Introduction to Calculus from Coursera

The focus and themes of the Introduction to Calculus course address the most important foundations for applications of mathematics in science, engineering and commerce. The course emphasises the key ideas and historical motivation for calculus, while at the same time striking a balance between theory and application, leading to a mastery of key threshold concepts in foundational mathematics.

Students taking Introduction to Calculus will:

• gain familiarity with key ideas of precalculus, including the manipulation of equations and elementary functions (first two weeks),

• develop fluency with the preliminary methodology of tangents and limits, and the definition of a derivative (third week),

• develop and practice methods of differential calculus with applications (fourth week),

• develop and practice methods of the integral calculus (fifth week).

What's inside

Syllabus

Precalculus (Setting the scene)

This module begins by looking at the different kinds of numbers that fall on the real number line, decimal expansions and approximations, then continues with an exploration of manipulation of equations and inequalities, of sign diagrams and the use of the Cartesian plane.

Functions (Useful and important repertoire)

This module introduces the notion of a function which captures precisely ways in which different quantities or measurements are linked together. The module covers quadratic, cubic and general power and polynomial functions; exponential and logarithmic functions; and trigonometric functions related to the mathematics of periodic behaviour. We create new functions using composition and inversion and look at how to move backwards and forwards between quantities algebraically, as well as visually, with transformations in the xy-plane.

Introducing the differential calculus

This module introduces techniques of differential calculus. We look at average rates of change which become instantaneous, as time intervals become vanishingly small, leading to the notion of a derivative. We then explore techniques involving differentials that exploit tangent lines. The module introduces Leibniz notation and shows how to use it to get information easily about the derivative of a function and how to apply it.

Properties and applications of the derivative

This module continues the development of differential calculus by introducing the first and second derivatives of a function. We use sign diagrams of the first and second derivatives and from this, develop a systematic protocol for curve sketching. The module also introduces rules for finding derivatives of complicated functions built from simpler functions, using the Chain Rule, the Product Rule, and the Quotient Rule, and how to exploit information about the derivative to solve difficult optimisation problems.

Introducing the integral calculus

This fifth and final module introduces integral calculus, looking at the slopes of tangent lines and areas under curves. This leads to the Fundamental Theorem of Calculus. We explore the use of areas under velocity curves to estimate displacement, using averages of lower and upper rectangular approximations. We then look at limits of approximations, to discover the formula for the area of a circle and the area under a parabola. We then develop methods for capturing precisely areas under curves, using Riemann sums and the definite integral. The module then introduces indefinite integrals and the method of integration by substitution. Finally, we discuss properties of odd and even functions, related to rotational and reflectional symmetry, and the logistic function, which modifies exponential growth.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Develops key ideas of precalculus and strengthens preliminary methodology for intermediate learners

Taught by David Easdown, who is recognized for their work in mathematics

Introduces techniques of differential and integral calculus, strengthening an existing foundation

Examines applications of calculus, which are highly relevant in engineering, science, and commerce

Covers foundational calculus concepts similar to an accredited institute, building a strong foundation for beginners

Uses Leibniz notation, which is standard in the field of engineering

Reviews summary

Introduction to calculus

Learners say this introduction to Calculus course is largely positive, with engaging assignments, helpful discussions, and deadlines that were easy to manage. Many learners found that the course provided a thorough overview of the important foundational concepts of Calculus and prepared them well for more advanced studies. They were able to engage with the material through a variety of resources, including lecture videos, readings, exams, quizzes, and homework assignments. The course instructor was highly praised for their passion, clear explanations, and responsiveness to learner questions.

Learners appreciated the manageable deadlines that allowed them to complete the course at their own pace without feeling overwhelmed.

"The course is incredible. Perfect balance between theoretical concepts and their implication in practice. I hope professor Easdown continues teaching courses on Calculus (now more on Advanced Calculus, please:)). Unfortunately, I couldn't find any online courses from him."

"I started studying first after like 15 years, but the lectures were great to refresh from high school math and lectures were really well-made to understand complex ideas."

