Calculus 1, part 2 of 2: Derivatives with applications
Differential calculus in one variable: theory and applications for optimisation, approximations, and plotting functions
What you will learn
How to solve problems concerning derivatives of real-valued functions of 1 variable (illustrated with 330 solved problems) and why these methods work.
Definition of derivatives of real-valued functions of one real variable, with a geometrical interpretation and many illustrations.
Write equations of tangent lines to graphs of functions.
Derive the formulas for the derivatives of basic elementary functions.
Prove, apply, and illustrate the formulas for computing derivatives: the Sum Rule, the Product Rule, the Scaling Rule, the Quotient and Reciprocal Rule.
Prove and apply the Chain Rule; recognise the situations in which this rule should be applied and draw diagrams helping in the computations.
Use the Chain Rule in problem solving with related rates.
Use derivatives for solving optimisation problems.
Understand the connection between the signs of derivatives and the monotonicity of functions; apply first- and second-derivative tests.
Understand the connection between the second derivative and the local shape of graphs (convexity, concavity, inflection points).
Determine and classify stationary (critical) points for differentiable functions.
Use derivatives as help in plotting real-valued functions of one real variable.
Main theorems of Differential Calculus: Fermat's Theorem, Mean Value Theorems (Lagrange, Cauchy), Rolle's Theorem, and Darboux Property.
Formulate, prove, illustrate with examples, apply, and explain the importance of the assumptions in main theorems of Differential Calculus.
Formulate and prove l'Hospital's rule; apply it for computing limits of indeterminate forms; algebraical tricks to adapt the rule for various situations.
Higher order derivatives; an intro to Taylor / Maclaurin polynomials and their applications for approximations and for limits (more in Calculus 2).
Classes of functions: C^0, C^1, ... , C^∞; connections between these classes, and examples of their members.
Implicit differentiation with some illustrations showing horizontal and vertical tangent lines to implicit curves.
Logarithmic differentiation: when and how to use it.
A sneak peek into some future applications of derivatives.
Why take this course?
¡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."
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