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BAC2017: tried proofs

This commit is contained in:
Chantal Keller 2018-06-06 08:37:06 +02:00
parent 3b7a029d35
commit 80f92c168c
1 changed files with 37 additions and 8 deletions
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45
examples/math_exam/BAC2017.thy
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@ -9,6 +9,10 @@ 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*}
subsection*[header::Header,examSubject= "[algebra,geometry]",
examTitle="''BACCALAUREAT GENERAL MATHEMATIQUES''",
date="''21-06-2017''",
@ -24,7 +28,7 @@ timeAllowed="240::int"]
*}
subsubsection*[exo1 :: Exercise,Exercise.concerns= "{examiner,validator,student}",
Exercise.content="[q1::Task]"]
Exercise.content="[q1::Task,q2,q3a]"]
{*
On considère la fonction h définie sur l’intervalle [0..+\<infinity>] par : @{term "h(x) = x * exponent (-x)"}
*}
@ -38,7 +42,7 @@ level="oneStar", mark="1::int", type="formal"]
{* Déterminer la limite de la fonction @{term h} en +\<infinity>. *}
text*[a1::Answer_Formal_Step]
{* First Step: Fill in term and justification*}
{* Fill in term and justification*}
lemma q1 : "(h \<longlongrightarrow> (0::real)) at_top"
sorry
@ -47,25 +51,50 @@ lemma q1 : "(h \<longlongrightarrow> (0::real)) at_top"
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} *}
find_theorems exp
de variation *}
text*[a2::Answer_Formal_Step]
{* First Step: Fill in term and justification*}
{* Fill in term and justification*}
lemma q2_a : "DERIV h x :> (1 - x) * exp (- x)"
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_minus has_derivative_compose Transcendental.DERIV_exp)
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)
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*[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)} *}
lemma q3a : "h x = (exp (- x)) - (h' x)"
by (simp add : h_def h'_def left_diff_distrib)
text*[a3a::Answer_Formal_Step]
{* Fill in term and justification*}
close_monitor*[exam]
end