Single Variable Calculus from Coursera

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 fifth part--part five of five--we cover a calculus for sequences, numerical methods, series and convergence tests, power and Taylor series, and conclude the course with a final exam. Learners in this course can earn a certificate in the series by signing up for Coursera's verified certificate program and passing the series' final exam.

What's inside

Syllabus

A Calculus for Sequences

It's time to redo calculus! Previously, all the calculus we have done is meant for functions with a continuous input and a continuous output. This time, we are going to retool calculus for functions with a discrete input. These are sequences, and they will occupy our attention for this last segment of the course. This first module will introduce the tools and terminologies for discrete calculus.

Introduction to Numerical Methods

That first module might have seemed a little...strange. It was! In this module, however, we will put that strangeness to good use, by giving a very brief introduction to the vast subjects of numerical analysis, answering such questions as "how do we approximate solutions to differential equations?" and "how do we approximate definite integals?" Perhaps unsurprisingly, Taylor expansion plays a pivotal role in these approximations.

Series and Convergence Tests

In "ordinary" calculus, we have seen the importance (and challenge!) of improper integrals over unbounded domains. Within discrete calculus, this converts to the problem of infinite sums, or series. The determination of convergence for such will occupy our attention for this module. I hope you haven't forgotten your big-O notation --- you are going to need it!

Power and Taylor Series

This course began with an exploration of Taylor series -- an exploration that was, sadly, not as rigorous as one would like. Now that we have at our disposal all the tests and tools of discrete and continuous calculus, we can finally close the loop and make sense of what we've been doing when we Talyor-expand. This module will cover power series in general, from we which specify to our beloved Taylor series.

Concluding Single Variable Calculus

Are we at the end? Yes, yes, we are. Standing on top of a high peak, looking back down on all that we have climbed together. Let's take one last look down and prepare for what lies above.

Good to know

Know what's good

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Explores calculus with an emphasis on conceptual understanding and applications, making it suitable for students in engineering, physical, and social sciences

Provides a novel synthesis of discrete and continuous forms of calculus, offering a unique perspective on the subject

Taught by Robert Ghrist, recognized for their work in mathematics and engineering

Offers a clear, dynamic, and unified approach to calculus, making it easy to understand

Involves the use of Taylor series and approximations from the beginning, providing a strong foundation for further studies in calculus

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 Single Variable Calculus with these activities:

Calculate limits from table data

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Taking time to look at some basic limits and reviewing basic techniques will reduce friction during the course.

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Find a table with values approaching infinity
Apply the concept of a limit to the table values

Practice recognizing type of discontinuity

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Practice is needed to be able to effectively identify discontinuity of functions.

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Go through various examples with different discontinuity types
Try identifying discontinuity types for different function types

Peer discussion on Mathematical Induction

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Discussing how to prove statements using Mathematical Induction with classmates will reinforce the understanding of the technique.

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Form a study group
Choose a statement to prove using Mathematical Induction
Work together to prove the statement

Six other activities

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Explore online resources for numerical methods

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Looking outside the course at different numerical method resources provides a more robust understanding of the topic.

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Search for online tutorials on numerical methods
Watch a few tutorials and take notes
Try out some of the numerical methods on your own

Convergence Test Examples

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Gathering convergence test examples will aid in grasping various test applications.

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Find real world examples of convergent series
Find real world examples of divergent series
Apply different convergence tests to the examples

Create a visual representation of a series expansion

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Generating a visual representation of Taylor Series will provide a deeper understanding beyond the formula.

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Choose a function to expand
Calculate the first few terms of the Taylor Series
Use a graphing calculator or software to plot the function and the Taylor Series approximation
Compare the graph of the function to the graph of the Taylor Series approximation

Derive Taylor Series

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Creating the Taylor Series of sin(x) from scratch will solidify the foundational understanding of Taylor Series.

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Review the general formula for deriving a Taylor Series
Calculate the derivatives of sin(x) at x=0
Substitute the values into the Taylor Series formula

Limit Practice Problems

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Continuously working through limit problems will increase understanding and comfort with the topic.

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Find practice problems online
Solve the practice problems
Check your answers against the solutions

Derivative Practice Problems

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Regular practice with derivative problems strengthens problem-solving skills.

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Find practice problems online
Solve the practice problems
Check your answers against the solutions

Career center

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

Statistician

This course will be useful to aspiring and practicing Statisticians. By covering modern techniques in both discrete and continuous domains, as well as introducing methods of approximation for more complex statistical ideas, this course is useful for building a foundation in contemporary statistical techniques.

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Research Analyst

Many roles in research, social and scientific alike, rely on a strong understanding of statistical techniques to gather meaningful data. This course covers many essential tools and techniques, useful both for planning and executing research, and presenting and drawing conclusions from collected data.

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Reliability Engineer

Reliability Engineering often relies on statistical techniques to evaluate the performance of a given system. This course provides a solid background in statistical modeling, with a focus on probability distributions.

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Data Analyst

The ability to analyze and understand trends in data is critical to success as a Data Analyst. This course covers statistical techniques commonly used in data analytics, and is useful both as a refresher or a bridge into more advanced topics in the field.

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Operations Research Analyst

In the design of complex systems, such as manufacturing or logistics networks, it is essential to understand and model a variety of stochastic processes. This course provides foundational understanding of continuous and discrete probability distributions, essential knowledge for any Operations Research Analyst.

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Market Researcher

The results of market research are made much more useful through the application of statistical techniques. This course provides a foundation in statistical modeling, including hypothesis testing and regression analysis, which are critical techniques for Market Researchers.

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Teacher

For teachers of math and statistics, this course may be useful for refreshing their understanding of the modern techniques in both discrete and continuous domains, as well as introducing methods of approximation for more complex statistical ideas.

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Biostatistician

The exploration and summarization of data are critical to success in biostatistics. Understanding how to analyze and make sense of data in a meaningful and efficient way is a valuable skill set for any Biostatistician, and this course can provide a strong foundation.

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Underwriter

In determining the likelihood of an event, such as a car accident, and thus determining the terms of an insurance policy, Underwriters rely on statistical techniques. This course provides a solid foundation in continuous probability distributions, which is helpful when beginning a career as an Underwriter.

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Software Engineer

In developing large software systems, the ability to analyze and model the size and complexity of the system is essential. This course provides a strong foundation in the mathematical techniques used to analyze the size and complexity of a software system, which can be helpful to Software Engineers.

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Quantitative Analyst

Many financial models rely on statistical estimation, such as the use of continuous-time Markov models to model price movement of assets. This course provides a solid foundation in continuous probability distributions, which is helpful when beginning a career as a Quantitative Analyst.

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Risk Manager

Risk management, such as in banking, often relies on statistical techniques to understand how risk factors interact and affect the probability of a given outcome. This course provides a solid foundation in continuous probability distributions, which is helpful when beginning a career as a Risk Manager.

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Systems Engineer

Systems, such as supply chains or telecommunication networks, can be modeled using statistical techniques. This course covers many essential tools and techniques, useful both for planning and executing system designs, and presenting and drawing conclusions from collected data.

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Data Scientist

Data science is a field in which statistical modeling, particularly in the form of inferential statistics, is critical for uncovering truths from gathered data. This course provides a strong foundation in the philosophies and techniques of inferential statistics that Data Scientists rely on.

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Financial Analyst

In financial markets, understanding the statistical forces that drive prices is a critical skill. This course covers many fundamentals of statistical inference, which is essential for understanding and predicting these forces.---.

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