## MATH1010 Mathematics 1

### 25 creditsClass Size: 290

Module manager: Dr Philip Walker; Dr Kevin Houston
Email: k.houston@leeds.ac.uk; P.Walker@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2019/20

### Pre-requisite qualifications

Grade B in A-level Mathematics or equivalent.

### This module is mutually exclusive with

 MATH1050 Calculus and Mathematical Analysis MATH1055 Numbers and Vectors MATH1060 Introductory Linear Algebra MATH1331 Linear Algebra with Applications MATH1400 Modelling with Differential Equations

This module is not approved as a discovery module

### Module summary

This module introduces students to several fundamental topics of mathematics. Calculus is an essential tool in many areas of mathematics. This module consolidates basic calculus material from A-level, extends the syllabus to include more advanced techniques, and introduces elements of the analysis required to formalise the subject. These techniques lead to methods for solving simple ordinary differential equations. Linear algebra provides a basis for wide areas of mathematics and this module provides the essential foundation for this topic. Students will complement theoretical work with projects and assignments using a mathematical programming package.

### Objectives

On completion of this module, students should:
- be able to differentiate functions of one variable and determine the location and nature of turning points;
- be able to compute the Taylor series of functions of one variable;
- be comfortable with the calculus of several variables;
- be able to use a variety of methods to integrate simple functions;
- be aware of the analytical basis of calculus as expressed in rigorous definitions and theorems such as the Fundamental Theorem of Calculus;
- be able to solve systems of equations by row reduction;
- be able to manipulate matrices and vectors and understand their basic properties;
- appreciate the value and limitations of computational methods, and be able to perform simple computational tasks using a mathematical programming package;
- have demonstrated problem solving and modelling, communication, and group-working skills.

### Syllabus

- Functions and their inverses. Continuity and discontinuity. Graphs of functions.
- Differentiation. Calculations from first principles. Non-differentiability.
- Chain rule, product rule, extrema, Taylor series.
- Intermediate value, Rolle's and Mean Theorems.
- Functions of several variables.
- Partial derivatives, directional derivatives, multivariable chain rule.
- Stationary points of functions of two variables. Conditions for a stationary point. Criteria for extrema. Lagrange multipliers.
- Gradients of scalar functions. Tangent planes.
- Implicit differentiation. Change of variables. Solution of exact equations.
- Integration. Areas under curves. Riemann integration. Calculations from first principles.
- Definite and indefinite integrals. Integration techniques.
- Fundamental theorem of the calculus.
- Systems of equations. Gaussian elimination. Echelon form.
- Vectors and matrices. Inverses. Transposes.
- Determinants. Computation. Cramer's rule.
- Introduction to a mathematical programming package as a tool for numerical, graphical and symbolic computation.
- Vectors in 2 and 3 dimensions. Dot and cross products. Geometrical interpretation.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Workshop 8 1.00 8.00 Lectures 55 1.00 55.00 Tutorial 11 1.00 11.00 Private study hours 176.00 Total Contact hours 74.00 Total hours (100hr per 10 credits) 250.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Weekly tutorials. Examples sheets marked and returned with feedback.

!!! In order to pass the module, students must pass the examination. !!!

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment Written Work Example sheets and project work 20.00 Total percentage (Assessment Coursework) 20.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 80.00 Total percentage (Assessment Exams) 80.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated