Exercise Sheets and Solutions

Exercise sheets should brought to classes. These may be downloaded and printed from the links below. They also appear in printed form at the end of the notes available for purchase from Kopystop. Solutions to exercises will be made available at the end of the relevant week.

• Week 2: Exercises - Solutions
• Week 3: Exercises - Solutions
• Week 4: Exercises - Solutions
• Week 5: Exercises - Solutions
• Week 6: Exercises - Solutions
• Week 7: (First Quiz held in tutorials) Exercises - Solutions
• Week 8: Exercises - Solutions
• Week 9: Exercises - Solutions
• Week 10: Exercises - Solutions
• Week 11: Exercises - Solutions
• Week 12: (Second Quiz held in tutorials) Exercises - Solutions (not yet available)
• Week 13: Exercises - Solutions (not yet available)

Additional Materials Related to Lectures

All lecture notes and some external links related to lectures are posted here.

Aims and Learning Outcomes

The unit of study MATH1901 is the beginning of a natural hierarchy, the first continuation being MATH1903 (Integral Calculus and Modelling) in second semester, followed in second year by MATH2961 and MATH2962, which also build upon MATH1902 (Linear Algebra).

The unit of study begins with an introduction to the field of complex numbers, where, for example, every quadratic equation has two roots or a double root and every number except zero has a logarithm. All of modern mathematics and its applications to physics and engineering would grind to a screeching halt without complex numbers.

Next, the familiar notation for functions gets an upgrade. Then the main theme of the unit is introduced, namely, the concept of limit. Limits will be treated rigorously (the epsilon-delta definition), although enough rules will be developed to allow the student to calculate and manipulate limits without having to reach for the epsilons and deltas except on special occasions.

With the concept of limit in place, one can now start endowing functions with nice properties such as continuity or differentiability, and derive consequences of these properties. In particular, a strong form of l'Hopital's rule for calculating certain types of limits will be proved. You are now in the land of differential calculus.

The later part of the unit deals with the powerful theory of Taylor polynomials and the extension of continuity and differentiability to functions of two or more variables and their graphs in space, borrowing some vector theory from MATH1902.

By the end of the semester, students should be able to

• apply mathematical logic and rigour to solving problems and express mathematical ideas coherently in written and oral form;
• demonstrate fluency in manipulating complex numbers, functions of one or more variables, inverse functions, limits, derivatives, maxima and minima, and polynomial approximations;
• understand and know how to use the theorems that apply specifically to continuous functions (intermediate value theorem, extreme value theorem) and to differentiable functions (chain rule, Rolle's theorem, mean value theorem, l'Hopital's rule, Taylor's formula with remainder);
• understand the differential calculus of functions of two or more variables, continuity, partial differentiation, directional differentiation, full differentiability, chain rule, implicit differentiation;
• be able to represent surfaces in space by contour lines (level curves) in the plane, and be able to construct tangent planes on surfaces as well as tangent lines on surfaces in given directions and in the steepest direction.

Assessment Information

Information relating to the assessments for this course will appear here when it becomes available.

• Quiz 1. Held in tutorials on Monday of Week 7.
  • Practice Questions and Solutions
• Assignment. Due in Week 10 (by 4 pm Thursday 16 May)
  • Click here to download the Cover Sheet
  • Assignment Questions
  • Solutions (not yet available)
• Quiz 2. Held in tutorials in Week 12.
  • Practice Questions and Solutions

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