Calculus is the mathematical study of continuous change, primarily divided into differential and integral calculus. It provides the essential framework for modeling dynamic systems, optimizing functions, and understanding the physical world, making it indispensable in physics, engineering, and data science. Mastering the relationship between derivatives and integrals—as formalized by the Fundamental Theorem—transforms calculus from rote memorization into a unified computational system for analyzing rates of change and accumulation.
What This Cheat Sheet Covers
This topic spans 13 focused tables and 99 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Fundamental LimitsTable 2: Continuity and Key TheoremsTable 3: Core Derivative RulesTable 4: Common DerivativesTable 5: Applications of DerivativesTable 6: Core Integration RulesTable 7: Common IntegralsTable 8: Advanced Integration TechniquesTable 9: Applications of IntegrationTable 10: Sequences and Series TestsTable 11: Differential EquationsTable 12: Multivariable CalculusTable 13: Series Approximations and Power Series
Table 1: Fundamental Limits
| Method | Example | Description |
|---|---|---|
\lim_{x \to 2} (x^2 + 1) = 5 | • Plugs the value directly into a continuous function • the first step to try for any limit. | |
\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2}(x+2) = 4 | Simplifies rational functions producing a 0/0 indeterminate form via algebraic factorization. | |
\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1 | Differentiates numerator and denominator when a limit yields 0/0 or ∞/∞ indeterminate form. | |
-x^2 \le x^2\sin(1/x) \le x^2\Rightarrow \lim_{x \to 0} x^2\sin(1/x) = 0 | Bounds a complex function between two simpler ones that share the same limit at the target point. |