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We offer a selection of improper integral exercises with detailed answers. Our main objective is to show you how to prove that an improper integral is convergent: how to employ the integration by parts; how to make a change of variables; how to apply the dominated convergence theorem; and how to integrate terms by terms. … Read more

We propose exercises on functions defined by integrals. We mainly prove the regularity of such a function, like continuity and differentiability. These functions can also be considered parameter-dependent integrals. In many cases, we deal with a function $f(x,t)$ of two variables and we take the integral of this function with respect to one of its … Read more

We propose simple techniques to learn you how to find the limit of a function. We first recall what a limit of a function is. Then give some properties of the limit as well as several examples in the form of exercises with detailed solutions. Generalities on limits of functions. let $a$ a real number … Read more

We offer practical tips for learning basic to advanced math. Above all, mathematics is a difficult and abstract subject, while a part of some students feels very good in this area. Some tips for learning basic to advanced math Mathematics is not created for itself, but to help other sciences such as physics, biology, chemistry, … Read more

Parameter-dependent integral is studied in this article. We propose a selection of exercises on continuity, limit, and differentiability of such parametric integrals. Some known transformations in mathematics such as Fourier transform and Laplace transform are parametric integrals. What is a parametric integral?  Suppose we have a function of two variables $f:I\times J\to\mathbb{R}$ where $I$ and … Read more

In this article, we offer exercises on convex functions. In fact, our objectives are: to be able to show that a function is convex; to exploit the convexity to show inequality; to use the derivatives to show inequality; to exploit the Cauchy-Schwarz inequality to show inequalities. What is a convex function? A function $f:I\mapsto \mathbb{R}$ … Read more

Recurring sequences are generally involved in the modeling of discrete systems. In fact, such sequences replace the differential equation by discretizing the time variable. To well understand the material of this page, properties of convergent sequences are needed. Facts about recurring sequences Definition: Let $I$ be an interval of $\mathbb{R}$ and $f$ a function on … Read more

We discuss the convergence of sequences and how to calculate the limit of a sequence. This subject is fundamental in real analysis because many proofs of theorems rely on the convergence of an appropriate sequence. We also offer several exercises with detailed solutions to make good use of the material in this course. This course … Read more

We offer exercises on differential calculus with detailed answers. In fact, our objectives are as follows: knowing how to calculate the derivative of a function according to a vector; calculating the differential of a function at a point; and showing that a function is continuously differentiable. We recall that differential calculus is very important in … Read more