MATH104 - Differential and Integral Calculus

Year

Credit points

Campus offering

Prerequisites

Nil

Teaching organisation

Unit rationale, description and aim

Calculus was developed to study quantities and processes that are continuously changing and so is crucial for modelling and understanding most physical processes. To support later units in the Mathematics sequence it is important that all students have a known and relatively advanced understanding of basic calculus. The study of calculus is fundamental in all Mathematics and is a requirement of all Initial Teacher Education (ITE )Mathematics courses.

This unit builds upon a basic knowledge of basic calculus obtained in high school to provide a solid base for further study by providing a brief review and extension of those concepts: functions, limits, continuity, differentiation and integration. This unit accepts that students who have studied calculus at high school may have differing levels of knowledge. The aim of this unit is to consolidate and extend students’ knowledge and understanding and to ensure that students have, at minimum, a known level of competence with the basic ideas of calculus that may then be applied in later units in the Mathematics sequence, especially for those units. 

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Use the properties of limits and infinite series to solve simple problems (GA4, GA5, GA8) 

LO2 - Use knowledge of circular and hyperbolic functions including their identities to solve problems (GA4, GA5, GA8, GA10) 

LO3 - Apply knowledge of the derivative, including the use of first principles, to derive rules of differentiation (GA4, GA5, GA8, GA9) 

LO4 - Use the derivative to problem solving, including curve sketching, related rates, optimization and other related problems. (GA4, GA5, GA8) 

LO5 - Anti-differentiate functions and use these to solve problems (GA4, GA5, GA8, GA10) 

LO6 - Use anti-differentiation and definite integrals to problems in areas, volumes (GA4, GA5, GA8) 

Graduate attributes

GA4 - think critically and reflectively 

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA9 - demonstrate effective communication in oral and written English language and visual media 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics will include: 

• Review of functions. Circular Functions and their inverses, and Hyperbolic functions, their inverses. 
• Sequences and series. 
• Gradients, limits, tangents and normals. 
• Differentiation by rules. 
• Applications, including curve sketching, concavity, related rates, optimization, and other relevant topics. 
• Antidifferentiation and applications of antiderivatives. 
• First order linear differential equations. 
• The definite integral and Fundamental theorems of calculus. 
• Applications of integration. 
• Improper integrals. 

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology. 

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study. 

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The continuous assessment component allows for the early detection of problems a student might be having and so ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be required to submit responses to questions dealing with simple problems in calculus. As specified times, through the semester, students will submit responses to some of the questions to allow for the provision of feedback and learning support to students. The examination components ensures that students have fully integrated the learning and can bring a variety of strategies to bear under pressure. 

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit. 

Overview of assessments

Representative texts and references

Betounes, D & Betounes, M.R. (2019) Calculus: Concepts and Computation, 3rd Edition Kendall/Hurt Publications. 

Anton H, Bivens I, Davis S (2016) Calculus: Early Transcendentals, 11th Edition New York: John Wiley & Sons. 

Courant, R., Robbins, H. & Stewart, I. (1996) What is Mathematics? Oxford: Oxford University Press. 

Edwards, C.H. & Penney D.E. (2007) Calculus and early trancendentals. Prentice-Hall Inc. 

Hughes-Hallett, D. et al (2005) Calculus Single and Multivariable, New York: John Wiley & Sons 

Spivak, M. (2008) Calculus, 4th Edition, Publish or Perish 

Stein, S.K. & Barcellos, A. (1992) A Calculus and Analytic Geometry, New York: McGraw Hill 

Stewart, J. (2015) Single Variable Calculus: Volume 1 8th Edition, CENGAGE Learning Custom Publishing 

Strang, G. (2017) Calculus 3rd Edition, Wellesley-Cambridge