x)y = Q(x). Integrating factor μ(x) = e^(∫P dx). Exact solution: y(x) = e^(−∫P dx)·[∫Q·e^(∫P dx) dx + C].
Bernoulli & Riccati ODEs
1. Bernoulli: y′ + Py = Qyⁿ linearised via z = y^(1−n) into z′ + (1−n)P z = (1−n)Q. 2. Riccati: y′ = Py² + Qy + R with known solution y₁ uses y = y₁ + 1/z to yield a first-order linear ODE.
Exact ODEs & Integrating factors
P dx + Q dy = 0 is exact iff ∂P/∂y = ∂Q/∂x with potential U(x, y) = C. Non-exact ODEs use integrating factor μ(x) = e^(∫(Py−Qx)/Q dx) or μ(y) = e^(∫(Qx−Py)/P dy).
Second-order linear ODEs & Lagrange variation
y″ + ay′ + by = f(x) uses characteristic roots r² + ar + b = 0. General solution y = y_th + y_r. Lagrange variation uses Wronskian W = y₁y₂′ − y₁′y₂ ≠ 0 with u₁′ = −y₂f/W and u₂′ = y₁f/W.
Euler-Cauchy ODEs, RK4 & PDEs
1. Euler-Cauchy: ax²y″ + bxy′ + cy = f(x) uses x = e^t. 2. Runge-Kutta 4 (RK4): order O(h⁴) via 4 slope evaluations. 3. PDEs: 1D Wave equation (D′Alembert), 1D Heat equation (Fourier) and 2D Laplace equation.
Modern matrix factorizations: LU, Cholesky, QR & SVD
1. LU: A = LU solves Ax = b in O(n²) after O(n³/3) factorization. 2. Cholesky: A = L·Lᵀ for symmetric positive-definite matrices (2x faster than LU). 3. QR: A = QR (Q orthogonal, R upper-triangular) solves least-squares problems. 4. SVD: A = U·Σ·Vᵀ with orthogonal U, V and singular values in Σ; fundamental to image compression, PCA dimensionality reduction and AI recommender systems.
Worked example
Example 1 (Linear): y′ + 2y = 4, y(0) = 3 gives integrating factor μ = e^(2x) ⇒ y = 2 + e^(−2x). Example 2 (Maclaurin): cos(2x) = 1 − 2x² + (2/3)x⁴ − ... Example 3 (Cholesky): Symmetric positive-definite A = [[4, 2], [2, 10]] factorizes into L = [[2, 0], [1, 3]] with L·Lᵀ = A.
Misconceptions to avoid
- f′(a)=0 identifies a stationary point but does not by itself classify it as extremum.
- A definite integral is signed area; total geometric area requires splitting at roots and taking magnitudes.
- Forgetting the constant C before dividing by integrating factor μ(x) produces invalid general solutions.
- The Maclaurin series of 1/(1−x) converges only for |x| < 1; evaluating at x = 2 produces invalid divergent sums.
- Cholesky factorization A = L·Lᵀ requires the matrix to be both symmetric and positive definite (all eigenvalues λ > 0).
Virtual experiment procedure: Calculus and differential equations explorer
- Identify the exact problem type (calculus, Taylor series, ODE, or matrix factorization).
- Compute integrating factor μ(x), solve characteristic equation, or compute higher derivatives at x₀.
- Apply integration, Lagrange variation via Wronskian, matrix factorization or RK4, then verify initial conditions.
Safety and cautions
- Too large a step size h in numerical Euler/RK4 causes instability; verify against analytical formulas or reduce h.
Evidence to collect
- A structured ODE solution identifying equation type, integrating factor/characteristic roots, Wronskian, general solution and initial value check.
Diagnostic check
1. What is the geometric meaning of f′(a)?
Show answer and explanation
Correct answer: Tangent gradient at x = a
At a differentiable point, the derivative is the tangent gradient.
2. What is the integrating factor of linear ODE y′ + P(x)y = Q(x)?
Show answer and explanation
Correct answer: μ(x) = e^(∫P(x) dx)
Multiplying by e^(∫P(x) dx) collapses the left side into [y · e^(∫P dx)]′.
3. What is the 3rd-order Maclaurin polynomial of f(x) = e^x?
Show answer and explanation
Correct answer: 1 + x + x²/2 + x³/6
e^x = 1 + x/1! + x²/2! + x³/3! = 1 + x + x²/2 + x³/6.
4. In Singular Value Decomposition SVD A = U·Σ·Vᵀ, what property do matrices U and V possess?
Show answer and explanation
Correct answer: Orthogonal matrices (UᵀU = I and VᵀV = I)
In SVD, U and V are orthogonal matrices representing rotations/reflections, while Σ is a diagonal matrix containing non-negative singular values.
Application practice
1. For f(x) = x², what is f′(3)?
Show answer and explanation
Correct answer: 6
f′(x) = 2x, so f′(3) = 6.
2. What is the general solution of y″ − 4y = 0?
Show answer and explanation
Correct answer: y = C₁·e^(2x) + C₂·e^(−2x)
Characteristic equation r² − 4 = 0 gives r = ±2, yielding general solution y = C₁·e^(2x) + C₂·e^(−2x).
3. What is the coefficient of x² in the Maclaurin expansion of f(x) = cos(2x)?
Show answer and explanation
Correct answer: −2
cos(u) = 1 − u²/2 + ... With u = 2x, cos(2x) = 1 − (2x)²/2 = 1 − 2x². The coefficient of x² is −2.
4. What mandatory condition must a real square matrix A satisfy for Cholesky factorization A = L·Lᵀ?
Show answer and explanation
Correct answer: Symmetric and positive definite
Cholesky factorization exists uniquely if and only if A is symmetric (A = Aᵀ) and positive definite (xᵀAx > 0 for all x ≠ 0).
5. In Lagrange variation for y″ + y = f(x), what is the Wronskian W(cos x, sin x)?
Show answer and explanation
Correct answer: 1
W = (cos x)(cos x) − (−sin x)(sin x) = cos²x + sin²x = 1 ≠ 0.
Academic references
- Cambridge IGCSE Mathematics 0580 — official syllabus and resources
- Cambridge AS & A Level Mathematics 9709 — official syllabus
- OpenStax Precalculus 2e — functions, geometry and probability
This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.
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