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@ -5,6 +5,7 @@ begin
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open_monitor*[exam::MathExam]
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section*[idir::Author,affiliation="''LRI, CentraleSupelec''",
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email="''idir.aitsadoune@centralesupelec.fr''"]
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{*Idir AIT SADOUNE*}
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@ -13,7 +14,8 @@ section*[keller::Author,affiliation="''LRI, Univ. Paris-Sud''",
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email="''Chantal.Keller@lri.fr''"]
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{*Chantal KELLER*}
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subsection*[header::Header,examSubject= "[algebra,geometry]",
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section*[header::Header,examSubject= "[analysis,geometry]",
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examTitle="''BACCALAUREAT GENERAL MATHEMATIQUES''",
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date="''21-06-2017''",
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timeAllowed="240::int"]
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@ -27,8 +29,10 @@ timeAllowed="240::int"]
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\end{itemize}
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*}
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subsubsection*[exo1 :: Exercise,Exercise.concerns= "{examiner,validator,student}",
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Exercise.content="[q1::Task,q2,q3a]"]
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subsection*[exo1 :: Exercise,
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Exercise.concerns= "{examiner,validator,student}",
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Exercise.content="[q1::Task,q2,q3a]"]
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{*
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On considère la fonction h définie sur l’intervalle [0..+\<infinity>] par : @{term "h(x) = x * exponent (-x)"}
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*}
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@ -36,7 +40,7 @@ On considère la fonction h définie sur l’intervalle [0..+\<infinity>] par :
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definition h :: "real \<Rightarrow> real"
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where "h x \<equiv> x * exp (- x)"
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subsubsection*[q1::Task, Task.concerns= "{examiner,validator,student}",
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level="oneStar", mark="1::int", type="formal"]
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{* Déterminer la limite de la fonction @{term h} en +\<infinity>. *}
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@ -47,17 +51,19 @@ text*[a1::Answer_Formal_Step]
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lemma q1 : "(h \<longlongrightarrow> (0::real)) at_top"
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sorry
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subsubsection*[v1::Validation,
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proofs="[q1::thm]"]
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{* See lemma q1 *}
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subsubsection*[q2::Task, Task.concerns= "{examiner,validator,student}",
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level="oneStar", mark="1::int", type="formal"]
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{* Étudier les variations de la fonction @{term h} sur l'intervalle [0..+\<infinity>] et dresser son tableau
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de variation *}
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text*[a2::Answer_Formal_Step]
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{* Fill in term and justification*}
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definition h' :: "real \<Rightarrow> real"
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where "h' x \<equiv> (1 - x) * exp (- x)"
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@ -73,26 +79,40 @@ proof -
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by (metis "***" left_diff_distrib mult_minus_right uminus_add_conv_diff)
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qed
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lemma q2_b : "0 \<le> x \<and> x \<le> y \<and> y \<le> 1 \<Longrightarrow> h x \<le> h y"
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sorry
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lemma q2_c : "1 \<le> x \<and> x \<le> y \<Longrightarrow> h x \<ge> h y"
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sorry
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subsubsection*[v2::Validation,
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proofs="[q2_b::thm, q2_c]"]
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{* See lemmas q2_b and q2_c *}
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subsubsection*[q3a::Task, Task.concerns= "{examiner,validator,student}",
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level="oneStar", mark="1::int", type="formal"]
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{* Vérifier que pour tout nombre réel x appartenant à l'intervalle [0..+\<infinity>], on a :
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@term{h x = (exp (- x)) - (h' x)} *}
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text*[a3a::Answer_Formal_Step]
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{* Fill in term and justification*}
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lemma q3a : "h x = (exp (- x)) - (h' x)"
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by (simp add : h_def h'_def left_diff_distrib)
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subsubsection*[v3a::Validation,
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proofs="[q3a::thm]"]
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{* See lemma q3a *}
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subsection*[sol1 :: Solution,
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Solution.content="[exo1::Exercise]",
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Solution.valids = "[v1::Validation,v2,v3a]"]
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{*
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See validations.
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*}
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text*[a3a::Answer_Formal_Step]
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{* Fill in term and justification*}
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close_monitor*[exam]
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