Séance Contact 13, Lundi 18 déc
Communications:
Aujourd'hui:
Exercice 1:
Le rayon de convergence de la série entière
\(\displaystyle\sum_{k=1}^{\infty}(k+\tfrac{1}{k})x^k\) vaut
- \(+\infty\)
- \(0\)
- \(1\)
- \(\frac{1}{2}\)
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
- \(\frac{e^2-1}{2}\)
- \(\frac{e+1}{2}\)
- \(\frac{e^2+1}{2}\)
- \(\frac{e-1}{2}\)
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
- \(4\leqslant I\leqslant 40\)
- \(I<0\)
- \(0\leqslant I\leqslant 4\)
- \(I\geqslant 40\)
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
- \(\frac{2}{3}\)
- \(\frac{\sqrt\pi}{2}\)
- \(\frac{\pi^{3/2}}{12}\)
- \(\frac{\pi^{3/2}}{8}\)
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
- \(f'(1)=e-\frac{1}{e}\)
- \(f'(1)=5e-\frac{3}{e}\)
- \(f'(1)=4e\)
- \(f'(1)=8e\)
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:
Réponse à la même question, 15 décembre 2023
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:
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.