Calculus 1, part 1 of 2: Limits and continuity
Single variable calculus with elements of Real Analysis: from axioms and proofs to illustrations and computations
What you will learn
How to solve problems concerning limits and continuity of real-valued functions of 1 variable (illustrated with 491 solved problems) and why these methods work.
The structure and properties of the set of real numbers as an ordered field with the Axiom of Completeness, and consequences of this definition.
Arithmetic on the extended reals, and various types of indeterminate forms.
Supremum, infimum, and a reformulation of the Axiom of Completeness in these terms.
Number sequences and their convergence or divergence; the epsilon-definition of limits of sequences, with illustrations and examples; accumulation points.
Getting new limits from old limits: limit of the sum, difference, product, quotient, etc, of two sequences, with illustrations, formal proofs, and examples.
Squeeze Theorem for sequences
Squeeze Theorem for functions
The concept of a finite limit of a real-valued function of one real variable in a point: Cauchy's definition, Heine's definition; proof of their equivalence.
Limits at infinity and infinite limits of functions: Cauchy's definition (epsilon-delta) and Heine's definition (sequential) of such limits; their equivalence.
Limit of the sum, difference, product, quotient of two functions; limit of composition of two functions.
Properties of continuous functions: The Boundedness Theorem, The Max-Min Theorem, The Intermediate-Value Theorem.
Limits and continuity of elementary functions (polynomials, rational f., trigonometric and inverse trigonometric f., exponential, logarithmic and power f.).
Some standard limits in zero: sin(x)/x, tan(x)/x, (e^x-1)/x, ln(x+1)/x and a glimpse into their future applications in Differential Calculus.
Some standard limits in the infinity: a comparison of polynomial growth (more generally: growth described by power f.), exponential, and logarithmic growth.
Continuous extensions and removable discontinuities; examples of discontinuous functions in one, several, or even infinitely many points in the domain.
Starting thinking about plotting functions: domain, range, behaviour around accumulation points outside the domain, asymptotes (vertical, horizontal, slant).
An introduction to more advanced topics: Cauchy sequences and their convergence; a word about complete spaces; limits and continuity in metric spaces.
Why take this course?
π Course Title: Calculus 1, Part 1 of 2: Limits and Continuity π
π Course Overview:
"Single variable calculus with elements of Real Analysis: from axioms and proofs to illustrations and computations."
π Week 1-3: Introduction to the Course You're about to embark on a mathematical adventure through the world of limits, continuity, and beyond. This course is designed to give you a deep understanding of Calculus, its applications, and the underlying principles that make it work. π
π Preliminaries: Basic Notions and Elementary Functions (S2) We'll start with a refresher on Precalculus concepts to ensure you're ready for Calculus. Don't worry if some topics feel daunting; we're here to guide you through every step. π€«β¨
π The Nature of the Set of Real Numbers (S4) Explore the intriguing properties of the real numbers, including the Axiom of Completeness and its profound implications for limits and beyond. π
π Sequences and Their Limits (S5) Dive into sequences with a focus on finding and understanding their limits. We'll cover everything from basic concepts to advanced topics like extended reals, and even touch upon the convergence of sequences using Cauchy's criterion. π¬
π Limit of a Function in a Point (S6) Understand the precise definition of a function's limit at a point and the concept of continuity, accompanied by practical examples and computational rules for limits. π
π Infinite Limits and Limits in the Infinities (S7) Learn to handle limits involving rational functions as they approach zero or infinity, and how these relate to asymptotes. π
βοΈ Continuity Properties and Graphing Functions (S8-S9) Explore the properties of continuous functions and start graphing real-valued functions by identifying domains, discontinuities, and asymptotes. This sets the stage for our next course segment on derivatives. π
π Extras (S11) Beyond this course, we offer a variety of courses that build upon each other. Keep an eye out for our future offerings and what might be on the horizon! π§ β¨
π Additional Resources (S11) For those looking to dive deeper or prepare for exams, make sure to check out the comprehensive list of all videos and solved problems in "001 List_of_all_Videos_and_Problems_Calculus_1_p1.pdf" linked in video 1. This resource will accompany your learning journey throughout the course. π
Why Take This Course?
- Understand the Fundamentals: Master the core concepts of limits and continuity that are essential for advanced calculus topics.
