BSc/MEnvSci Environmental Science with an Integrated Foundation Year
Year of entry: 2023
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Course unit details:
|Unit level||Level 1|
|Teaching period(s)||Semester 1|
|Available as a free choice unit?||No|
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.
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.
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
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)
|Scheduled activity hours|
|Independent study hours|
|Colin Steele||Unit coordinator|
This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.
Please see Blackboard / course unit related emails for any further updates