Engaging assignments helped build a comprehensive understanding of calculus. Practice quizzes after every topic helped learners solidify their understanding before taking the module quiz.

"The practice quiz has been extremely helpful in getting the concepts right before taking the Module quiz."

"I really love to know the topics starting from its origin,its need. Thank You Professor.Thank You Coursera."

Learners appreciated the responsive instructor and helpful discussions that fostered a sense of community and learning.

"Overall, it was a wonderful experience, and I hope to pursue further courses in mathematics."

"This was an excellent way to brush up on pre-calculus mathematics and learn the reasoning behind calculus."

Learners found the course provided a thorough overview of important foundational concepts in Calculus.

"Calculus is the foundation of many scientific and all engineering disciplines. I hope that you will also pass this course."

"I highly recommend this course to others. Lastly I will say a sincerely thanks to the professor, my tutor, my parents and my students for helping me build up the idea of calculus."

Learners highly praised the instructor for their passion, clear explanations, and responsiveness to learner questions.

"Professor Easdown is one of the best educators I have ever had the privilege to study with."

"I wish to thank David and other people who helped to create this course. I look forward to see new courses coming soon :D"

While some learners found the practice materials helpful, they noted that the exams were significantly more difficult.

"With all the video lectures and the notes accompanying the video lectures, I was able to retain a good portion of the material. The last 3 weeks made me feel as though I had a good foundation to learn more about calculus and higher levels of math."

"This course has been a good occasion to review some pre-calculus algebra and some calculus topics and to learn new items."

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 Introduction to Calculus with these activities:

Review precalculus concepts

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Ensure a strong foundation by reviewing fundamental precalculus concepts.

Browse courses on Precalculus

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Review basic algebra concepts, such as solving equations and inequalities.
Review basic trigonometry concepts, such as the definitions of the trigonometric functions and the unit circle.
Practice applying these concepts to solve problems.

Explore online tutorials on the chain rule

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The chain rule is a fundamental technique in calculus. Explore online tutorials to enhance your understanding.

Browse courses on Chain Rule

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Start by searching for online tutorials on the chain rule.
Watch the tutorials and follow along with the examples.
Pause the tutorials when necessary to take notes.
Practice applying the chain rule to your own problems.

Join a study group to discuss key concepts

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Engage with peers to clarify understanding, challenge perspectives, and reinforce learning.

Browse courses on Calculus

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Join a study group for the course.
Attend study group meetings regularly.
Participate actively in discussions.
Help other students with concepts they may be struggling with.

Four other activities

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Practice integration techniques

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Practice additional integration problems to improve your skills and build confidence.

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Start by reviewing the basic integration techniques.
Choose the appropriate integration technique for each problem.
Carry out the integration.
Check your answer by differentiating the result.

Practice solving related-rate problems

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Related-rate problems can be challenging to solve, but they are an important application of derivatives. Practice additional problems to build your skills.

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Start by reading the problem carefully to understand what is being asked.
Identify the variables that are changing and the relationship between them.
Write an equation that relates the variables.
Differentiate the equation with respect to time.
Solve the resulting equation for the unknown rate.

Create a study guide for the derivative

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To reinforce learning about derivatives, create a study guide that summarizes the key concepts.

Browse courses on Derivative

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Start by reviewing your notes and the textbook.
Identify the key concepts that you need to include in the study guide.
Write out clear and concise explanations of each key concept.
Include examples to illustrate the key concepts.
Review your study guide regularly to help you remember the material.

Develop a presentation on applications of calculus in real life

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To further explore the practical relevance of calculus, create a presentation that showcases its applications in various fields.

Browse courses on Applications of Calculus

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Start by brainstorming different fields where calculus is applied.
Research specific examples of how calculus is used in each field.
Create a presentation that explains these applications in a clear and engaging way.
Include visuals and examples to illustrate the concepts.
Practice presenting your presentation to get feedback.

Career center

Learners who complete Introduction to Calculus will develop knowledge and skills that may be useful to these careers:

Mathematician

Mathematicians use mathematical principles to solve problems and develop new theories. They work in a variety of fields, including academia, research, and industry. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in science, can help build a foundation for understanding the complex mathematical concepts used in this field.

