用Python学微积分 (it-ebooks)（Z-Library）

# 【One-Line Pitch】 A hands-on introduction to single-variable calculus where every concept — from functions and limits to derivatives and optimization — is explained through executable Python code using NumPy, Matplotlib, and SymPy. Ideal for self-learners who want to *see* math work, or programmers who never quite grasped calculus from abstract formulas alone. --- # 【Book Arc】 - **Opening (~0%–11%)**: The book opens by framing functions as machines with domains and ranges, then walks through polynomial, exponential, logarithmic, and trigonometric functions — each defined and plotted in Python. It also introduces function composition, inverse functions, and higher-order functions (functions that take functions as arguments), establishing the "functions as objects" mindset that carries through the entire book. - **Early (~11%–33%)**: This stage builds the theoretical backbone. Euler's formula is derived by plugging imaginary numbers into the exponential series, then verified numerically in Python. The Taylor series is introduced as the general mechanism behind those "magic" polynomial forms, with SymPy used to compute partial sums and visualize how approximation accuracy improves as more terms are added and as the expansion point moves. - **Middle (~33%–56%)**: Limits are defined rigorously (epsilon-delta), then immediately demonstrated through plotting and SymPy's `.limit()`. Continuity, l'Hôpital's rule, and Big-O notation follow — with Big-O shown as a tool for both growth-rate comparison and error analysis of Taylor approximations. The connection between Taylor series and l'Hôpital's rule is made explicit. - **Late (~56%–78%)**: Derivatives are presented through three equivalent definitions, including the "linear coefficient" view that makes linear approximation natural. Newton's method for root-finding is developed from the linear approximation idea, with a step-by-step visual walkthrough of how tangent lines converge to the root. - **Ending (~78%–100%)**: The final stage applies derivatives to optimization — critical points, the second-derivative test (explained via Taylor series rather than memorized), and a worked box-volume maximization problem. Linear regression is framed as an optimization problem, and the book closes with indefinite integrals and ordinary differential equations, showing integration as the inverse of differentiation. --- # 【Key Takeaways】 - **Functions are computational objects, not just formulas** (Opening): The book treats functions as machines you can define, compose, shift, and pass around in Python — including higher-order functions that return new functions. This reframing makes later concepts like derivatives-as-operators feel natural. - **Euler's formula is verified, not just stated** (Early): By substituting imaginary numbers into the exponential series, the book derives \( e^{ix} = \cos x + i\sin x \) and confirms it numerically in both NumPy and SymPy. This demystifies a formula that most textbooks present as magic. - **Taylor series is the master key** (Early): Nearly every "special" function's polynomial form comes from expanding at \( x=0 \). The book shows how to compute partial sums in SymPy and demonstrates — with colored plots — that more terms mean better approximation, especially near the expansion point. - **Big-O notation is a practical error-analysis tool** (Middle): Beyond computer science's "algorithm complexity" usage, Big-O describes truncation error in Taylor approximations. The book shows how to compute orders with SymPy and even do algebra with Big-O terms directly. - **Derivatives have three equivalent definitions** (Late): The geometric tangent-slope view, the limit definition, and the "coefficient of the first-order change" view. The third definition is what makes linear approximation and Newton's method almost obvious. - **Newton's method is linear approximation applied repeatedly** (Late): Starting from an initial guess, each iteration replaces the function with its tangent line and finds where that line crosses zero. The book walks through the iteration visually and in code, then shows SymPy's `solve()` as the "cheat" alternative. - **The second-derivative test makes sense via Taylor series** (Ending): At a critical point, the function behaves like a quadratic whose curvature is determined by \( f''(x) \). This explains *why* the test works — a rare payoff for readers who memorized it in high school. - **Optimization unifies calculus and data fitting** (Ending): The box-volume problem and linear regression are both solved by finding critical points of a function. The regression example even previews how this extends to multivariable calculus. --- # 【Reading Tips】 1. **Run the code as you read** — this book's value is in the execution. The plots (tangent lines zooming in, Taylor approximations converging, Newton's method iterating) are where the intuition lives. Set up a Jupyter notebook and type out the examples rather than copying. 2. **Skim the epsilon-delta limit section** (~33%–44%) if you're not a math purist. The formal definition is presented, but the real takeaway is the SymPy `.limit()` call and the continuity concept. Come back to the proof only if you want the rigor. 3. **Deep-read the Taylor series and Big-O chapters** (~11%–56%). These two ideas recur throughout the rest of the book — l'Hôpital's rule, error analysis, linear approximation, and the second-derivative test all build on them. Understanding these early pays compound interest. 4. **Don't skip the "boring" function review** (Opening). The higher-order function examples (`horizontal_shift`, `composite`) introduce a programming pattern that the book reuses for derivatives and integrals later. It's short but foundational. 5. **The ending is a preview, not a conclusion** — the ODE and integration sections are brief and assume you'll review integration techniques elsewhere. Treat them as motivation for what's next, not as a complete treatment. --- # 【Coverage Limits】 This guide covers the single-variable calculus portion (Part 1) of the book. The excerpts do not include the multivariable calculus content (Part 2), nor do they cover integration techniques in depth — the book explicitly leaves substitution and integration-by-parts to the reader's prior knowledge. --- #