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Calculus

Single Variable Part 1 - Functions

Robert Ghrist

Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.

In this first part--part one of five--you will extend your understanding of Taylor series, review limits, learn the *why* behind l'Hopital's rule, and, most importantly, learn a new language for describing growth and decay of functions: the BIG O.

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What's inside

Syllabus

Introduction

Welcome to Calculus: Single Variable! below you will find the course's diagnostic exam. if you like, please take the exam. you don't need to score a minimal amount on the diagnostic in order to take the course. but if you do get a low score, you might want to readjust your expectations: this is a very hard class...

A Review of Functions

This module will review the basics of your (pre-)calculus background and set the stage for the rest of the course by considering the question: just what is the exponential function?

Taylor Series

This module gets at the heart of the entire course: the Taylor series, which provides an approximation to a function as a series, or "long polynomial". You will learn what a Taylor series is and how to compute it. Don't worry! The notation may be unfamiliar, but it's all just working with polynomials....

Limits and Asymptotics

A Taylor series may or may not converge, depending on its limiting (or "asymptotic") properties. Indeed, Taylor series are a perfect tool for understanding limits, both large and small, making sense of such methods as that of l'Hopital. To solidify these newfound skills, we introduce the language of "big-O" as a means of bounding the size of asymptotic terms. This language will be put to use in future Chapters on Calculus.

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Explores the core ideas of single-variable Calculus, including Taylor series, limits, and asymptotics

Taught by Robert Ghrist, a recognized expert in the field

Ideal for students beginning in the engineering, physical, and social sciences

Emphasizes conceptual understanding over computational methods

Introduces a novel synthesis of discrete and continuous forms of Calculus

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Reviews summary

Challenging but thorough calculus course

Learners say this Calculus course is challenging but thorough. Students in this course rave about the instructor and the engaging assignments.

Students highly recommend this course for its in-depth coverage of Calculus topics.

"This course covers a lot of ground and is very thorough in its explanations."

"This course is a great way to get a deep understanding of Calculus."

"The instructor does a great job of breaking down complex concepts in a way that is easy to understand."

Students highly praise the instructor for their expertise and helpfulness.

"The instructor is very knowledgeable and helpful."

"The instructor is very patient and takes the time to answer questions."

"The instructor is very passionate about the subject and makes the material interesting."

Be prepared for difficult exams that require a deep understanding of the material.

"The exams are very difficult and require a lot of studying."

"The exams are not very fair and do not accurately reflect what is taught in class."

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 Calculus: Single Variable Part 1 - Functions with these activities:

Review logarithmic and exponential functions

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Review logarithms and exponents to prepare for this course's focus on Taylor series.

Browse courses on Logarithms

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Review the properties of logarithms and exponential functions, such as their inverses and derivatives.
Practice solving problems involving logarithms and exponential functions.

Review functions and limits

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Review foundational Calculus concepts to ensure a strong foundation for this course's topics.

Browse courses on Functions

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Review the definition of a function.
Practice finding the domain and range of a function.
Review the definition of a limit.
Practice finding the limit of a function.

Practice solving limits using l'Hopital's rule

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Revisit and practice using l'Hopital's rule to solve challenging limits.

Browse courses on Limits

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Review the statement of l'Hopital's rule.
Practice applying l'Hopital's rule to solve limits.
Complete practice problems involving l'Hopital's rule.

Three other activities

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Follow tutorials on Taylor series and applications

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Understand the concepts and techniques involved in Taylor series through guided tutorials.

Browse courses on Taylor Series

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Watch video tutorials on Taylor series.
Read articles and online resources about Taylor series.
Complete practice problems involving Taylor series.

Create a summary of the key concepts of Taylor series

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Summarize the key concepts of Taylor series to reinforce understanding and review the material.

Browse courses on Taylor Series

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Review your notes and the course materials on Taylor series.
Identify the key concepts of Taylor series.
Write a summary of the key concepts.

Practice solving Calculus problems using Taylor series

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Deepen understanding of calculus by applying Taylor series to solve problems in the field.

Browse courses on Taylor Series

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Review the different types of Calculus problems that can be solved using Taylor series.
Practice solving Calculus problems using Taylor series.
Complete practice problems involving Calculus and Taylor series.

Career center

Learners who complete Calculus: Single Variable Part 1 - Functions will develop knowledge and skills that may be useful to these careers:

Economist

Economists analyze economic data and statistics to explore economic trends and make predictions. They use their knowledge of mathematical and statistical modeling techniques in their work. Calculus is heavily leveraged in econometrics, the quantitative analysis of economic data, and understanding Taylor series, limits, and asymptotics are all critical to building a foundation in this subfield of economics. This course can be especially helpful for students preparing for graduate study in economics or those interested in careers as economic consultants.

