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Require Import Reals.
Axiom y : R -> R.
Axiom d_y : derivable y.
Axiom n_y : forall x : R, y x <> 0%R.
Axiom dy_0 : derive_pt y 0 (d_y 0%R) = 1%R.
Lemma essai0 : continuity_pt (fun x : R => ((x + 2) / y x + x / y x)%R) 0.
assert (H := d_y).
assert (H0 := n_y).
reg.
Qed.
Lemma essai1 : derivable_pt (fun x : R => (/ 2 * sin x)%R) 1.
reg.
Qed.
Lemma essai2 : continuity (fun x : R => (Rsqr x * cos (x * x) + x)%R).
reg.
Qed.
Lemma essai3 : derivable_pt (fun x : R => (x * (Rsqr x + 3))%R) 0.
reg.
Qed.
Lemma essai4 : derivable (fun x : R => ((x + x) * sin x)%R).
reg.
Qed.
Lemma essai5 : derivable (fun x : R => (1 + sin (2 * x + 3) * cos (cos x))%R).
reg.
Qed.
Lemma essai6 : derivable (fun x : R => cos (x + 3)).
reg.
Qed.
Lemma essai7 :
derivable_pt (fun x : R => (cos (/ sqrt x) * Rsqr (sin x + 1))%R) 1.
reg.
apply Rlt_0_1.
red in |- *; intro; rewrite sqrt_1 in H; assert (H0 := R1_neq_R0); elim H0;
assumption.
Qed.
Lemma essai8 : derivable_pt (fun x : R => sqrt (Rsqr x + sin x + 1)) 0.
reg.
rewrite sin_0.
rewrite Rsqr_0.
replace (0 + 0 + 1)%R with 1%R; [ apply Rlt_0_1 | ring ].
Qed.
Lemma essai9 : derivable_pt (id + sin) 1.
reg.
Qed.
Lemma essai10 : derivable_pt (fun x : R => (x + 2)%R) 0.
reg.
Qed.
Lemma essai11 : derive_pt (fun x : R => (x + 2)%R) 0 essai10 = 1%R.
reg.
Qed.
Lemma essai12 : derivable (fun x : R => (x + Rsqr (x + 2))%R).
reg.
Qed.
Lemma essai13 :
derive_pt (fun x : R => (x + Rsqr (x + 2))%R) 0 (essai12 0%R) = 5%R.
reg.
Qed.
Lemma essai14 : derivable_pt (fun x : R => (2 * x + x)%R) 2.
reg.
Qed.
Lemma essai15 : derive_pt (fun x : R => (2 * x + x)%R) 2 essai14 = 3%R.
reg.
Qed.
Lemma essai16 : derivable_pt (fun x : R => (x + sin x)%R) 0.
reg.
Qed.
Lemma essai17 : derive_pt (fun x : R => (x + sin x)%R) 0 essai16 = 2%R.
reg.
rewrite cos_0.
reflexivity.
Qed.
Lemma essai18 : derivable_pt (fun x : R => (x + y x)%R) 0.
assert (H := d_y).
reg.
Qed.
Lemma essai19 : derive_pt (fun x : R => (x + y x)%R) 0 essai18 = 2%R.
assert (H := dy_0).
assert (H0 := d_y).
reg.
Qed.
Axiom z : R -> R.
Axiom d_z : derivable z.
Lemma essai20 : derivable_pt (fun x : R => z (y x)) 0.
reg.
apply d_y.
apply d_z.
Qed.
Lemma essai21 : derive_pt (fun x : R => z (y x)) 0 essai20 = 1%R.
assert (H := dy_0).
reg.
Abort.
Lemma essai22 : derivable (fun x : R => (sin (z x) + Rsqr (z x) / y x)%R).
assert (H := d_y).
reg.
apply n_y.
apply d_z.
Qed.
(* Pour tester la continuite de sqrt en 0 *)
Lemma essai23 :
continuity_pt
(fun x : R => (sin (sqrt (x - 1)) + exp (Rsqr (sqrt x + 3)))%R) 1.
reg.
left; apply Rlt_0_1.
right; unfold Rminus in |- *; rewrite Rplus_opp_r; reflexivity.
Qed.
Lemma essai24 :
derivable (fun x : R => (sqrt (x * x + 2 * x + 2) + Rabs (x * x + 1))%R).
reg.
replace (x * x + 2 * x + 2)%R with (Rsqr (x + 1) + 1)%R.
apply Rplus_le_lt_0_compat; [ apply Rle_0_sqr | apply Rlt_0_1 ].
unfold Rsqr in |- *; ring.
red in |- *; intro; cut (0 < x * x + 1)%R.
intro; rewrite H in H0; elim (Rlt_irrefl _ H0).
apply Rplus_le_lt_0_compat;
[ replace (x * x)%R with (Rsqr x); [ apply Rle_0_sqr | reflexivity ]
| apply Rlt_0_1 ].
Qed.
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