- Real-World Applications: Learn how calculus is applied in various fields such as physics, engineering, economics, and more.
- Step-by-Step Guidance: With detailed video lectures and problem-solving examples, you'll have a clear path to learning.
- Flexible Learning: Study at your own pace, with resources available 24/7 for review and practice.
Join us on this journey through the heart of mathematics. With expert instruction, engaging content, and a supportive community, you'll be well-equipped to conquer the world of calculus. Enroll now and let's discover the wonders of limits and continuity together! ππ
Important Notes:
- Course Adaptation: The content of this course may vary depending on your institution or country's standards. Always confirm with your professor which parts of the course are required for your exam.
- Comprehensive Course Material: A detailed description of all videos and problems is provided in the resource file linked at the beginning of the course. Make sure to utilize these materials for a deeper understanding of the subjects covered. ππ
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Our review
Course Review for "Understanding Calculus with Real Analysis" by Hania Uscka-Wehlou
Overview
The global course rating stands at an impressive 4.93, with all recent reviews being overwhelmingly positive. The course, "Understanding Calculus with Real Analysis," is taught by Dr. Hania Uscka-Wehlou and has received high praise for its comprehensive content, clear explanations, and the teacher's dedication to providing a thorough yet humane introduction to calculus.
Pros
π Comprehensiveness and Organization
- The course is described as having an "astounding comprehensiveness", with care and organization worthy of more than just five stars. (Review 2)
- It is praised for its thorough revisiting of prerequisite material before delving into technical subjects, making it accessible to both newcomers and students familiar with the instructor's previous courses. (Review 4)
- The course structure is highly appreciated, with a big document guiding students through the material non-linearly if preferred. (Review 10)
π Pedagogical Excellence
- Hania's teaching style is lauded for its detail and patience, with videos that are clear and step-by-step, enhancing understanding and providing intuition for key concepts. (Review 5)
- The course is structured in a way that allows students to build an intuition for calculus, which is a unique strength highlighted by several reviews. (Review 12)
π Detail and Depth
- The explanations are very clear, and the depth of knowledge displayed in the course is described as one-of-a-kind on the platform. (Review 9)
- The approach to real analysis is commended for not treating it as an afterthought but integrating it seamlessly into the course content. (Review 14)
π€ Engagement and Support
- Dr. Hania engages with students' questions promptly, offering support that is both encouraging and stimulating. (Review 8)
- Students feel empowered and confident in their ability to learn math independently after taking this course, thanks to the comprehensive material and guidance provided. (Review 13)
π Impact on Learning
- The course is considered better than working through a textbook or YouTube videos, as it integrates theory with practice and provides an excellent balance of concept and problems. (Review 15)
Cons
? Minor Observations
- Some reviews mention that other calculus courses might skip over topics covered in detail in this course, such as real analysis. (Reviews 2 & 9)
- A minor concern is the potential fear of math among students, but this is addressed positively by the instructor. (Review 16)
Student Testimonials
π "Astounding Comprehensiveness"
"The course's comprehensiveness and organization are astounding. The care and attention to detail ensure that students who have taken previous courses from Dr. Hania will not be disappointed. It's a must-take for anyone interested in delving deeper into the world of calculus." (Review 2)
π "Pedagogical Excellence"
"Hania's pedagogical excellence shines through in her ability to explain complex topics with clarity and depth. Her course is the gold standard for learning calculus with real analysis." (Review 9)
π "Better Than Textbooks and YouTube"
"This course stands out as a more comprehensive and integrated approach compared to textbooks or YouTube videos. It's an excellent resource for anyone looking to understand calculus beyond the basics." (Review 15)
π "Empowering Students"
"Taking this course has given me the confidence to learn math independently, which I never thought possible. Dr. Hania's dedication to delivering quality content is evident and truly impactful." (Review 13)
Conclusion
Hania Uscka-Wehlou's "Understanding Calculus with Real Analysis" is a standout course that receives high marks for its comprehensive content, clear explanations, and the instructor's commitment to providing an exceptional learning experience. It is a valuable resource for students looking to deepen their understanding of calculus and real analysis, offering both theoretical depth and practical problem-solving skills. With near-perfect ratings and glowing testimonials, this course is highly recommended for anyone interested in furthering their mathematical education.