Mathematics for Biotechnology

1. Preliminaries. Sets. Real numbers. Sets of real numbers. Upper bound and lower bound of sets of real numbers. Intervals. Distance. Functions of a real variable, graph, domain, image. Injectivity. Composition of functions. Operations on graphs: translations, symmetries. Inverse functions. Monotone functions. Maxima and minima. Sign and zeros of a function.

2. Elementary functions. Absolute values. Powers with natural exponent, integer, rational and real. Power functions and exponential functions. Logarithmic functions. Trigonometric functions. Algebraic inequalities (second degree, irrational, exponential, logarithmic). Systems of inequalities.

3. Limits and continuous functions. Distance and neighborhoods. Limits of functions. Continuity. Elementary limits. Algebra of limits. Limits of composite functions. Comparison theorems. Asymptotes: horizontal, vertical and oblique. Continuous functions and their basic properties. Theorem of zeros. Weierstrass Theorem.

4. Derivatives. Definition of derivative at a point. Tangent line to a graph. Derivatives of elementary functions. Rules of derivation of sums, products, quotients, composite finctions inverse functions. Differentiability and continuity. Relative maxima and minima. Fermat, Rolle and Lagrange theorems. Consequences of Lagrange's theorem: differentiable functions with zero derivative, differentiable functions with the same derivative, sign of the first derivative and monotonicity intervals of the function. Searching for maxima and minima by using the sign of derivatives. Second derivative, its sign and convexity. Qualitative study of the graph of a function. Higher order derivatives.

5. Integrals. Primitive functions (indefinite integrals). Elementary integrals. Definition of definite integrals. Areas. The fundamental theorem of calculus.

6. Linear algebra. Geometric vectors in R^n. Matrices with real coefficients and their properties. Linear systems in matricial form
Ax = b. Systems resolution with the Gauss method. Rank of a matrix. Determinant of square matrices. Kronecker theorem. Rouché-Capelli theorem. Inverse of a square matrix. Cramer theorem. The scalar product and its properties. Orthogonal vectors. Reference systems in E^2 and E^3. Elements of analytical geometry. Vector product in R^3.

7. Differential equations. Definitions of differential equation (in normal and non-normal form) and of order of a differential equation. Solution of a differential equation. Examples of differential equations. Cauchy's problem.

1. Preliminaries. Sets. Real numbers. Sets of real numbers. Upper bound and lower bound of sets of real numbers. Intervals. Distance. Functions of a real variable, graph, domain, image. Injectivity. Composition of functions. Operations on graphs: translations, symmetries. Inverse functions. Monotone functions. Maxima and minima. Sign and zeros of a function.

2. Elementary functions. Absolute values. Powers with natural exponent, integer, rational and real. Power functions and exponential functions. Logarithmic functions. Trigonometric functions. Algebraic inequalities (second degree, irrational, exponential, logarithmic). Systems of inequalities.

3. Limits and continuous functions. Distance and neighborhoods. Limits of functions. Continuity. Elementary limits. Algebra of limits. Limits of composite functions. Comparison theorems. Asymptotes: horizontal, vertical and oblique. Continuous functions and their basic properties. Theorem of zeros. Weierstrass Theorem.

4. Derivatives. Definition of derivative at a point. Tangent line to a graph. Derivatives of elementary functions. Rules of derivation of sums, products, quotients, composite finctions inverse functions. Differentiability and continuity. Relative maxima and minima. Fermat, Rolle and Lagrange theorems. Consequences of Lagrange's theorem: differentiable functions with zero derivative, differentiable functions with the same derivative, sign of the first derivative and monotonicity intervals of the function. Searching for maxima and minima by using the sign of derivatives. Second derivative, its sign and convexity. Qualitative study of the graph of a function. Higher order derivatives.

5. Integrals. Primitive functions (indefinite integrals). Elementary integrals. Definition of definite integrals. Areas. The fundamental theorem of calculus.

6. Linear algebra.
a) Vectors in R^n. Matrices with real coefficients and their properties. Linear systems in matricial form Ax = b. Systems resolution with the Gauss method.
b) Rank of a matrix. Rouché-Capelli theorem.
c) Invertibility of a square matrix. Determinant of square matrices. Cramer theorem.
d)The scalar product and its properties. Orthogonal vectors. Elements of analytical geometry. Vector product in R^3.

7. Differential equations. Definitions of differential equation (in normal and non-normal form) and of order of a differential equation. Solution of a differential equation. Examples of differential equations. Cauchy's problem.

Note: the lessons will be accompanied by numerous hours of exercises carried out by the teachers themselves.