Differential Calculus
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This is the first installment in a comprehensive treatment of core mathematics courses typical in undergraduate mathematics, physics, and natural science programs. Future volumes—covering multivariable calculus, linear algebra, differential equations, and more—will be published when ready.
This resource develops single-variable differential calculus in the spirit of how a “real” mathematics course treats these concepts: through definitions, theorems, and proofs. While accessible to motivated beginners, the approach is mathematically honest, incorporating foundational ideas—such as linear maps and differentials (introduced in § Section 8.1)—to clarify the underlying structure and build habits for further mathematical study.
Contents
Differential Calculus: Limits, continuity, derivatives, linear approximations, differentiation rules, and applications such as optimization and related rates.
Sequences: Definitions and convergence, used to motivate and support the later development of limits and continuity.
Linear Foundations: A minimal introduction to vector spaces, linear maps, and the differential as a linear transformation to clarify the structural meaning of differentiation.
Structure and Approach
Each chapter follows a consistent progression:
- Definitions and intuition — Concepts are motivated geometrically or through simple linear-algebraic ideas
- Theorems and examples — Formal statements with representative examples
- Proofs — Collapsible proofs for optional deeper study
Visualizations and diagrams appear throughout, and linear structures are highlighted where they naturally arise, especially through the differential as a linear map.
Why We Begin With Sequences
Sequences provide a concrete and intuitive setting for understanding the idea of approaching a value. Studying convergence first lets us develop limit intuition in a setting that avoids the technicalities of full \varepsilon–\delta definitions while still preparing the ground for them.
Many standard Calculus 1 courses introduce limits heuristically before giving precise definitions. Here, we take a more conceptual route: sequences offer a simpler but genuinely rigorous first encounter with limiting behavior. This approach frames limits and derivatives in a way that is closer in spirit to real analysis—formal yet accessible—without requiring topology or the construction of \mathbb{R}.
Note for the Reader. If you prefer a lighter, more traditional start to calculus, you may skip the sequence material on a first pass and begin at § Section 4.1. The sequence chapter enriches understanding, but is not strictly required for the standard development of differential calculus.
Prerequisites
Familiarity with algebra, trigonometry, and basic function behavior is assumed. No prior knowledge of linear algebra is required beyond what is introduced in this text.
Using This Text
Readers may approach the material according to their goals:
- For intuition: Focus on motivations, examples, and visualizations
- For rigor: Work through proofs and verify intermediate steps
- For problem-solving: Attempt examples before consulting solutions
Begin with the Introduction to Sequences or proceed to Limits and Continuity.