Calculus I: From Functions to Differential Equations from edX

A strong foundation in mathematics is critical for success in all science and engineering disciplines. Whether you want to make a strong start to a master’s degree, prepare for more advanced courses, solidify your knowledge in a professional context or simply brush up on fundamentals, this course will get you up to speed.

This course allows you to get a solid basis by refreshing and reviewing your bachelor-level calculus.

The course focuses on functions of one variable. In the first 5 weeks you will learn all the basic integration, differentiation and approximation techniques required in a first calculus course of an engineering bachelor education. In the final week these topics will all come together as you solve and analyze several ordinary differential equations.

We use examples that are based on real-life applications so you can practice your mathematical skills in an engineering context. Our courses in calculus offer enough depth to cover what you need to succeed in your engineering master’s or profession in areas such as modeling, physics, fluid dynamics, dynamical systems and more.

This is a review courseThis self-contained course is modular, so you do not need to follow the entire course if you wish to focus on a particular aspect. As a review course you are expected to have previously studied or be familiar with most of the material. Hence the pace will be higher than in an introductory course.

This format is ideal for refreshing your bachelor level mathematics and letting you practice as much as you want. Through the Grasple platform, you will have access to plenty of exercises and receive intelligent, personal and immediate feedback.

This course is part of our series Mastering Mathematics for Engineers , and together with the course Calculus II part of the program Mastering Calculus.

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What's inside

Learning objectives

Apply differentiation techniques such as the chain rule and implicit differentiation.
Apply integration techniques such as integration-by-parts and substitution.
Solve ordinary differential equations that are important in engineering like a damped, forced harmonic oscillator.

Compute horizontal asymptotes to find equilibria and growth rates.
Analyze challenging engineering problems using these techniques

Apply differentiation techniques such as the chain rule and implicit differentiation.
Apply integration techniques such as integration-by-parts and substitution.
Solve ordinary differential equations that are important in engineering like a damped, forced harmonic oscillator.
Compute horizontal asymptotes to find equilibria and growth rates.
Analyze challenging engineering problems using these techniques

Syllabus

Week 1:

functions

graphs

inverse function

inverse trigonometric functions

compositions of functions

Week 2:

differentiation

tangent lines

implicit differentiation

differentiation of inverses

Week 3:

approximation errors

linear approximation

differentials

Taylor polynomials

Taylor’s inequality

Week 4:

horizontal asymptotes

growth rates

computing horizontal asymptotes

Week 5:

integration

integration by parts

substitution method

integration by Taylor polynomial

integrals over unbounded domains

Week 6:

differential equations

direction fields

first order separable and linear equations

forced and damped harmonics oscillators

approximating solutions to differential equations using Taylor polynomials

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Builds a strong foundation for learners who need review and refresher work

Covers a range of fundamental calculus concepts, suitable for a self-contained review

Emphasizes useful examples based on real-life engineering applications

Taught by reputable and well-known instructors in the field of engineering mathematics

Provides plenty of exercises and intelligent feedback through the Grasple platform

Recommended for those with a background in calculus who need to brush up on their knowledge

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 I: From Functions to Differential Equations with these activities:

Organize Course Materials

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Proper organization will help you stay on top of your coursework and easily find what you need.

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Create a system for organizing your notes, assignments, quizzes, and exams.
Review your materials regularly and identify areas where additional reinforcement is needed.
Consider creating a digital or physical repository for your materials.

Create a Calculus Resource Compilation

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Organizing your resources can help you stay organized and locate information quickly.

Browse courses on Calculus

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Gather a collection of useful calculus resources, including lecture notes, textbooks, online tutorials, and practice problems.
Organize these resources into a logical structure, such as topics or difficulty levels.
Create a document or online repository to store and access your compilation.

Review Ordinary Differential Equations

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Ensure you have a solid understanding of ordinary differential equations.

Browse courses on Ordinary Differential Equations

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Revisit notes and textbooks from previous courses or online resources.
Review the basic concepts, such as solving first-order linear and separable equations.
Practice solving problems to strengthen your skills.

Four other activities

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Follow Online Calculus Tutorials

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Online tutorials can provide additional explanations and demonstrations of calculus concepts.

Browse courses on Calculus

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Search for reputable online platforms or educators offering calculus tutorials.
Choose tutorials that cover specific topics you need to strengthen.
Follow the tutorials, take notes, and practice the examples provided.

Calculus Exercise Problems

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Regular practice of calculus problems is key to mastering the concepts.

Browse courses on Calculus

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Find online exercises or textbooks with a collection of calculus problems.
Set aside dedicated time for practicing these problems.
Review your solutions and identify areas where you need improvement.

Join Calculus Discussion Groups

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Engaging in discussions with peers can enhance your understanding and problem-solving skills.

Browse courses on Calculus

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Join online forums or discussion groups dedicated to calculus.
Participate in discussions, ask questions, and share your insights.
Collaborate with peers on solving challenging problems.

Calculus Project: Engineering Application

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Applying calculus concepts to a real-world engineering project deepens your understanding.

Browse courses on Calculus

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Choose an engineering problem that can be modeled using calculus.
Develop a mathematical model and solve it using calculus techniques.
Present your findings and explain the engineering significance of your solution.

Career center

Learners who complete Calculus I: From Functions to Differential Equations will develop knowledge and skills that may be useful to these careers:

Quantitative Analyst

Quantitative Analysts (Quants) use their advanced math skills to analyze financial data to make informed investment decisions and manage risk. This Calculus I course helps build a foundation in the mathematical concepts, including integration and ordinary differential equations, that are essential for success as a Quant. The course also introduces applications of calculus in finance, such as the calculation of option prices and the analysis of stock movements. These are concepts that Quants must master to perform complex financial modeling and make optimal investment decisions.

