Calculation methods

This UE is divided into two parts. 

The first is designed to consolidate the skills acquired in secondary school that are essential for higher studies in the sciences: 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 covered will be illustrated with concrete examples from biology. 

Provide the basic computational tools needed to pursue studies in the life sciences.  

second-level mathematics

Recommended prerequisites*: first-year maths speciality

A continuous assessment CC grade that takes 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))

• 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 equation ax=b

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

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

Illustrative examples: determining bacterial concentration from calibration data, converting degrees Celsius to degrees Fahrenheit. 

• Linear regression

1.b) Fractions

• what is a fraction (proportionality ratio 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 range 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) Function vocabulary through examples

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

Illustrative examples: half-life times, epidemiological models, allometry.

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

2.b) Limits and application

• notion of limit (examples using common functions already studied). 
• general results: Gendarmes theorem, comparative growths, limits of rational fractions. 
• continuity of standard functions.

Illustrative examples: predictive use of a functional model, Verhulst model and load capacity.

2.c) Rate of increase and derivative number

• notion of derivative number as instantaneous rate of increase. Graphical representation and equation of the tangent. 
• instantaneous speed

Examples of illustrations: all kinds of speeds. 

N.B. The calculation of derivative functions is part of the optional UE in the second semester.

Hourly volumes* :

            CM :12h

            TD :21h

            TP :

            Terrain :