Program Overview

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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.

Unit Details and Rules

• Academic unit: Mathematics and Statistics Academic Operations
• Credit points: 3
• Prerequisites: None
• Corequisites: None
• Prohibitions: MATH1011 or MATH1901 or MATH1906 or ENVX1001 or MATH1001 or MATH1921 or MATH1931
• Assumed knowledge: HSC Mathematics Extension 1 or equivalent
• Available to study abroad and exchange students: Yes

Teaching Staff

• Coordinator: Mary Myerscough
• Lecturer(s): Mary Myerscough
• Tutor(s): Nathan Brownlowe

Assessment

• Type:
  • Final exam (Record+): 65%
  • Assignment: 4%
  • Online task: 15%
  • Assignment: 8%
  • Webwork Online Quizzes: 8%
• Details:
  • Final exam: multiple choice and written calculations, 1.5 hours
  • Assignment 1: written calculations, due Week 04
  • Quiz: multiple choice or written calculations, due Week 08
  • Assignment 2: written calculations, due Week 11
  • Webwork Online Quizzes: online task, weekly

Assessment Summary

• Assignments: two assignments, each submitted electronically as a single typeset or scanned PDF file via Canvas
• Quiz: one quiz, held online through Canvas, 40 minutes
• Webwork Online Quizzes: ten weekly online quizzes, each consisting of a set of randomized questions
• Final Exam: one examination during the examination period at the end of Semester

Learning Support

• Simple extensions: available for five calendar days through a simple extension application process
• Special consideration: available for longer periods of time or for essential commitments that impact performance in an assessment
• Using AI responsibly: guidance on the use of generative AI tools to support learning

Weekly Schedule

Week	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)	LO2 LO3
Week 02	1. Complex numbers in polar form; 2. De Moivre’s theorem; 3. Complex powers and nth roots	Lecture and tutorial (3 hr)	LO3 LO4 LO5
Week 03	1. Definition of e^iθ and e^z for z complex; 2. Applications to trigonometry; 3. Revision of domain and range of a function	Lecture and tutorial (3 hr)	LO5
Week 04	1. Limits and continuity; 2. Vertical and horizontal asymptotes	Lecture and tutorial (3 hr)	LO6
Week 05	1. Differentiation and the chain rule; 2. Implicit differentiation; 3. Hyperbolic and inverse functions	Lecture and tutorial (3 hr)	LO8
Week 06	1. Optimising and sketching functions of one variable; 2. Linear approximations and differentials; 3. L’Hopital’s rule	Lecture and tutorial (3 hr)	LO7 LO9
Week 07	1. Taylor polynomials; 2. The remainder term	Lecture and tutorial (3 hr)	LO10
Week 08	Taylor series	Lecture and tutorial (3 hr)	LO10
Week 09	1. Riemann sums; 2. Definition of definite integral; 3. Non-positive functions	Lecture and tutorial (3 hr)	LO11 LO12
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)	LO14
Week 11	1. Integration by substitution; 2. Integration by parts; 3. Trigonometric substitutions	Lecture and tutorial (3 hr)	LO13
Week 12	1. Areas and volumes by slicing; 2. The disk and shell methods	Lecture and tutorial (3 hr)	LO12
Week 13	Revision/further applications	Lecture and tutorial (3 hr)	LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14

Learning Outcomes

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

Graduate Qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course.

• GQ1: Depth of disciplinary expertise
• GQ2: Critical thinking and problem solving
• GQ3: Oral and written communication
• GQ4: Information and digital literacy
• GQ5: Inventiveness
• GQ6: Cultural competence
• GQ7: Interdisciplinary effectiveness
• GQ8: Integrated professional, ethical, and personal identity
• GQ9: Influence

Outcome Map

Learning outcomes are aligned with the University's graduate qualities and are assessed as part of the curriculum.

• GQ1: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ2: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ3: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ4: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ5: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ6: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ7: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ8: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14
• GQ9: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10 LO11 LO12 LO13 LO14

Responding to Student Feedback

This section outlines changes made to this unit following staff and student reviews. No changes have been made since this unit was last offered.

Additional Information

• Science student portal
• Mathematics and Statistics student portal
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• The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.