Séance Contact 13, Lundi 18 déc

Exercice 1: Le rayon de convergence de la série entière \(\displaystyle\sum_{k=1}^{\infty}(k+\tfrac{1}{k})x^k\) vaut

On utilise la version ''d'Alembert'' du critère vu dans le cours: \[ \sigma =\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right| =\lim_{k\to\infty}\left|\frac{(k+1)+\frac{1}{k+1}}{k+\frac{1}{k}}\right| =1\,, \] donc le rayon de convergence est égal à \(R=\frac{1}{\sigma}=1\).

Exercice 2: Vrai ou faux? Si \(f\colon[a,b]\to\mathbb{R}\) est intégrable, alors elle est continue en tout point \(x_0\in]a,b[\).

C'est faux. Une fonction peut être intégrable tout en ayant des discontinuités (voir par exemple la deuxième partie de l'Exercice 13-05).

Exercice 3: Vrai ou faux? Soit \(f\colon[0,1]\to \mathbb{R}\) une fonction continue. Alors \[ \int_0^1\Big( f(x)-f(1-x)\Big)\,dx=0\,. \]

C'est vrai, puisque par le changement de variable \(u:= 1-x\), on peut écrire \[ \int_0^1 f(1-x)\,dx =\int_1^0 f(u)(-1)\,dx=\int_0^1 f(u)\,du\,. \]

Exercice 4: L'intégrale \(\displaystyle\int_{1}^{e}2x\log(x)\,dx\) vaut

Après une intégration par parties, on trouve que la réponse correcte est \(\frac{e^2+1}{2}\).

Exercice 5: Soit \(\displaystyle I=\int_0^4 e^{\sqrt x}\,dx\). Alors

On peut répondre sans connaître la primitive de \(e^{\sqrt{x}}\). En effet, on peut remarquer que pour tout \(x\in [0,4]\), \[ 1\leqslant e^{\sqrt{x}}\leqslant e^{\sqrt{4}}=e^2=2.718\cdots^2 \lt 10 \,, \] et donc \[ 1\cdot(4-0)\leqslant \int_0^4 e^{\sqrt x}\,dx \leqslant 10(4-0)\,. \] Donc la réponse correcte est: \(4\leqslant I\leqslant 40\)

Exercice 6: L'intégrale \(\displaystyle I=\int_{0}^{1} \frac{\sqrt{\arctan(x)}}{x^2+1}\,dx \) vaut

En posant \(u=\arctan(x)\), on a ''\(du=\frac{1}{x^2+1}\,dx\)'', et donc \[ \int_{0}^{1} \frac{\sqrt{\arctan(x)}}{x^2+1}\,dx = \int_{0}^{\pi/4} \sqrt{u}\,du=\frac{\pi^{3/2}}{12}\,. \]

Exercice 7: Soit \(f:\mathbb{R}\to\mathbb{R}\) définie par \[ f(x)=\int_{-x^3}^{x^5}e^{t^4}\,dt \] Alors

Puisque \(g(t)=e^{t^4}\) est continue, elle possède une primitive, que l'on note \(G\). On a donc, par le Théorème Fondamental de l'analyse, \[ f(x)=G(x^5)-G(-x^3)\,, \] et donc \[ f'(x)=g(x^5)(5x^4)-g(-x^3)(-3x^2)\,, \] qui donne \(f'(1)=8e\).

chatGPT, décembre 2022:

chatGPT: The role of educators, including math teachers, is unlikely to become obsolete. While technology, including artificial intelligence and educational tools, continues to advance, the role of teachers remains crucial in fostering critical thinking, problem-solving skills, and a deeper understanding of subjects.

Here are a few reasons why math teachers are likely to remain essential:

  1. Guidance and Support: Teachers provide guidance and support tailored to individual students. They can address students' unique learning styles, strengths, and challenges, offering personalized assistance.
  2. Motivation and Engagement: Teachers play a key role in motivating and engaging students. They create a positive learning environment, encourage curiosity, and inspire a love for learning that goes beyond rote memorization.
  3. Social and Emotional Learning: Teachers contribute to students' social and emotional development. They help students navigate challenges, build resilience, and develop interpersonal skills, which are essential for success in various aspects of life.
  4. Adaptability: While technology can assist in delivering content and assessing certain skills, teachers bring adaptability and creativity to the learning process. They can adjust their teaching methods based on real-time feedback and the evolving needs of their students.
  5. Facilitating Collaboration: Teachers facilitate collaborative learning experiences, encouraging students to work together, discuss ideas, and learn from one another. These collaborative skills are valuable in the workplace and beyond.
  6. Instilling Values: Beyond academic knowledge, teachers play a role in instilling values, ethics, and a sense of responsibility in students. These qualities are essential for well-rounded individuals.
While technology can enhance education and provide additional resources, the human element of teaching, which includes mentorship, inspiration, and emotional support, remains irreplaceable. Teachers who integrate technology into their teaching practices can leverage its benefits to enhance the learning experience.

As education evolves, the role of teachers may adapt, incorporating new tools and methodologies, but the core value of educators in shaping the next generation's minds is likely to persist.