Complete Mathematics Masterclass: College & University Level from Udemy

What's inside

Syllabus

[Pilot Section] If you just finished school: What you know & what you will learn

Overview of the course

Overview image

Section intro

Overview of this pilot section: What you know and what you don't know yet

What exactly is a function?

Linear functions

Quadratic functions and solving quadratic equations

Factorizing polynomials using their roots & Outlook: Complex numbers

Exponential function: How is it defined exactly?

Vector algebra in polar coordinate system

Vector rotation using matrices

Section outro

Download the slides of this section

Summary of integration rules

[Part 1] College-level: What you are expected to know at university

Overview of the first part of this course

Limits of functions

Limits: Dealing with infinity

Limits versus function values

Polynomial fractions

Asymptotic behavior

Useful rules

Your turn! About exercises

Exercises

Integration & Limits - Improper integrals

Check your solutions

Solutions of Tasks 1 & 2 - Limits of functions

Solutions of Tasks 3 & 4 - Limits of functions

Add on: Two advanced lectures & results that we need later on

Squeeze theorem: Limit of sin(x)/x for x to 0

L'Hospital's rule (L'Hôpital's rule)

Integration versus differentiation

Derivatives in one dimension

What is a derivative?

Derivative of constant, linear and quadratic functions

Sum rule for derivatives

Derivative of polynomials

Product rule for derivatives

Derivative of the 1 / x function

Chain rule for derivatives

Quotient rule for derivatives

Derivative of the inverse function

Derivative of root functions

Derivative of power functions

Derivative of exponential & logarithm functions

Derivative of trigonometric functions: Sine & Cosine

Outlook: Derivative of trigonometric functions by series expansion

Higher derivatives

Extrema of functions & Inflection points

Curve sketching

Summary of the rules for derivatives

Planes & Points: Calculating the distance

Solutions Task 1 - Derivatives

Solutions Tasks 2 & 3 - Derivatives

Add on: How to calculate derivatives numerically

Numerics with Python: Calculating derivatives

Integrals in one dimension

What is an integral?

Integration by parts

Integration by substitution

Alternative way of using integration by substitution

Integration of cosine square & sine square

Exploiting symmetry

Solutions Task 1 - Integrals

Solutions Task 2 - Integrals

Add on: How to calculate integrals numerically

Numerics with Python: Calculating integrals

Vectors in Cartesian coordinates

What is a vector?

Basic vector operations

Dot product or scalar product

Cross product or vector product

Triple product

Lines in parametric form

Lines & Points: Calculating the distance

Lines & Lines: Identical, parallel, intersecting or skew lines

Planes in parametric form

Planes in coordinate form

Planes in Hesse normal form

Planes & Lines: Included, parallel or intersecting

Planes & Planes: Identical, parallel or intersecting line

Solutions Task 1 - Vectors: Two points

Solutions Task 2 - Vectors: Two lines

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 Complete Mathematics Masterclass: College & University Level with these activities:

Review Algebra Fundamentals

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Strengthen your understanding of core algebraic concepts. A solid foundation in algebra is crucial for success in calculus and other advanced math topics covered in this course.

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Review key concepts and formulas.
Work through practice problems.
Identify areas of weakness and seek additional resources.

Review Calculus Textbook

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Reinforce your understanding of calculus concepts. This book will help you to master the calculus concepts presented in the course.

View Biocalculus: Calculus, Probability, and... on Amazon

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Read the relevant chapters on limits, derivatives, and integrals.
Work through the examples and exercises in the book.
Compare the book's explanations with the course lectures.

Practice Differentiation and Integration

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Sharpen your calculus skills through repetitive exercises. Consistent practice with differentiation and integration will improve your speed and accuracy in solving problems.

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Find online resources with calculus practice problems.
Work through a set number of problems each day.
Check your answers and review any mistakes.

Four other activities

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Create a Cheat Sheet for Integration Techniques

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Summarize key integration techniques in a concise format. Creating a cheat sheet will help you to consolidate your knowledge and quickly recall important formulas and methods.

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Review the different integration techniques covered in the course.
Organize the techniques into categories.
Write down the formulas and steps for each technique.
Design a visually appealing and easy-to-use cheat sheet.

