Äîêóìåíò LATEX
Wild Mathing
Çàäà÷è, ôîðìóëû è òåîðåìû
Àëãåáðà
1.
2.
3.
4.
5.
Íàéäèòå ïðîèçâåäåíèå x1 x2 êîðíåé ìíîãî÷ëåíà f (x) = x2 + 7x + 1.
Ðåøèòå íåðàâåíñòâî log22 x < 4.
Ðåøèòå óðàâíåíèå sin√2x = 1.
Ðåøèòå íåðàâåíñòâî 2x ⩽ x + 1.
(
) Íàéäèòå íàèìåíüøåå çíà÷åíèå âûðàæåíèÿ
ÌÃÓ, 2021
log22 (ax)
+
log22
1−a
x
è âñå çíà÷åíèÿ ïàðàìåòðà a, ïðè êîòîðûõ îíî äîñòèãàåòñÿ.
Ãåîìåòðèÿ
1.
2.
3.
4.
5.
.
△ABC ∼ △A1 B1 C1 ⇒ ∠ABC = ∠A1 B1 C1
BC ∥ AD ⇒ ∠DAB + ∠ABC = 180◦
(l ⊥ β, l ⊂ α) ⇒ α ⊥ β
γ ∩ α = a, γ ∩ β = b α ∥ β
a∥b
.
.
Ïóñòü
è
. Òîãäà .
Ôîðìóëà êîñèíóñà óãëà ìåæäó âåêòîðàìè ⃗a è ⃗b:
⃗a · ⃗b
c
.
cos (⃗a, ⃗b) =
|⃗a| · |⃗b|
Àíàëèç
1.
Îïðåäåëåíèå ÷èñëà e:
x
1
1+
.
x→∞
x
e = lim
2.
Ñóììà ðÿäà îáðàòíûõ êâàäðàòîâ:
∞
X
1
1
1
1
1
π2
= 2 + 2 + 2 + ... + 2 + ... =
.
2
n
1
2
3
n
6
n=1
3.
4.
Òàáëè÷íûé èíòåãðàë:
Z
1
dx = arctg x + C.
1 + x2
Ãàììà-ôóíêöèÿ ÷åðåç àáñîëþòíî ñõîäÿùèéñÿ èíòåãðàë:
+∞
Z
Γ(z) =
tz−1 e−t dt, z ∈ C, Re(z) > 0.
0
5.
Ïðåêðàñíàÿ ôîðìóëà Ðàìàíóäæàíà:
1+
1
1
1
1
+
+
+
+ ... +
1·3 1·3·5 1·3·5·7 1·3·5·7·9
r
1
=
1
1+
e·π
.
2
2
1+
3
1+
4
1+
1+
5
1 + ...
1
Äîêóìåíò LATEX
Wild Mathing
Ðåøåíèÿ
1.
2.
Ïîñêîëüêó äèñêðèìèíàíò êâàäðàòíîãî òðåõ÷ëåíà ïîëîæèòåëåí, òî ïî òåîðåìå Âèåòà x1 x2 = 1.
Èñïîëüçóÿ, ôîðìóëó ðàçíîñòè êâàäðàòîâ, ïîëó÷èì
log22 x < 4 ⇔ (log2 x − 2) (log2 x + 2) < 0.
Äàëüøå âñå ïðîñòî:
−2 < log2 x < 2 ⇔
3.
1
< x < 4.
4
Ðåøèì ïðîñòåéøååå òðèãîíîìåòðè÷åñêîå óðàâíåíèå:
sin 2x = 1,
π
+ 2πn, n ∈ Z,
2
π
x = + πn, n ∈ Z.
4
2x =
4.
Íå çàìåòèâ áîëåå ïðîñòîãî ïóòè, âûïîëíèì ðàâíîñèëüíûå ïðåîáðàçîâàíèÿ.
x + 1 ⩾ 0,
2x ⩽ x2 + 2x + 1,
2x ⩾ 0
x ⩾ −1,
⇔ x2 ⩾ −1,
x⩾0
⇔ x ⩾ 0.
Âûðàæåíèå èìååò ñìûñë òîëüêî ïðè x > 0 è 0< a < 1, è äàëüíåéøèå ðàññóæäåíèÿ ïðîâåäåíû
. Òîãäà, èñïîëüçóÿ íåðàâåíñòâî ìåæäó ñðåäíèì
ñ ó÷åòîì ýòîãî. Ïóñòü b = log2 (ax), c = log2 1−a
x
êâàäðàòè÷åñêèì è ñðåäíèì àðèôìåòè÷åñêèì, à òàêæå ñâîéñòâî ñóììû ìîäóëåé, ïîëó÷èì
5.
r
log2 (ax) + log2
b2 + c2
|b| + |c|
|b + c|
⩾
⩾
=
2
2
2
2
1−a
x
=
log2 (−a2 + a)
log2
⩾
2
2
1
4
= 1.
Òàêèì îáðàçîì, äîêàçàíà îöåíêà:
log22 (ax) + log22
1−a
x
⩾ 2,
ïðè÷åì ðàâåíñòâî äîñòèãàåòñÿ òîëüêî ïðè a = 12 .
Äîïîëíåíèÿ
Ñèìâîëû
∅ ∀ ∃ % {a} φ ε
Q
B ̸∈ ± × ̸= ≡ ∼
=
¾¿ · · ·
Çàäà÷à
Ðåøèòå íåðàâåíñòâî |log3 x| ⩾ 1.
Èñïîëüçóÿ ðàâíîñèëüíûå ïåðåõîäû, ïîëó÷èì
Ðåøåíèå.
"
Îòâåò:
0; 13 ∪ [3; +∞)
x ⩾ 3,
⇔
1
log3 x ⩽ −1
0<x⩽ .
3
log3 x ⩾ 1,
.
2