See salaries and explore the career path for Mathematician

Statistician

Statisticians use mathematical and statistical techniques to collect, analyze, and interpret data. They work in a variety of fields, including academia, research, and industry. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in science, can help build a foundation for understanding the complex statistical models used in this field.

See salaries and explore the career path for Statistician

Data Scientist

Data Scientists use mathematics, statistics, and computer science to extract insights from data. They work in a variety of fields, including finance, healthcare, and marketing. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in science, can help build a foundation for understanding the complex data analysis techniques used in this field.

See salaries and explore the career path for Data Scientist

Quantitative Analyst

Quantitative Analysts use mathematics and statistics to analyze financial data and make investment decisions. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in commerce, can help build a foundation for understanding the complex financial models used in this field.

See salaries and explore the career path for Quantitative Analyst

Financial Planner

Financial Planners use mathematical and statistical techniques to develop financial plans for individuals and families. They work in a variety of settings, including banks, investment firms, and insurance companies. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in commerce, can help build a foundation for understanding the complex financial models used in this field.

See salaries and explore the career path for Financial Planner

Economist

Economists use mathematical and statistical techniques to analyze economic data and make predictions about the economy. They work in a variety of fields, including academia, research, and government. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in commerce, can help build a foundation for understanding the complex economic models used in this field.

See salaries and explore the career path for Economist

Financial Analyst

Financial Analysts utilize mathematics, statistics, and computer models to make sound investment decisions. This course's emphasis on key ideas and historical motivation for calculus can help build a foundation for understanding the complex financial models used in this field. Additionally, its focus on applications of mathematics in commerce may be particularly relevant to this role.

See salaries and explore the career path for Financial Analyst

Market Researcher

Market Researchers use mathematical and statistical techniques to collect and analyze data about consumer behavior. They work in a variety of fields, including marketing, advertising, and product development. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in commerce, can help build a foundation for understanding the complex data analysis techniques used in this field.

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Actuary

Actuaries use mathematical and statistical skills to assess risk and uncertainty. They work in a variety of fields, including insurance, finance, and healthcare. This course's emphasis on key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in commerce, can help build a foundation for understanding the complex statistical models used in this field.

See salaries and explore the career path for Actuary

Business Analyst

Business Analysts use mathematical and statistical techniques to analyze business data and make recommendations for improvement. They work in a variety of industries, including consulting, finance, and healthcare. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in commerce, can help build a foundation for understanding the complex data analysis techniques used in this field.

See salaries and explore the career path for Business Analyst

Operations Research Analyst

Operations Research Analysts apply analytical methods to solve complex business problems. They use techniques such as linear programming, queuing theory, and simulation to optimize processes and improve efficiency. This course's focus on the development of differential calculus with applications may be particularly relevant to this role.

See salaries and explore the career path for Operations Research Analyst

Operations Manager

Operations Managers plan and oversee the production of goods and services. They work in a variety of industries, including manufacturing, healthcare, and retail. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in commerce, can help build a foundation for understanding the complex operations management techniques used in this field.

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Teacher

Teachers use mathematical and statistical techniques to teach students about math and science. They work in a variety of settings, including elementary schools, high schools, and colleges. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in science, can help build a foundation for understanding the complex mathematical concepts used in this field.

See salaries and explore the career path for Teacher

Technical Writer

Technical Writers use mathematical and statistical techniques to write technical manuals, reports, and other documents. They work in a variety of industries, including software development, engineering, and manufacturing. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in science, can help build a foundation for understanding the complex technical concepts used in this field.

See salaries and explore the career path for Technical Writer

Software Engineer

Software Engineers design, develop, and test software systems. They work in a variety of industries, including technology, finance, and healthcare. This course's emphasis on the key ideas and historical motivation for calculus, as well as its focus on applications of mathematics in science, can help build a foundation for understanding the complex algorithms and data structures used in this field.

See salaries and explore the career path for Software Engineer