The aim of the course is to provide an introduction to the Differential Calculus, studying the methods and some of the applications of the subject. The course starts from the beginning of the subject, but it is advantageous to be familiar with the material on Calculus contained in the Scottish Highers syllabus or equivalent.
By the end of the course the student should:
- be familiar with the basic mathematical languages, such as sets and functions, the basic logic underpinning mathematical reasoning, and methods of proof.
- be familiar with the concepts of limits. Understand their usage for defining continuity.
- be familiar with the concept of the derivative of a function and its geometrical significance. Know the basic rules of differentiation and be able to apply them to find first and higher derivatives of functions.
- know the elementary properties of the trigonometric functions, the inverse trigonometric functions, the exponential and logarithmic functions. Be able to differentiate expressions involving these functions.
- be able to find the equation of a tangent to a curve given explicitly or implicitly. Know the significance of the first and second derivatives for the shapes of graphs.
- know about critical points of differentiable functions and their use in determining maxima and minima. Be able to apply these ideas to simple problems in optimization and curve sketching.