Analysis Course 1 Part 2 of 2: Derivatives with Applications. Calculus 1, Part 2 of 2: Derivatives with Applications Calculus of one variable
S1. Introduction to the course: You will learn the contents of this course and the importance of differential calculus. The purpose of this section is not to teach you all the details (which come later in the course), but to show you the bigger picture.
S2. Definition of derivatives with some examples and examples: You will learn: Formal definition of derivatives and differentiability. Terminology and notation; Geometric interpretation of the derivative at a point. Tangent lines and their equations. How to calculate some derivatives directly from the definition and see the result along with the graph of the function in the coordinate system. Continuity vs. differentiation; Higher order derivatives; Fractions and their geometric interpretation. Linearization
S3. “Derivation of derivatives of elementary functions” You will learn: How to derive formulas for derivatives of basic elementary functions: constant function, Mooney monomials, roots, trigonometric and inverse functions, exponential functions, logarithmic functions and some power functions (more in the next section); how to prove and apply the sum law, the scaling law, the product law and the quotient law to derivatives and how to use these laws to differentiate a large number of new elementary functions formed from basis functions. Differentiation of continuous piecewise functions are defined using elementary functions.
S4. The Chain Law and Related Rates: You will learn: How to calculate the derivatives of complex functions using the chain law. Some pictures and proofs of the chain law. Derivatives Derivative formulas are a more general type of power functions and exponential functions with a different base than, for example, How to solve some types of related rate problems (problems that can be solved using the chain law).
S5. Derivatives of inverse functions: You will learn: to formulate the derivative of an inverse function with respect to a separable inverse function defined on an interval (with a very good geometric/trigonometric intuition behind it). We will review some of the formulas derived earlier in the course and show how they can be motivated by the new theorem, but you will also see other examples of applications of this theorem.
S6. Mean value theorems and other important theorems: You will learn: various theorems that play an important role in later applications: mean value theorems (Lagrange, Cauchy), Darbox property, Rolle’s theorem, Fermat’s theorem. You will learn formulations, proofs, intuitive/geometric interpretations, application examples and the meaning of various assumptions. You will learn some new terms such as CP (critical point, also stationary point) and singular point. The definitions of maximum/relative minimum and global/absolute maximum/minimum are repeated from Calculus 1 so that we can use them in the calculus context (they are discussed more practically in Sections 7 and 17 and 18).
S7. Applications: Uniformity and Optimization: You will learn: How to apply the results of the previous section in more practical situations, such as checking the uniformity of differentiable functions and optimizing (mainly continuous) functions. First derivative test and second derivative test to classify CP (critical points) of different functions.
S8. Convexity and second derivatives: You will learn: How to use the second derivative to determine if a function is convex on a concave-convex interval. Milestones and how to see them in the graph of a function. The concept of convexity is general, but we apply it here only to doubly differentiable functions.
S9. L’Hôpital’s Law with Applications: You will learn: Use l’Hôpital’s Law to calculate the limits of indeterminate shapes. A very detailed proof is provided in the article attached to the first video in this section.
S10. Higher order derivatives and an introduction to the Taylor formula: You will learn: about the classes of real-valued functions of a real variable: C^0, C^1, … , C^∞ and some prominent members of these classes. Importance of Taylor/Maclarin polynomials and their form for exponential functions, for sine and for cosine. You will only get a small insight into these topics as they are normally part of Account 2.
S11. Implicit Differentiation: You will learn: How to find the derivative y'(x) of an implicit relation F(x,y)=0 by combining various rules for differentiation. You will find examples of curves described by implicit relations, but the study of these is not included in this course (normally studied in “Algebraic Geometry”, “Differential Geometry” or “Geometry and Topology”; the subject is also covered in “Limit Calculus 3 (Calculus of Multivariables), Part 1 of 2”: The Implicit Function Theorem).
S12. Logarithmic differentiation: You will learn how to perform logarithmic differentiation and in which cases it is practical to use it.
S13. A quick look at partial derivatives: You will learn: How to calculate partial derivatives for multivariate functions (just an introduction).
S14. Briefly on antiderivatives: You will learn about the unusual application of integrals and about basic integration techniques.
S15. A very short introduction to the topic of ODEs: You will learn: Some very basic things about ordinary differential equations.
S16. More advanced concepts based on the concept of derivative: You will learn: about some more advanced concepts based on the concept of derivative: partial derivative, gradient, Jacobian, Hessian derivative, derivative of vector-valued functions, divergence, rotation (skew).
S17. Problem Solving: Optimization: You will learn: How to solve optimization problems (exercise for Section 7).
S18. Problem Solving: Graphing Functions: You will learn: How to create a table of changes (signs) of a function and its derivatives. You will be given many practice functions for graphing (the topic is partially covered in “Calculus 1, Part 1 of 2: Limits and Continuity” and completed in Parts 6-8 of this course).
S19. Extras: You will learn about all the courses we offer. You will also get a glimpse into our plans for future courses with approximate (very hypothetical!) release dates.
Be sure to check with your professor which course components you will need for your final exam. Such cases vary from country to country, university to university, and may even vary from year to year within the same university.