# BSc/MEng Materials Science with an Integrated Foundation Year / Course details

Year of entry: 2023

## Course unit details:Mathematics 0B1

Unit code MATH19801 10 Level 1 Semester 1 Department of Mathematics No

### Aims

The course unit unit aims to provide a basic course in calculus and algebra to students in Foundation studies with AS-level mathematics or equivalent.

### Learning outcomes

On completion of this unit successful students will be able to:

1 - Express a (proper or improper) rational function in terms of simpler (partial) fractions. Isolate parts of a non-rational expression

which can be turned into partial fractions form.

2 - Find information (e.g. centres, crossing-points) from the equations of straight lines and circles. On the basis of information relevant to straight-lines and circles, find the equations of straight-lines and circles.

3 - Combine, as a single trigonometric ratio multiplied by a constant, two or more expressions of the form a \cos x or b \sin x.

4 - Convert the coordinates of a point from plane-polar to cartesian (rectangular) coordinates or from cartesian to plane-polar coordinates.

5 - Carry out simple differentiation and (indefinite and definite) integration using tables of derivative and integrals.

6 - Use differentiation to locate and to classify (maximum, minimum or point of inflection) stationary points and to find the maximum or minimum value that a given function takes on a given interval.

7 - Carry out differentiation of functions using the product, quotient and chain (function of a function) rules. Carry out differentiation using implicit, logarithmic and parametric differentiation.

8 - Sketch simple curves seen previously. Sketch curves on the basis of their relation with curves sketched previously or on the basis of specific values of the function. Sketch a curve using locations of axis-crossings, stationary points and asymptotes. Sketch curves for different values of a parameter.

9 - Use definite integration to find the areas between curves or between curves and the axes.

10 - Evaluate integrals using integration by parts, integration by substitution or by re-arrangement e.g. integration using partial fractions.

11 - Write down terms in a series based on the formula for a general term. Find possible general forms for a series based on a small number of terms. Find information (specific terms, sums of terms)

on the basis of other information for arithmetic and geometric series. Expand a function of the form $(a + x)^n$ as a binomial series for negative or non-integer values of n. Determine whether a (simple) series will converge or diverge.

12 - Write down the series expansion of a function around a given point as a Maclaurin or Taylor series.

13 - Determine physical behaviour of a system by means of an derivative, integral or other quantity.

### Syllabus

Rational Functions and Partial Fractions (3 lectures)

• Simple Rational Functions (including distinction of proper / improper)
• Forms for Partial Fractions
• Techniques for finding partial fraction coefficients
• Limitations of partial fractions (combination with non-rational functions etc)

Geometry and Trigonometry (3 lectures)

• Straight Lines and Conic Sections
• Combining Trigonometric Ratios ( a cos x + b sin x = r cos (x - \alpha) etc)
• Polar Coordinates of points

Differentiation (4 lectures)

• Reminder of simple differentiation
• Stationary Points
• Product, quotient and chain rules
• Implicit, logarithmic and parametric differentiation

Curve Sketching (4 lectures)

• Some simple curves e.g. trig, exponentials,
• Functions of the form f(ax+b)
• Curve sketching by using function values
• Curve sketching using axis-crossings, stationary points and asymptotes
• Curves and a parameter.

Integration (4 lectures)

• Reminder of simple indefinite and definite integration
• Integration and areas under / between curves
• Integration by parts
• Integration by substitution
• Integration by partial fractions

Sequences and Series (4 lectures)

• The notation of series
• Arithmetic and Geometric Series
• The role of convergence
• Binomial Series
• Maclaurin and Taylor Series

### Assessment methods

Method Weight
Other 30%
Written exam 70%

Quizzes during tutorials in weeks 5, 7, 9, 11. Weighting within unit 20%

Diagnostic Followup: (week 3). Weighting within unit 10%

Examination. Weighting within unit 70%

CROFT, A & DAVISON, R. 2010. Foundation Maths (5th ed.) Pearson Education, Harlow. (ISBN9780273730767)

BOSTOCK, L., & CHANDLER, S. 1981. Mathematics - the core course for A-level. Thornes, Cheltenham. (ISBN0859503062)

BOSTOCK, L., CHANDLER, S., & ROURKE, C. 1982. Further pure mathematics. Thornes, Cheltenham. (ISBN0859501035)

### Study hours

Scheduled activity hours
Lectures 24
Tutorials 11
Independent study hours
Independent study 65

### Teaching staff

Staff member Role
Colin Steele Unit coordinator