• ECTS

    4 credits

  • Component

    Faculty of Science

Description

This EU is divided into two parts. 

The first one aims at consolidating the high school knowledge that is essential for the pursuit of higher studies in science: understanding proportionality and linearity, calculating with powers, manipulating fractions, and solving simple equations. 

The second part will be devoted to the study of functions of one real variable: the emphasis will be on the usual functions, the graphical representation of functions, and the mathematical notion of derivative (or instantaneous rate of increase). 

Most of the concepts discussed will be illustrated with concrete examples from biology. 

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Objectives

To provide the basic computational tools necessary for further study in the life sciences.  

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Necessary pre-requisites

second level mathematics

Recommended prerequisites*: first year math speciality

 

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Knowledge control

A continuous assessment CC grade that will take into account: 

    - participation and investment in TD. 

    - the results of two interim evaluations (on each of the two parts)

A final CT test on the entire program. 

Max rule: the score is calculated using the formula Max(CT,Average(CT,CC))

 

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Syllabus

  • Basic mathematical techniques

 

1.a) Proportionality, linearity, and their different representations : 

  • table of values, cross product, proportionality coefficient
  • graphic representation (abscissa, ordinate, slope and equation of a vector line)
  • solving the equation ax=b

Examples of illustrations: conversion between units of measurement (joules vs. kilocalories for example), voltage/intensity relationship, etc.

  • linear variations: affine concepts, y-intercept, equation y=ax+b

Illustrative examples: determination of a bacterial concentration from calibration data, conversion degrees Celsius degrees Fahrenheit. 

  • Linear regression

 

1.b) Fractions

  • what is a fraction (proportionality between integers, simplification rule, notion of PGCD)
  • calculation rules (sum and product, notion of PPCM)
  • inequalities (operations that preserve or reverse inequalities)

Examples of illustrations: concentration calculation after mixing, parallel resistance associations, diagnostic tests (sensitivity, specificity, PPV, NPV, to be compared with prevalence)

 

1.c) Powers and orders of magnitude

  • integer powers (calculation rules, definition domain for negative powers, scientific notation)
  • fractional powers and n-th root (definition domain, equation x^n=c)
  • orders of magnitude
  • geometric growth

Illustrative examples: dilution calculations, conversions (%, ‰, ppm, cubic liter-meters, etc...), order of magnitude estimation "à la Fermi", reproduction number of an infectious disease. 

 

    2) Functions of one real variable

        

2.a) The vocabulary of functions by examples

  • basic notions (function, definition set, graph, image, antecedent). Examples from part 1: affine, power and polynomial functions. 
  • the notion of bijection. Detailed study of the logarithm and exponential functions (logarithmic scale). 

Examples of illustrations: half-life times, epidemiological models, etc...

  • properties of functions and their visualization on graphs (parity, monotonicity, periodicity: trigonometric functions). 

 

2.b) Limits and continuity

  • notion of limit (examples via the usual functions already studied). 
  • general results: gendarmes theorem, comparative growths, limits of rational fractions. 
  • continuous functions (intermediate values, existence of extrema, reciprocal bijection). 

 

2.c) Local study of functions of order 1: rate of increase and derivative

  • notion of derivative as an instantaneous rate of increase. Graphical representation and equation of the tangent. 
  • Examples of illustrations: all kinds of speeds. 
  • First properties: calculation rules and table of variations. Illustration on usual functions. 
  • Search for extrema. 

Examples of illustrations: optimization problems, search for equilibrium points, etc. .... 

 

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Additional information

Hourly volumes* :

            CM :12h

            TD :21h

            TP:

            Terrain:

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