Model a Real-World Phenomenon with Differential Equations

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Apply your knowledge of differential equations to a practical problem. This project will allow you to see how mathematical concepts can be used to model and understand real-world phenomena.

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Choose a phenomenon that can be modeled with differential equations.
Develop a mathematical model of the phenomenon.
Solve the differential equation and analyze the results.
Present your findings in a report or presentation.

Explore Linear Algebra

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Deepen your understanding of linear algebra concepts. This book will help you to master the matrix algebra concepts presented in the course.

View Linear Algebra and Its Applications on Amazon

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Read the relevant chapters on matrices, vectors, and linear transformations.
Work through the examples and exercises in the book.
Relate the book's explanations to the course lectures.

Develop a Fourier Transform Visualizer

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Create a tool to visualize Fourier transforms. This project will solidify your understanding of Fourier transforms and their applications.

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Learn about Fourier transforms and their properties.
Choose a programming language and libraries for visualization.
Implement the Fourier transform algorithm.
Create a user interface to input signals and display their Fourier transforms.

Career center

Learners who complete Complete Mathematics Masterclass: College & University Level will develop knowledge and skills that may be useful to these careers:

Research Scientist

A research scientist conducts scientific research, often in a specific field such as physics, chemistry, or biology. They design experiments, collect data, and analyze results. This career typically requires an advanced degree. Many scientific disciplines rely heavily on advanced mathematics for modeling and analysis. This course covers a wide range of mathematical topics relevant to research scientists, like differential equations, Fourier transforms, and statistical analysis. The course may be particularly useful to those working in quantitative fields of research.

See salaries and explore the career path for Research Scientist

Bioinformatician

A bioinformatician develops and applies computational tools and methods to analyze biological data. This requires a strong foundation in mathematics, statistics, and computer science. This course may be useful because it covers essential mathematical concepts like linear algebra, calculus, and stochastic processes, which are relevant to analyzing genomic data, building predictive models, and understanding complex biological systems. The sections on statistics and stochastic processes, as well as matrix algebra, help the bioinformatician.

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Quantitative Analyst

A quantitative analyst, often working in the finance industry, develops and implements mathematical models for pricing, trading, and risk management. This career frequently requires a master's degree, or even a PhD. The mathematical concepts covered in this course, such as calculus, differential equations, and matrix algebra, are directly applicable to the models used by quantitative analysts. Understanding Taylor expansions and Fourier transforms, as taught in this course, helps the quantitative analyst analyze financial data and develop sophisticated trading strategies. Aspiring quantitative analysts may find this course useful as a refresher.

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Statistician

A statistician collects, analyzes, and interprets data to identify trends and patterns. Statisticians apply mathematical and statistical techniques to solve problems in a variety of fields, including healthcare, business, and government. This course offers a great introduction to many of the mathematical concepts that will be used by statisticians on a daily basis. A future statistician may appreciate learning about probability distributions and matrix algebra, as well as stochastic.

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Machine Learning Engineer

A machine learning engineer designs, develops, and deploys machine learning models. This role requires a strong foundation in mathematics, particularly linear algebra, calculus, and statistics. This course covers these topics, providing aspiring machine learning engineers with the mathematical tools and knowledge needed to understand and implement machine learning algorithms. The course may be especially useful for learning about derivatives, integrals, and probability distributions.

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Actuary

An actuary assesses and manages financial risks, typically for insurance companies and pension funds. Actuaries rely heavily on mathematical and statistical models to predict future events and determine appropriate pricing and reserves. A strong understanding of probability distributions, calculus, and financial mathematics is essential for this role. This course provides a foundational knowledge of these areas, including stochastic and probability distributions. Aspiring actuaries may find this course a helpful refresher.

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Operations Research Analyst

An operations research analyst uses mathematical and analytical methods to help organizations make better decisions. They develop models and algorithms to optimize processes, improve efficiency, and solve complex problems. This role requires a strong understanding of linear algebra, calculus, and optimization techniques. This course may be useful because it provides a comprehensive overview of these topics, including matrix algebra, differential equations, and derivatives and integrals in multiple dimensions.

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Data Scientist

A data scientist analyzes complex datasets to extract meaningful insights and develop predictive models. This often requires a strong foundation in mathematics, particularly statistics and calculus. This course touches on calculus, algebra, and stochastics. With a solid grasp of these concepts, a data scientist is better equipped to understand the underlying principles of machine learning algorithms and interpret data effectively. The course gives potential data scientists exposure to limits, derivatives, and integrals, all of which help build a good footing in the field.

