School of Mathematics and Statistics (SoMaS)

## MAS170 Practical Calculus

 Semester 2, 2010/11 10 Credits Lecturer: Dr Neil Dummigan Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

In this course we learn how to define and evaluate derivatives and integrals for functions which depend on more than one variable, with an emphasis on functions of two variables, for which the main ideas already appear. We also think about what it means to approach a limit or to add up a sum with infinitely many terms, but throughout the emphasis is on explicit examples and getting answers.

Prerequisites: MAS100 (Mathematics with Maple)

The following modules have this module as a prerequisite:

 MAS173 Probability and Inference MAS174 Applications of Probability and Statistics MAS202 Advanced Calculus MAS207 Continuity and Integration

## Outline syllabus

• Inequalities.
• Limits.
• Separation of variables.
• Partial derivatives.
• Double integrals.
• Infinite series.

Thursday 4-5.

## Aims

• To introduce the basic techniques of calculus of functions of more than one variable, and to gain expertise in calculating partial derivatives and double integrals, and in using the Chain Rule.
• To install mental images of limits and the convergence of infinite series, and to become familiar with important examples and techniques.

## Learning outcomes

• manipulate inequalities;
• use standard limit formulas;
• solve separable first-order ordinary differential equations;
• differentiate functions of more than one variable, including use of the Chain Rule;
• evaluate double integrals;
• find Maclaurin series and their radii of convergence.

## Teaching methods

Lectures, problem solving

22 lectures, 10 tutorials

## Assessment

One formal two-hour written examination. Format: 4 questions from 4.

## Full syllabus

1. Inequalities
(2 lectures)

Basic properties and examples. Modulus. Triangle and arithmetic-geometric mean inequalities.
2. Limits
(4 lectures)
Idea of a limit, including at infinity. Sandwich rule, standard limit formulas, compound interest, differentiation.
3. Separation of variables
(2 lectures)
Differential equation for continuous compound interest. Solution by inspection and by separation of variables. Radioactive decay, half-life. Newton's law of cooling. Other examples of separable equations.
4. Partial derivatives
(4 lectures)
Functions of two variables, their graphs, level curves and tangent planes. Partial derivatives, their graphical interpretation and evaluation. Jacobians, higher derivatives. Increments, the Chain Rule and its applications, including to Laplace's equation.
5. Double integrals
(5 lectures)
Review of the Fundamental Theorem of Calculus. Two-dimensional integrals as volumes under graphs, their evaluation by double integration, in either order. Change of variables, including to polar coordinates. ∫−∞ e−[1/2]x2 dx.
6. Infinite series
(5 lectures)
Infinite series of positive terms. Basic examples including geometric and harmonic series. Sum as a limit of partial sums. Numerical and graphical illustration. Absolute convergence. Manipulating Maclaurin series. Finding the radius of convergence.