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Mechanical Engineer

Mechanical Engineers apply the principles of physics and mathematics to design and build a wide range of systems and machines. Calculus I provides a solid foundation in the mathematical techniques, such as integration and differential equations, that are essential for Mechanical Engineering. These techniques are used to analyze stresses on structures, calculate fluid flow, and optimize energy efficiency. The course also introduces modeling and simulation techniques that are used to design and test engineering systems.

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Aerospace Engineer

Aerospace Engineers design and build aircraft, spacecraft, and other vehicles that fly. Calculus I provides a strong foundation in the mathematical techniques, such as integration and differential equations, that are essential for Aerospace Engineering. These techniques are used to analyze the forces and stresses on aircraft and spacecraft and to design propulsion systems. The course also provides an introduction to aerodynamics and thermodynamics, which are essential for understanding the behavior of aircraft and spacecraft.

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Biomedical Engineer

Biomedical Engineers apply engineering principles to solve healthcare problems. Calculus I provides a strong foundation in the mathematical techniques, such as integration and differential equations, that are essential for Biomedical Engineering. These techniques are used to model biological systems, design medical devices, and understand drug interactions. The course also introduces biomechanics and physiology, which are essential for understanding the human body and developing new medical treatments.

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Civil Engineer

Civil Engineers design, construct, and maintain structures and infrastructure. Calculus I provides a strong foundation in the mathematical techniques, such as integration and differential equations, that are essential for Civil Engineering. These techniques are used to analyze forces on structures, design bridges and buildings, and plan transportation systems. The course also provides an introduction to fluid mechanics and geotechnical engineering.

See salaries and explore the career path for Civil Engineer

Software Engineer

Software Engineers design, build, and maintain computer systems. Calculus I provides a strong foundation in the mathematical techniques, such as differentiation and integration, that are essential for Software Engineering. These techniques are used to solve problems in areas such as computer graphics, robotics, and artificial intelligence. The course also helps develop problem-solving and analytical skills necessary for success in this field.

See salaries and explore the career path for Software Engineer

Data Scientist

Data Scientists use their knowledge of math, statistics, information science, and computer science to make sense of complex data. They apply their skills to analyze data, solve problems, and build systems that improve business outcomes. This course in Calculus I can help build a foundation in the mathematical skills, including functions and derivatives, that are essential for success in Data Science. The course also introduces differential equations, which are used to model dynamic systems like the spread of a virus or responses to climate change. These are concepts that Data Scientists must master to build models and extract accurate insights from data sets.

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Chemical Engineer

Chemical Engineers design, build, and operate plants and processes for the chemical industry. Calculus I provides a strong foundation in the mathematical techniques, such as integration and differential equations, that are essential for Chemical Engineering. These techniques are used to model chemical reactions, design reactors, and optimize processes. The course also introduces mass transfer and thermodynamics, which are essential for understanding and controlling chemical reactions.

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Actuary

Actuaries evaluate financial risks and develop insurance policies and retirement plans. Calculus I may be useful for Actuaries who want to build a strong foundation in the mathematical techniques that are used to calculate insurance premiums and benefits. The course covers concepts such as differentiation and integration, which are used to model mortality rates and other actuarial factors. The course also provides an introduction to statistics and probability, which are essential for understanding risk and making sound financial decisions.

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Operations Research Analyst

Operations Research Analysts use mathematical and analytical techniques to solve problems in a variety of industries, including manufacturing, transportation, and healthcare. Calculus I may be useful for Operations Research Analysts who want to build a strong foundation in the mathematical techniques that are used to optimize systems and processes. The course covers concepts such as differentiation and integration, which are used to model and analyze operations. The course also provides an introduction to probability and statistics, which are essential for understanding risk and making optimal decisions.

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Statistician

Statisticians collect, analyze, and interpret data to provide insights for businesses and organizations. Calculus I may be useful for Statisticians who want to build a strong foundation in the mathematical techniques that are used to collect, analyze, and interpret data. The course covers concepts such as differentiation and integration, which are used to calculate probabilities and other statistical measures. The course also provides an introduction to probability and statistics, which are essential for understanding data and making informed decisions.

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Data Analyst

Data Analysts collect, analyze, and interpret data to provide insights for businesses and organizations. Calculus I may be useful for Data Analysts who want to build a strong foundation in the mathematical techniques that are used to collect, analyze, and interpret data. The course covers concepts such as differentiation and integration, which are used to calculate probabilities and other statistical measures. The course also provides an introduction to probability and statistics, which are essential for understanding data and making informed decisions.

See salaries and explore the career path for Data Analyst

Financial Analyst

Financial Analysts evaluate financial information and make recommendations to investors and businesses. Calculus I may be useful for Financial Analysts who want to build a strong foundation in the mathematical techniques that are used in financial modeling and analysis. The course covers concepts such as differentiation and integration, which are used to calculate returns, risk, and other financial metrics. The course also provides an introduction to statistics and probability, which are essential for understanding financial markets and making sound investment decisions.

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Teacher

calculus I could be especially helpful for those who plan to teach mathematics at the high school or college level. The course covers the fundamental concepts of calculus, including functions, limits, derivatives, and integrals. This knowledge is essential for effectively teaching these concepts to students and preparing them for higher-level mathematics courses. Additionally, the course includes a focus on problem-solving and critical thinking skills, which are essential for success in teaching.

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Economist

Economists study the production, distribution, and consumption of goods and services. Calculus I provides a strong foundation in the mathematical techniques, such as differentiation and integration, that are essential for Economics. These techniques are used to model economic systems, analyze market behavior, and forecast economic trends. The course also introduces probability and statistics, which are essential for understanding economic data and making informed decisions.

See salaries and explore the career path for Economist