Unit outline_

# MATH1021: Calculus Of One Variable

## Overview

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates differential calculus and integral calculus of one variable and the diverse applications of this theory. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include complex numbers, functions of a single variable, limits and continuity, differentiation, optimisation, Taylor polynomials, Taylor's Theorem, Taylor series, Riemann sums, and Riemann integrals. Students are strongly recommended to complete MATH1021 or MATH1921 Calculus Of One Variable (Advanced) before commencing MATH1023 Multivariable Calculus and Modelling or MATH1923 Multivariable Calculus and Modelling (Adv).

### Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 3 None None MATH1901 or MATH1906 or ENVX1001 or MATH1001 or MATH1921 or MATH1931 HSC Mathematics Extension 1 or equivalent Yes

### Teaching staff

Coordinator Haotian Wu, haotian.wu@sydney.edu.au Haotian Wu Alexander Fish

## Assessment

Type Description Weight Due Length
Supervised exam

Exam
multiple choice and written calculations
60% Formal exam period 1.5 hours
Outcomes assessed:
Assignment Assignment 1
written calculations
5% Week 04
Due date: 16 Mar 2023 at 23:59

Closing date: 26 Mar 2023
2-4 pages (as a guide)
Outcomes assessed:
multiple choice or written calculations
15% Week 08
Due date: 20 Apr 2023 at 23:59

Closing date: 20 Apr 2023
40 minutes
Outcomes assessed:
Assignment Assignment 2
written calculations
10% Week 11
Due date: 11 May 2023 at 23:59

Closing date: 21 May 2023
6-8 pages (as a guide)
Outcomes assessed:
Small test Webwork Online Quizzes
online task (may require written calculations)
8% Weekly Weeks 2-7, 9-12
Outcomes assessed:
Participation Tutorials
Participation in tutorials
2% Weekly 50 minutes/week
Outcomes assessed:

### Assessment summary

• Assignments:  There are two assignments. Each must be submitted electronically, as one single typeset or scanned PDF file only, via Canvas by the deadline. Note that your assignment will not be marked if it is illegible or if it is submitted sideways or upside down. It is your responsibility to check that your assignment has been submitted correctly and that it is complete (check that you can view each page). Late submissions will receive a penalty. A mark of zero will be awarded for all submissions more than 10 days past the original due date. Further extensions past this time will not be permitted.
• Quiz: One quiz will be held online through Canvas. The quiz is 40 minutes and has to be submitted by the closing time of 23:59 on the due date. The quiz can be taken any time during the 24 hour period before the closing time.The better mark principle will be used for the quiz so do not submit an application for Special Consideration or Special Arrangements if you miss a quiz. The better mark principle means that the quiz counts if and only if it is better than or equal to your exam mark. If your quiz mark is less than your exam mark, the exam mark will be used for that portion of your assessment instead.
• Webwork Online Quizzes: There are ten weekly online quizzes (equally weighted) and the marks for the best eight count. Each online quiz consists of a set of randomized questions. You should not apply for special consideration for the quizzes. The better mark principle will apply for the total 8% - i.e. if your overall exam mark is higher, then your 8% for the Webwork quizzes will come from your exam. The deadline for completion of each quiz is 23:59 Thursday (starting in week 2). The precise schedule for the quizzes is found on Canvas. We recommend that you follow the due dates outlined above to gain the most benefit from these quizzes.
• Tutorial Participation: This is a satisfactory/non-satisfactory mark assessing whether or not you participate in class activities during the tutorials. It is 0.25 marks per tutorial class up to 8 tutorials (there are 12 tutorials).
• Final Exam: The final exam for this unit is compulsory and must be attempted. Failure to attempt the final exam will result in an AF grade for the course. Further information about the exam will be made available at a later date on Canvas. If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

