BAC2017: tried proofs
This commit is contained in:
parent
3b7a029d35
commit
80f92c168c
|
@ -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
|
||||
|
|
Loading…
Reference in New Issue