Advanced Calculus/Real Analysis with the Math Sorcerer from Udemy

What's inside

Syllabus

Introduction to Advanced Calculus

Sets and Notation

Triangle Inequality Proof

Reverse Triangle Inequality Proof

Bernoulli's Inequality Proof

Inequality Assignment

Solution to Problem #1

Solution to Problem #2

Solution to Problem #3

The Archimedean Principle

Introduction to the Archimedean Property

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Properties of Sequences

The Limit of a Sequence is Unique

Every Convergent Sequence is Bounded

The Squeeze Theorem for Sequences

Sequences

Convergence Proof Example 1

Convergence Proof Example 2

Convergence Proof Example 3

Proving a Sequence Converges without the Archimedean Property(Harder)

Proof sin(1/n) converges

Proving a Sequence Diverges

Sequence Convergence Proof Problem

Convergence of Cesaro Means

A Divergent Sequence with a Convergent Subsequence

Properties of Cauchy Sequences

Every Cauchy Sequence is Bounded

Every Convergent Sequence is Cauchy

Cauchy Sequences

Cauchy Sequence Convergence Example 1

Cauchy Sequence Convergence Example 2

Cauchy Sequence Convergence Example 3

Cauchy Sequence Convergence Example 4

Limits

Delta Epsilon Proof Example

Delta Epsilon Proof Linear Example

Delta Epsilon Proof Quadratic Example

Delta Epsilon Proof with Trig Function

Delta Epsilon Proof with Cubic Function

Sum of Limits Delta Epsilon Proof

Squeeze Theorem Proof Example

Limit Proof involving Bounded Function

An Interesting Example

Continuous Functions

Continuity Proof Example 1

Continuity Proof Example 2

Continuity Proof Example 3

Dirichlet Function

Uniformly Continuous Functions

Uniform Continuity versus Continuity

Uniform Continuity Proof Example 1

Uniform Continuity Proof Example 2

Uniform Continuity Proof Example 3

Uniform Continuity Proof Example 4

Uniform Continuity Proof Example 5

Uniform Continuity Proof Example 6

Uniform Continuity Proof Example 7

Differentiation

Every Differentiable Function is Continuous Proof

Proving a Function is Not Differentiable

Proving a Function is Differentiable Example 1

Proving a Function is Differentiable Example 2

Trigonometric Function Differentiability Proof Example 1

Trigonometric Function Differentiability Proof Example 2

Derivatives of Even and Odd Functions Proofs

Piecewise Function Differentiability Proof

Infinite Series

Series Convergence Explanation and Zeno's Paradox

Dirichlet's Test Example

Multivariable Limits

Delta Epsilon Multivariable Limit Proof Example 1

Delta Epsilon Multivariable Limit Proof Example 2

Pointwise and Uniform Convergence

Pointwise Versus Uniform Convergence

Pointwise Convergence Proof Example 1

Pointwise Convergence Proof Example 2

Finding a Pointwise Limit

Uniform Convergence Proof Example 1

Uniform Convergence Proof Example 2

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Example 1

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Advanced Calculus/Real Analysis with the Math Sorcerer with these activities:

Review Proof Techniques

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Strengthen your understanding of fundamental proof techniques to better grasp the concepts and proofs presented in the course.

Browse courses on Mathematical Induction

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Review definitions of proof methods.
Practice writing simple proofs.
Identify proof structures in examples.

Read 'How to Prove It' by Velleman

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Improve your ability to construct and understand mathematical proofs, a core skill emphasized in the Advanced Calculus course.

View How to Prove It: A Structured Approach on Amazon

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Read the book chapter by chapter.
Work through the examples and exercises.
Apply the techniques to course material.

Practice Delta-Epsilon Proofs

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Reinforce your understanding of limits and continuity by working through delta-epsilon proofs, a challenging but essential topic in Advanced Calculus.

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Review the definition of a limit.
Solve delta-epsilon problems from textbooks.
Check solutions and understand errors.

Four other activities

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Collaborate on Proofs

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Enhance your understanding of course material by discussing and working through proofs with peers, fostering a deeper comprehension of the concepts.

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Form a study group with classmates.
Select problems from course assignments.
Present and critique each other's proofs.

Create a Proof Wiki

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Solidify your knowledge by creating a wiki page explaining a specific theorem or concept from the course, including its proof and applications.

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Choose a theorem from the course.
Research the theorem and its proof.
Write a clear and concise explanation.
Include examples and applications.

Read 'Understanding Analysis' by Abbott

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Deepen your understanding of real analysis concepts and proof techniques by studying this well-regarded textbook.

