MATHEMATICS I B
Updated A.Y. 2019-2020
Lecture by lecture Program
Sequences: monotonic sequences. Existence of the limit for monotonic sequences.
Euler sequence: definition, properties (without proofs), Limit of the Euler sequence. Exercises.
Subsequences. Necessary and sufficent condition for the existence of the limit of a sequence.
Recursive sequences: Existence of the limit.
Again on recursive sequences: Existence of the limit and its computation. Use of the Induction Principle.
Series. Geometric, Harmonic and Euler series: conditions for convergence
Idea of limits. Four case studies: Finite limit at a point, infinite limit at a point, Finite limit at infinity, Infinite limit at infinity. Discussion of the definition on function plots.
Notable Limits and Undetermined forms.
Exercises on computations of limits for undetermined forms 0/0.
Finite limit at a point: Definition and verification of a limit using the definition.
Infinite limit at a point: Definition and verification of a limit using the definition. Vertical asymptotes.
Finite limit at infnity: Definition and verification of a limit using the definition. Horizontal asymptotes.
Exercises on computations of limits.
Infinite limits at infinity: Definition and verification of a limit using the definition.
Notable limits and Undetermined forms: exercises.
Classification of discontinuities.
Theorems on continuous functions (sum, product ratio and composition of continuous functions is continuous).
Weierstrass theorem: statement and meaning.
Intermediate value theorem: statement and meaning.
Theorem, on existence of zeros: statement and meaning. Examples and exercises.
Derivatives: Definition as limit of the incremental ratio.
Interpretation as the slope of the tangent line to the graph of a function at a given point.
Discontinuity of derivatives: angle point, cusp point, inflection point with vertical tangent.
Differentiability and continuity (A differentiable function is also continuous).
Example of a continuous function that is not differentiable.
Derivatives of elementary functions.
Operations with derivatives.
Theorems on differentiable functions: Rolle, Lagrange (statement and interpretation).
Differentiability and monotonicity.
Local extrema: local minima, local maxima, inflection points with horizontal tangent.
De L'Hopital rule
Convex and concave functions
Second-order conditions for maxima and minima
Derivative of the inverse of a function (exponential and trigonometric functions)
Taylor formula: derivation and applications.
Program (The final program will only be available at the end of the course)
Monotone sequences. Conditions for convergence of monotonic sequences. The Euler sequence. Subsequences. Recursive sequences: Existence of the limit.
Series: formal definition. Necessary condition for convergence. Convergence of armonic series. Convergence of geometric series.
Limits of real functions of one real variable: Definition and uniquness of the limit. Continuity: definition and fundamental theorems. Formal proof of the continuity of the square root and of the sin functions. Characterization of the discontinuity points of a function: jumps and removable (or type-I) and Infinite (or type-II) discontinuities.
Derivative of a real functions of one real variable. Necessary condition for derivability (continuity of the function).
Counter-examples of continuous functions that are not derivable. Continuity Theorem of a derivative of a function (the derivative of a function cannot have jump discontinuities). Example of a function with discontinuous first derivative with type-II discontinuity. Maxima and minima of a function: necessary and sufficient conditions.
Fundamental theorems on derivatives: Rolle, Lagrange. Convex and concave functions. Second-order conditions for maxima and minima. De L'Hopital rule.
Taylor formula: derivation and applications to the approximation of irrational numbers with rational numbers.
Complete study of a function: sketch graph.
Knowing and understanding the basic principles of mathematical analysis, starting from the foundations of topology. Applying those principles to econocmic modeling with a rigorous approach.
Lectures and tutorials