Calculus 1, part 2 of 2: Derivatives with applications

¡Claro! The course "Calculus 1, part 2 of 2" (S0) by Professor Robert Lewis from the University of Texas at Austin, covers advanced topics in differential calculus and introduces integral calculus. Here's a breakdown of the course content:

S0. Introduction to Calculus 1, part 2

• Overview of what the course will cover
• A brief recapitulation of the key concepts from Calculus 1, part 1

S1. Derivatives of higher order and applications

• Second derivatives and their geometric meaning (concavity and convexity)
• Higher-order derivatives up to the nth derivative
• Applications of higher-order derivatives (e.g., finding points of inflection)

S2. The Mean Value Theorem for Derivatives

• Statement and proof of the Mean Value Theorem for Derivatives
• Geometric interpretation of the theorem
• Applications of the Mean Value Theorem for Derivatives

S3. L'Hôpital's Rule

• Explanation and justification of l'Hôpital's Rule
• Examples and applications of l'Hôpital's Rule to indeterminate forms

S4. Sequences and Series

• Convergence and divergence of sequences and series
• The Nth Term Test, the Root Test, and the Ratio Test for series
• Geometric series and the formula for summing an infinite geometric series

S5. Infinite Sequences and Series - Advanced

• Convergence of sequences using epsilon-delta definition
• Compare tests for convergence of series (Ratio Test, Root Test, Integral Test)

S6. Introduction to Integral Calculus

• Definition of the definite integral
• Fundamental Theorem of Calculus
• Applications of the Fundamental Theorem of Calculus to evaluation of definite integrals

S7. Indefinite Integrals (Antiderivatives)

• Techniques for finding antiderivatives
• Basic integration formulas and substitution methods
• Partial fraction decomposition

S8. Definite Integrals - Applications

• The Mean Value Theorem for integrals
• Area between the curve and the x-axis, area between two curves, and composite functions
• Average value of a function over an interval

S9. Techniques of Integration

• Rational and radical simplification before integration
• Integration by parts
• Trigonometric integrals and trigonometric substitutions

S10. Improper Integrals

• Definition and evaluation of improper integrals
• Comparison Test and Limit Test for improper integrals

S11. Applications of Integration

• Area, volume, and work applications of integration
• Arc length, surface area, and volume of revolution applications

S12. Calculus in Several Variables

• Partial derivatives (introduction)
• The Mean Value Theorem for partial derivatives
• Chain rule for partial derivatives

S13. Multivariable Functions - Applications

• Gradient vector and directional derivative
• Divergence and curl in vector calculus (brief introduction)

S14. Optimization Problems

• Methods of finding absolute extrema (critical points, second derivative test)
• Lagrange multipliers for optimization with constraints

S15. Applications of Calculus to Ordinary Differential Equations (ODE)

• Introduction to ordinary differential equations and their solutions

S16. Advanced Concepts in Calculus

• Partial derivative, gradient, Jacobian, Hessian, and divergence, rotation (curl)

S17. Problem Solving: Optimization

• Practical exercises on optimization problems

S18. Problem Solving: Plotting Functions

• Sign variations and applications to plotting functions

S19. Extras

• Overview of other courses offered by Professor Robert Lewis
• Future course plans (very hypothetical!)

This course is designed for students who have already completed an introductory course in calculus and are ready to delve deeper into the subject. It covers both differential calculus and integral calculus, along with applications of these mathematical concepts. The content is organized into sections, each covering a different aspect of calculus. The course material can be found under "Introduction to Calculus 1, part 2."