Detailed information for each assessment can be found on Canvas

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

The Current Student website  provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 1. Set notation, the real number line; 2. Complex numbers in Cartesian form; 3. Complex plane, modulus Lecture (2 hr)
Week 02 1. Complex numbers in polar form; 2. De Moivre’s theorem; 3. Complex powers and nth roots 4. Roots of polynomials 5. The complex exponential. Lecture and tutorial (3 hr)
Week 03 1. Definition, domain and range of a function 2. Composite and inverse functions 3. Inverse trigonometric functions 4. Hyperbolic functions Lecture and tutorial (3 hr)
Week 04 1. Definition of a limit; 2. Vertical and horizontal asymptotes; 3. The Squeeze Law; 4. Continuity Lecture and tutorial (3 hr)
Week 05 1. The derivative as a rate of change; 2. Formal definition of the derivative; 3. The Chain Rule; 4. Implicit differentiation; 5. The Mean Value Theorem Lecture and tutorial (3 hr)
Week 06 1. Optimising and sketching functions of one variable; 2. Concavity; 3. Curve sketching; 4. L’Hopital’s rule Lecture and tutorial (3 hr)
Week 07 1. Taylor polynomials; 2. The remainder term Lecture and tutorial (3 hr)
Week 08 Taylor series Lecture and tutorial (3 hr)
Week 09 1. The integral as a measure of accumulation; 2. Riemann sums; 3. Definition of definite integral; 4. Properties of Riemann integrals Lecture and tutorial (3 hr)
Week 10 1. Fundamental theorem of calculus (parts 1 and 2); 2. Functions defined by integrals; 3. Natural logarithm and exponential functions Lecture and tutorial (3 hr)
Week 11 1. Integration by substitution; 2. Partial fraction decomposition; 3. Integration by parts; 4.Trigonometric integrals; 5. Reduction formulae. Lecture and tutorial (3 hr)
Week 12 1. Areas and volumes by slicing; 2. The disk and shell methods; 3. Using incremental lengths to find the arc length of a function. Lecture and tutorial (3 hr)
Week 13 Revision/further applications Lecture and tutorial (3 hr)

### Attendance and class requirements

• Lecture attendance: You are expected to attend lectures. If you do not attend lectures you should at least follow the lecture recordings available through Canvas.
• Tutorial attendance: Tutorials (one per week) start in Week 2. You should attend the tutorial given on your personal timetable. Attendance at tutorials and participation will be recorded to determine the participation mark. Your attendance will not be recorded unless you attend the tutorial in which you are enrolled. We strongly recommend you attend tutorials regularly to keep up with the material and to engage with the tutorial questions.

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 3 credit point unit, this equates to roughly 60-75 hours of student effort in total.

• Course Notes for MATH1021 Calculus of One Variable (available on Canvas)
• See the Canvas site for more reference material.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. apply mathematical logic and rigour to solve problems
• LO2. read and write basic set notation
• LO3. demonstrate competency in arithmetic operations with complex numbers in Cartesian, polar, and exponential form
• LO4. use de Moivre’s theorem to find powers and roots of complex numbers
• LO5. solve simple polynomial equations involving complex numbers
• LO6. apply an intuitive understanding of a limit and knowledge of basic limit laws to calculate the limits of functions
• LO7. use the differential of a function to calculate critical points and apply them to optimise functions of one variable
• LO8. find inverse functions
• LO9. use L’Hopital’s rule to find limits of indeterminate forms
• LO10. find Taylor polynomials and the Taylor series expansion of a function
• LO11. approximate definite integrals by finite sums and vice versa
• LO12. express areas, and volumes of revolution, as definite integrals
• LO13. apply standard integration techniques to find anti-derivatives and definite integrals
• LO14. determine properties of a function defined by an integral using the graph of its integrand
• LO15. express mathematical ideas and arguments coherently in written form.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

Small tutorial participation mark added with a consequent reduction in the weekly quiz weightings.

• Lectures: Lectures are face-to-face and streamed live with online access from Canvas.
• Tutorials: Tutorials are small classes in which you are expected to work through questions from the tutorial sheet in small groups on the white board. The role of the tutor is to provide support and to some extent give feedback on your solutions written on the board.
• Tutorial and exercise sheets: The question sheets for a given week will be available on the MATH1021 Canvas page. Solutions to tutorial exercises for week n will usually be posted on the web by the afternoon of the Friday of week n.
• Ed Discussion forum: https://edstem.org

### Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

### Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.