Deep Dive into Calculus: From Foundations to Mastery
MCreated by Michael EspaillatWhat you'll learn
Explain and contextualize the historical and conceptual foundations of calculus
Apply rigorous definitions of limits, derivatives, and integrals in problem solving
Compute derivatives and integrals using a range of techniques, including advanced methods
Analyze and sketch functions using calculus-based reasoning
Solve optimization and related rates problems from real-world contexts
Evaluate and interpret infinite series, including Taylor and power series
Extend calculus techniques to functions of several variables and vector fields
Model and solve basic differential equations arising in science and engineering
Before you start
- Comfort with algebraic manipulation
- Basic knowledge of trigonometric functions
- Familiarity with graphs and coordinate plane
- Arithmetic with fractions, exponents, and roots
Curriculum
1. The Origins and Foundations of Calculus
Explore the historical development and foundational concepts that set the stage for modern calculus, including intuitive methods, the evolution of notation, and the transition to rigorous definitions.
2. Functions, Limits, and Continuity
Build a solid understanding of functions, limits, continuity, and the epsilon-delta definition, forming the essential groundwork for all of calculus.
3. Derivatives and Differential Calculus
Master the concept of the derivative, its rigorous definition, differentiation rules, implicit and logarithmic differentiation, and their applications in analyzing function behavior.
4. Applications of Derivatives
Apply derivatives to real-world and mathematical problems, including optimization, related rates, curve sketching, linear approximation, and the Mean Value Theorem.
5. Integration and the Fundamental Theorem of Calculus
Understand integration as the inverse of differentiation, learn Riemann sums, definite and indefinite integrals, and the Fundamental Theorem of Calculus.
6. Advanced Integration Techniques and Applications
Develop proficiency with advanced integration techniques (substitution, integration by parts, trigonometric substitution, partial fractions, improper integrals) and explore their applications to area, volume, work, and probability.
7. Sequences, Series, and Power Series
Delve into sequences and series, convergence tests, power series, and Taylor/Maclaurin series, connecting infinite processes to function approximation.
8. Multivariable and Vector Calculus
Extend calculus concepts to functions of several variables, covering partial derivatives, gradients, tangent planes, multiple integrals, and the basics of vector calculus (vector fields, line integrals, Green’s, Stokes', and Divergence theorems).
9. Introduction to Differential Equations
Introduce the basics of differential equations, solution methods, and applications, laying a pathway to further study in mathematics and applied fields.
10. Synthesis and the Spirit of Calculus
Synthesize and connect all major concepts, emphasizing rigorous reasoning, proof, and the broad impact of calculus across disciplines and history.
Your instructor
5 courses published
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