View The Proof Stage: How Theater Reveals the Human... on Amazon

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Read the book chapter by chapter.
Work through the examples and exercises.
Compare the presentation to course material.

Tutor Underclassmen

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Reinforce your understanding of Advanced Calculus by tutoring students in introductory calculus courses, explaining concepts and helping them with problem-solving.

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Volunteer as a tutor at your university.
Prepare for tutoring sessions by reviewing material.
Explain concepts clearly and patiently.

Career center

Learners who complete Advanced Calculus/Real Analysis with the Math Sorcerer will develop knowledge and skills that may be useful to these careers:

Mathematical Modeler

Mathematical modelers create and analyze mathematical representations of real-world systems to gain insights and make predictions. This course on Advanced Calculus/Real Analysis helps build a firm foundation in the mathematical principles underlying modeling techniques. The emphasis on writing proofs helps ensure that the modeler rigorously validates their models and understands their limitations. The course covers a wide array of topics important to mathematical modelers, from limits to continuity to differentiation to series. The mathematical modeler might find that the concepts of pointwise and uniform convergence may be helpful in their career.

See salaries and explore the career path for Mathematical Modeler

Financial Quantitative Analyst

A financial quantitative analyst constructs mathematical models to understand and predict the behavior of financial markets. This course on Advanced Calculus/Real Analysis helps build a firmer and more nuanced understanding of the concepts these models rely on. The course will be particularly helpful for financial quantitative analysts, where the major focus is on writing proofs. Topics such as sequences, Cauchy sequences, and limits are all helpful for analyzing financial markets. For many, advanced calculus, or real analysis can be challenging. By just watching the videos and trying to follow along closely with pencil and paper, one can build a lot of knowledge.

See salaries and explore the career path for Financial Quantitative Analyst

Applied Mathematician

Applied mathematicians use mathematical theories and techniques to solve practical problems in various fields, often involving complex modeling and simulations. This course on Advanced Calculus/Real Analysis provides a deeper understanding of the fundamental principles that underpin these models. The course's focus on writing proofs may be especially useful, as it enables the applied mathematician to rigorously validate their models and ensure their reliability. A student benefits from the course's coverage of topics like limits, sequences, series, and continuity, all while gaining a firm grasp on the math beyond general courses.

See salaries and explore the career path for Applied Mathematician

Theoretical Physicist

Theoretical physicists develop mathematical models and theories to explain the fundamental laws of the universe. This course on Advanced Calculus/Real Analysis helps build a strong foundation in the mathematical tools needed to construct and analyze these theories. A key aspect of theoretical physics is proving the consistency and validity of new theories. The course’s emphasis on writing proofs provides a solid foundation for this task. Studying limits, continuity, and differentiation helps the theoretical physicist analyze the behavior of physical systems. The course is a step above, and for some, this provides the right mix of theory and practice.

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Data Scientist

A data scientist relies heavily on mathematical principles and rigorous proofing to develop models and extract insights from data. This course on Advanced Calculus/Real Analysis helps build a strong foundation in understanding the theoretical underpinnings of many data science algorithms. The ability to write proofs, which is a major focus of the course, is directly applicable to validating models and ensuring their reliability. Data scientists can leverage the knowledge gained in this course to tackle complex analytical problems and develop innovative solutions, especially given how the course steps above general mathematics. By working through the course's examples on limits, sequences, and continuity, learners can deeply understand the mathematical concepts essential for data analysis.

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Machine Learning Engineer

Machine learning engineers design, develop, and deploy machine learning models, requiring a solid understanding of the underlying mathematics. This course in Advanced Calculus/Real Analysis helps build a strong understanding of the mathematical foundations behind these models. For example, the course's treatment of limits, continuity, and differentiation is crucial for understanding optimization algorithms used in training machine learning models. The ability to write and understand proofs, emphasized throughout the course, enables a machine learning engineer to rigorously test and validate the performance of these models. Someone wishing to become a machine learning engineer benefits by diving deeply into the course, which covers topics like uniform convergence and multivariable limits.

See salaries and explore the career path for Machine Learning Engineer

Cryptographer

Cryptographers design and analyze encryption algorithms to secure data. This course on Advanced Calculus/Real Analysis helps build expertise in the mathematical techniques used in cryptography. The course's emphasis on writing proofs is directly applicable to verifying the security properties of cryptographic systems. Cryptographers may find the topics covered, like sequences and limits, valuable in analyzing the performance and robustness of encryption algorithms. By understanding mathematical notation, such as the notation described in the first syllabus item, the work of the Cryptographer can be even more impactful.