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Economist

An economist studies the production, distribution, and consumption of goods and services. Economists develop and use mathematical models to analyze economic trends, forecast future conditions, and advise policymakers. A solid foundation in calculus, statistics, and econometrics is essential for this role. This course may be useful by providing the learner with those tools, helping the economist better comprehend economic models and analyze data effectively. Aspiring economists may find the sections on derivatives, integrals, and probability distributions particularly relevant.

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Financial Analyst

A financial analyst analyzes financial data, provides investment recommendations, and helps businesses make informed decisions. Financial analysts use mathematical models and statistical techniques to evaluate investment opportunities and manage risk. This course provides a good foundation in calculus, algebra, and stochastics. Learning about sequences and series, as well as derivatives and integrals, can enhance a financial analyst's ability to interpret financial data and construct financial models.

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Game Developer

A game developer creates video games for various platforms. Game developers need a solid understanding of mathematics, particularly linear algebra and calculus, to create realistic graphics, physics simulations, and artificial intelligence. This course provides a strong foundation in these areas, including vector algebra, coordinate systems, and derivatives and integrals. The sections on linear algebra and calculus may be very practical for aspiring game developers.

See salaries and explore the career path for Game Developer

Software Engineer

Software engineers design, develop, and test software applications. While not always immediately apparent, mathematics plays a crucial role in many areas of software engineering, especially in fields like game development, computer graphics, and data science. This course covers essential mathematical concepts like linear algebra, calculus, and statistics, all of which help when developing efficient algorithms and solving complex problems. The topics on vector algebra and coordinate systems included in this course help software engineers working on graphics applications.

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Civil Engineer

Civil engineers design, construct, and maintain infrastructure projects such as roads, bridges, and buildings. A solid understanding of mathematics, including calculus, linear algebra, and differential equations, is essential for analyzing structural loads, designing stable structures, and managing projects effectively. This course may be useful by reinforcing these core mathematical concepts. Aspiring civil engineers may find the sections on vector algebra and differential equations particularly relevant.

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Mechanical Engineer

Mechanical engineers design, develop, and test mechanical devices and systems. Mechanical engineering relies heavily on mathematical principles, particularly calculus, differential equations, and linear algebra. This course provides a strong foundation in these areas, which may be key for mechanical engineers in modeling and analyzing complex systems. The course may be especially useful for mechanical engineers to brush up on differential equations and calculus.

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Financial Planner

A financial planner helps individuals and families manage their finances and achieve their financial goals. They provide advice on investments, retirement planning, insurance, and other financial matters. While this role is more client-facing, a solid grasp of mathematics is helpful for understanding financial products and developing effective financial plans. This course may be useful by giving the financial planner a good foundation in algebra and statistics.

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Physics Teacher

A physics teacher educates students in the principles of physics. They present information by lecture, demonstration, discussion, and assignments. This course may be helpful to teachers by providing them with a comprehensive understanding of the college and university level mathematics that underlies physics. This course helps physics teachers teach mathematical methods and reasoning necessary for solving physics problems. The thorough nature of this course may prove valuable in keeping educators prepared.

See salaries and explore the career path for Physics Teacher

Reading list

We've selected two books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Complete Mathematics Masterclass: College & University Level.

Biocalculus: Calculus, Probability, and Statistics...

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Provides a comprehensive introduction to calculus. It covers limits, derivatives, integrals, sequences, and series in detail. It valuable reference for understanding the calculus concepts presented in the course, and it is commonly used as a textbook in many universities.

Biocalculus: Calculus, Probability, and Statistics...

Loose Leaf

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Linear Algebra and Its Applications

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Provides a thorough introduction to linear algebra. It covers matrices, vectors, linear transformations, eigenvalues, and eigenvectors. It useful reference for understanding the matrix algebra concepts presented in the course, and it is commonly used as a textbook in many universities.

Linear Algebra and Its Applications

Paperback

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Linear Algebra and Its Applications

Kindle Edition

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Complete Mathematics Masterclass

College & University Level

What's inside

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Activities

Career center

Reading list

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