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This document comprises a collection of past exam papers for a university Mathematics course taught by Prof. Elisabetta Ulivi, spanning from 2004 to 2013. The exams consistently feature four types of problems designed to test fundamental calculus and analysis skills. Students are typically required to perform detailed function analysis and graph sketching, calculate complex limits, evaluate various types of integrals (definite and indefinite), and solve differential equations. This resource is ideal for students preparing for similar assessments in mathematical analysis, providing diverse examples of typical university-level problems in these areas. Key topics include logarithmic, exponential, trigonometric, and rational functions, as well as first- and second-order differential equations.
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This compilation of past examination papers for a university Mathematics course, curated by Prof. Elisabetta Ulivi, provides a comprehensive overview of typical assessment formats from December 2004 to February 2013. Each exam presents four core problems designed to test students' understanding and application of fundamental mathematical concepts.
The recurring problem types include:
The recurring problem types include:
- Function Analysis and Graphing: Students are asked to "Studiare la seguente funzione e tracciarne il grafico." This involves:
- Determining domain, limits, and asymptotes.
- Calculating first and second derivatives for monotonicity, extrema, concavity, and inflection points.
- Sketching the graph.
- Functions often include logarithmic, exponential, rational, radical, and trigonometric expressions.
- Limits, Continuity, and Differentiability: Problems like "Calcolare il limite," "Studiare la continuità," or "Studiare la derivabilità" are frequent. These focus on:
- Calculating limits of complex expressions, including indeterminate forms.
- Analyzing continuity of piecewise functions or at specific points.
- Investigating differentiability, especially where definitions change.
- Integral Calculus: "Calcolare l'integrale" problems require evaluating definite and indefinite integrals. Proficiency in various techniques is tested, such as:
- Substitution and integration by parts.
- Integration of rational, trigonometric, and exponential functions.
- Definite integrals with specified limits.
- Differential Equations: "Risolvere l'equazione differenziale" tasks typically involve:
- First-order differential equations (separable, linear).
- Second-order linear differential equations with constant coefficients (homogeneous and non-homogeneous).
- Equations may contain trigonometric, exponential, and polynomial terms.
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