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Financial Engineer

Financial engineers use mathematical and computational tools to solve complex financial problems. This course on Advanced Calculus/Real Analysis helps build a strong foundation in the underlying mathematical principles used in financial modeling. The course’s emphasis on writing proofs and mathematical rigor is directly applicable to ensuring the stability and accuracy of financial models. The course's treatment of limits, continuity, and differentiation can be applied to pricing derivatives and managing risk effectively. By learning to recreate these proofs, the financial engineer can reach the next level.

See salaries and explore the career path for Financial Engineer

Bioinformatics Scientist

Bioinformatics scientists analyze biological data using computational tools and mathematical models. This course on Advanced Calculus/Real Analysis helps build a firmer grasp of the mathematical principles underlying these models. Proof writing, which is emphasized throughout the course, helps the bioinformatician to rigorously validate the correctness of their methodology. Concepts such as sequences and limits, which are covered in the syllabus, can enable them to analyze biological processes and interpret experimental results more accurately. Spending even a little time on the course each day helps to make the journey more enjoyable.

See salaries and explore the career path for Bioinformatics Scientist

Operations Research Analyst

Operations research analysts use mathematical and analytical methods to help organizations make better decisions. This course on Advanced Calculus/Real Analysis helps build expertise in the optimization techniques used in operations research. The course’s emphasis on writing proofs strengthens the analytical skills required to model and solve complex problems. Operations research analysts may find the course's coverage of limits, sequences, and series particularly valuable in understanding optimization algorithms and their convergence properties. They may also find it useful when working through problems.

See salaries and explore the career path for Operations Research Analyst

Econometrician

Econometricians use statistical methods to analyze economic data and test economic theories. This course on Advanced Calculus/Real Analysis helps build expertise in the mathematical foundations of econometric models. The course's emphasis on writing proofs may come in handy, strengthening the rigorous thinking required to validate econometric methods. Concepts covered in the course, such as limits, continuity, and differentiation, contribute to understanding the properties of estimators and test statistics. An econometrician will find the course beneficial for understanding advanced topics in asymptotic theory and statistical inference.

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Statistician

Statisticians collect, analyze, and interpret data to solve real-world problems. A key component of the work is to establish a rigorous basis for statistical inference. This course on Advanced Calculus/Real Analysis may be useful to statisticians. The course's emphasis on proof-writing provides an understanding of the theoretical underpinnings of statistical methods. Statisticians might find the sections on sequences, limits, and continuity particularly helpful in understanding asymptotic properties and convergence results. The deep dive into mathematical notation helps statisticians communicate their findings effectively and defend their conclusions with mathematical rigor. The course helps build confidence in one's understanding.

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Quantitative Analyst

Quantitative analysts, often working in the finance industry, use mathematical and statistical methods to assess risk and develop trading strategies. This course in Advanced Calculus/Real Analysis may be useful to those looking to enter this career. It is an excellent resource for anyone who wants to have a much deeper understanding of the mathematical models that they will encounter. The focus on writing proofs throughout the course sharpens the rigorous thinking required for verifying the correctness and stability of financial models. A quantitative analyst can use the concepts covered in this course, such as sequences and limits, to analyze market trends and optimize investment portfolios more effectively. Understanding the proofs and mathematical notation taught in the course enables the analyst to build confidence in their approaches.

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Game Developer

Game developers, especially those working on physics engines or advanced graphics, benefit from a solid understanding of mathematical concepts. This course on Advanced Calculus/Real Analysis may be useful in grasping the underlying principles behind game physics and rendering algorithms. The course's focus on limits, continuity, and differentiation is applicable to creating smooth and realistic simulations. A game developer can use the proof-writing skills, which the course highlights, to verify the correctness of their algorithms and optimize performance. This is a valuable skillset whether the game developer focuses on 2D or 3D games, indie or AAA games. Even finishing 50% of this course is a big accomplishment.

See salaries and explore the career path for Game Developer

Software Engineer

While a software engineer may not directly use advanced calculus in their daily tasks, a deeper understanding of mathematical principles may still be useful. This course in Advanced Calculus/Real Analysis may be useful in developing more efficient algorithms and solving complex computational problems. The course's focus on writing proofs sharpens logical reasoning skills that are valuable in software development. Software engineers looking to specialize in areas such as graphics, simulations, or data analysis may find the concepts covered in this course particularly relevant. By working to learn the proofs as presented, a software engineer can reach the next level.

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Advanced Calculus/Real Analysis with the Math Sorcerer

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