See salaries and explore the career path for Economist

Financial Analyst

Financial analysts use their knowledge of mathematics and statistics to advise companies and individuals on investment decisions. They use complex mathematical and statistical techniques to evaluate the performance of stocks, bonds, and other financial instruments. Calculus is a cornerstone of the mathematical modeling used in financial analysis and this course will help students build a strong foundation in its core concepts, particularly in the use of Taylor series and limits to approximate and understand financial data.

See salaries and explore the career path for Financial Analyst

Data Analyst

Data analysts use their knowledge of mathematics and statistics to collect, clean, and analyze data. They use their findings to help businesses make better decisions. Calculus is a fundamental tool for data analysis, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for data analysts who need to be able to apply Calculus to real-world problems, such as understanding the growth and decay of functions.

See salaries and explore the career path for Data Analyst

Market Research Analyst

Market research analysts use their knowledge of mathematics and statistics to collect, analyze, and interpret data about markets and consumers. They use their findings to help businesses make better decisions about their products and services. Calculus is a valuable tool for market research analysts, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for market research analysts who need to be able to apply Calculus to real-world problems, such as understanding the growth and decay of market trends.

See salaries and explore the career path for Market Research Analyst

Operations Research Analyst

Operations research analysts use mathematical models to solve problems that arise in a variety of industries, such as manufacturing, transportation, and healthcare. Calculus is essential for operations research analysts, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for operations research analysts who need to be able to apply Calculus to real-world problems, such as optimizing production schedules or designing transportation routes.

See salaries and explore the career path for Operations Research Analyst

Physicist

Physicists use mathematics and statistics to study the laws of nature. They use their knowledge to develop new technologies and to understand the universe. Calculus is a cornerstone of physics, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for physicists who need to be able to apply Calculus to real-world problems, such as understanding the motion of objects or the behavior of light.

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Software Engineer

Software engineers use mathematics and statistics to design, develop, and test software systems. Calculus is occasionally used in software engineering, for example in computer graphics or in the analysis of algorithms, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for software engineers who need to be able to apply Calculus to real-world problems.

See salaries and explore the career path for Software Engineer

Actuary

Actuaries use mathematics and statistics to assess risk and uncertainty. They use their findings to help businesses and individuals make informed decisions about their finances. Calculus is a critical tool for actuaries, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for actuaries who need to be able to apply Calculus to real-world problems, such as calculating insurance premiums or designing pension plans.

See salaries and explore the career path for Actuary

Quantitative Analyst

Quantitative analysts use mathematics and statistics to develop and implement quantitative models for use in financial markets. Calculus is a cornerstone of quantitative finance, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for quantitative analysts who need to be able to apply Calculus to real-world problems, such as pricing financial instruments or developing trading strategies.

See salaries and explore the career path for Quantitative Analyst

Data Scientist

Data scientists use mathematics and statistics to extract knowledge from data. They use their findings to help businesses make better decisions. Calculus is a valuable tool for data scientists, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for data scientists who need to be able to apply Calculus to real-world problems, such as understanding the growth and decay of functions.

See salaries and explore the career path for Data Scientist

Statistician

Statisticians use mathematics and statistics to collect, analyze, and interpret data. They use their findings to help businesses and governments make informed decisions. Calculus is a critical tool for statisticians, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for statisticians who need to be able to apply Calculus to real-world problems, such as designing surveys or analyzing data.

See salaries and explore the career path for Statistician

Chemical Engineer

Chemical engineers use mathematics and statistics to design, develop, and operate chemical plants and processes. Calculus is a valuable tool for chemical engineers, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for chemical engineers who need to be able to apply Calculus to real-world problems, such as designing chemical reactors or optimizing production processes.

See salaries and explore the career path for Chemical Engineer

Materials Scientist

Materials scientists use mathematics and statistics to study the structure and properties of materials. Calculus is a valuable tool for materials scientists, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for materials scientists who need to be able to apply Calculus to real-world problems, such as understanding the behavior of materials under different conditions or designing new materials.

See salaries and explore the career path for Materials Scientist

Biostatistician

Biostatisticians use mathematics and statistics to analyze biomedical data. They use their findings to help researchers understand the causes and treatments of diseases. Calculus is a valuable tool for biostatisticians, and this course will help students build a strong foundation in its core concepts. The course's emphasis on conceptual understanding and applications will be particularly helpful for biostatisticians who need to be able to apply Calculus to real-world problems, such as designing clinical trials or analyzing the results of medical studies.

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Reading list

We've selected 41 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 Calculus: Single Variable Part 1 - Functions.

The Hitchhiker's Guide to Calculus

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Provides a rigorous treatment of single-variable calculus, with an emphasis on mathematical proofs. It valuable resource for students who are interested in learning the subject in a more advanced way.

Calculus

Single Variable Part 1 - Functions

What's inside

Syllabus

Good to know

Save this course

Reviews summary

Challenging but thorough calculus course

Activities

Career center

Reading list

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