## Module Content

The module covers
• Functions and Graphs (8 lectures),
• Differentiation (8 lectures),
• Max, Min and Related Rate Problems (8 lectures),
We will give applications of the mathematics to population dynamics, applied optimisation and data visualisation.

## Module Coordinates

• Lecturer: Angela Carnevale
• Lectures: Monday at 1 in IT125 and Tuesday at 10 in Tyndall
• Tutorials: The tutorials for MA133 will take place on Mondays at noon in AC213. The tutorials for MA160 will take place on Thursdays at 1pm in MY126, Classroom 3.
• Recomended text: Notes for the course will be posted online throughout the semester. I also recommend Chapters 1-4 of "Calculus: Early Transcendentals" by Stewart (available at MAIN LIB 515 STE) as a supplementary reference.
• Problem sheet: available here.
• Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard MA133/MA160 Calculus pages. Blackboard will also be used for announcements and for posting grades.

## Module Assessment

The assessment consists of a 2 hour exam (covering both algebra and calculus). The continuous assessment for this course is done as part of the module MA131. The continuous assessment for MA133 comprises 50% of the mark for MA131 (the other 50% is the CA for MA135 whch takes place in semester 2.

The continuous assessment is submitted online. A link will be posted here and on Blackboard when that site is available.

## Supplementary Material and News

Here you can find a model exam paper.

## Lecture Notes

 Lecture Notes (click on number) Lecture Summaries 1 Introduction to the course. Functions and the various ways to represent them. Linear functions. 2 Quadratic functions, their graphs and applications. Polynomials. 3 Rational and algebraic functions: domain and graph. Examples. 4 Sketching graphs of new functions from old functions. Asymptotic behaviour of functions: an introduction to limits at infinity. Examples. 5 Weekly recap. Practical rules for limits at infinity of rational functions. Trigonometric functions: definition, graphs and elementary properties of cosine, sine and tangent functions. 6 Trigonometric functions: an application. Composition of functions, absolute value and exercises. 7 Weekly recap. Exponential functions: definition, properties and examples. Applications to population growth. 8 Limits of functions, vertical asymptotes. Limit laws and examples. 9 Rate of change: average velocity and instant velocity. Definition of derivative at a point and examples. 10 Geometric interpretation of the derivative: slope of secants and tangents. The derivative as a function. Differentiation by rule: derivatives of powers and polynomials. 11 Recap on derivatives: definition, geometric interpretation, differentiation by rules. How to relate properties of the derivative to properties of the original function. You can practice with this puzzle. Derivative of the exponential function and special limits. Derivatives of products and quotients of functions. 12 Derivatives of trigonometric functions and special limits. Examples. Introduction to second derivatives and acceleration. Derivatives of compositions of functions: the chain rule. 13 Recap on differentiation by rule and examples. One-to-one functions and inverse functions. 14 Natural logarithm and logarithm in base a: definition, laws, examples. The natural logarithm as a function. Derivative of the natural logarithm and examples. 15 Weekly recap. Derivatives in use: Newton's Law of Cooling, marginal costs, velocity and acceleration. Examples. 16 Applications of differentiation: maximum and minimum values. Definitions, examples. The Extreme Value Theorem. 17 Fermat's Theorem and extreme values. Concavity of a function and sign of second derivative. Second derivative test for local extreme values. 18 Weekly recap, problems on (local) extreme values. Mean Value Theorem: statement, geometric interpretation and examples. 19 L'Hopital's Rule: statement and examples. Summary of graph sketching. Example. 20 Optimisation problems: examples. 21 Revision: functions and their graphs. Problems. 22 Revision: derivatives and limits. L'Hopital's rule. Problems. 23 Revision problems. 24