121 lines
3.6 KiB
Plaintext
121 lines
3.6 KiB
Plaintext
theory BAC2017
|
||
imports "../../ontologies/mathex_onto"
|
||
Deriv Transcendental
|
||
begin
|
||
|
||
open_monitor*[exam::MathExam]
|
||
|
||
|
||
section*[idir::Author,affiliation="''CentraleSupelec''",
|
||
email="''idir.aitsadoune@centralesupelec.fr''"]
|
||
{*Idir AIT SADOUNE*}
|
||
|
||
section*[keller::Author,affiliation="''LRI, Univ. Paris-Sud''",
|
||
email="''Chantal.Keller@lri.fr''"]
|
||
{*Chantal KELLER*}
|
||
|
||
|
||
section*[header::Header,examSubject= "[analysis,geometry]",
|
||
examTitle="''BACCALAUREAT GENERAL MATHEMATIQUES''",
|
||
date="''21-06-2017''",
|
||
timeAllowed="240::int"]
|
||
{*
|
||
\begin{itemize}
|
||
\item Les calculatrices électroniques de poche sont autorisées, conformément à la réglementation en vigueur.
|
||
\item Le sujet est composé de 4 exercices indépendants.
|
||
\item Le candidat doit traiter tous les exercices.
|
||
\item Le candidat est invité à faire figurer sur la copie toute trace de recherche, même incomplète ou non fructueuse, qu’il aura développée.
|
||
\item Il est rappelé que la qualité de la rédaction, la clarté et la précision des raisonnements entreront pour une part importante dans l’appréciation des copies.
|
||
\end{itemize}
|
||
*}
|
||
|
||
|
||
subsection*[exo1 :: Exercise,
|
||
Exercise.concerns= "{examiner,validator,student}",
|
||
Exercise.content="[q1::Task,q2,q3a]"]
|
||
{*
|
||
On considère la fonction h définie sur l’intervalle [0..+\<infinity>] par : @{term "h(x) = x * exponent (-x)"}
|
||
*}
|
||
|
||
definition h :: "real \<Rightarrow> real"
|
||
where "h x \<equiv> x * exp (- x)"
|
||
|
||
|
||
subsubsection*[q1::Task, Task.concerns= "{examiner,validator,student}",
|
||
level="oneStar", mark="1::int", type="formal"]
|
||
{* Déterminer la limite de la fonction @{term h} en +\<infinity>. *}
|
||
|
||
text*[a1::Answer_Formal_Step]
|
||
{* Fill in term and justification*}
|
||
|
||
lemma q1 : "(h \<longlongrightarrow> (0::real)) at_top"
|
||
sorry
|
||
|
||
subsubsection*[v1::Validation,
|
||
proofs="[q1::thm]"]
|
||
{* See lemma q1 *}
|
||
|
||
|
||
subsubsection*[q2::Task, Task.concerns= "{examiner,validator,student}",
|
||
level="oneStar", mark="1::int", type="formal"]
|
||
{* Étudier les variations de la fonction @{term h} sur l'intervalle [0..+\<infinity>] et dresser son tableau
|
||
de variation *}
|
||
|
||
text*[a2::Answer_Formal_Step]
|
||
{* Fill in term and justification*}
|
||
|
||
definition h' :: "real \<Rightarrow> real"
|
||
where "h' x \<equiv> (1 - x) * exp (- x)"
|
||
|
||
lemma q2_a : "DERIV h x :> h' x"
|
||
proof -
|
||
have * : "DERIV (exp \<circ> uminus) x :> - (exp (-x))"
|
||
by (simp add: has_derivative_compose)
|
||
have ** : "DERIV id x :> 1"
|
||
by (metis DERIV_ident eq_id_iff)
|
||
have *** : "DERIV h x :> x * (- (exp (- x))) + 1 * (exp (- x))"
|
||
by (simp add: * ** has_derivative_mult comp_def)
|
||
show ?thesis
|
||
by (metis "***" left_diff_distrib mult_minus_right uminus_add_conv_diff)
|
||
qed
|
||
|
||
lemma q2_b : "0 \<le> x \<and> x \<le> y \<and> y \<le> 1 \<Longrightarrow> h x \<le> h y"
|
||
sorry
|
||
|
||
lemma q2_c : "1 \<le> x \<and> x \<le> y \<Longrightarrow> h x \<ge> h y"
|
||
sorry
|
||
|
||
subsubsection*[v2::Validation,
|
||
proofs="[q2_b::thm, q2_c]"]
|
||
{* See lemmas q2_b and q2_c *}
|
||
|
||
|
||
subsubsection*[q3a::Task, Task.concerns= "{examiner,validator,student}",
|
||
level="oneStar", mark="1::int", type="formal"]
|
||
{* Vérifier que pour tout nombre réel x appartenant à l'intervalle [0..+\<infinity>], on a :
|
||
@term{h x = (exp (- x)) - (h' x)} *}
|
||
|
||
text*[a3a::Answer_Formal_Step]
|
||
{* Fill in term and justification*}
|
||
|
||
lemma q3a : "h x = (exp (- x)) - (h' x)"
|
||
by (simp add : h_def h'_def left_diff_distrib)
|
||
|
||
subsubsection*[v3a::Validation,
|
||
proofs="[q3a::thm]"]
|
||
{* See lemma q3a *}
|
||
|
||
|
||
subsection*[sol1 :: Solution,
|
||
Solution.content="[exo1::Exercise]",
|
||
Solution.valids = "[v1::Validation,v2,v3a]"]
|
||
{*
|
||
See validations.
|
||
*}
|
||
|
||
|
||
|
||
close_monitor*[exam]
|
||
|
||
end
|