Sơ đồ tư duy Toán 12 — tổng hợp kiến thức ôn thi THPT

$y = ax^3 + bx^2 + cx + d$ có hai cực trị $\iff b^2 - 3ac > 0$

$y = ax^4 + bx^2 + c$ có ba cực trị $\iff ab < 0$

$\max\limits_{[a;b]} f = \max\{f(a),\, f(b),\, f(x_i)\}$

$y = \dfrac{ax + b}{cx + d}$: tiệm cận đứng $x = -\dfrac{d}{c}$, tiệm cận ngang $y = \dfrac{a}{c}$

Tiệm cận xiên: $a = \lim\limits_{x \to \infty} \dfrac{f(x)}{x}$, $\; b = \lim\limits_{x \to \infty} [f(x) - ax]$

$v(t) = s'(t)$, $\; a(t) = v'(t)$

$\vec a \perp \vec b \iff \vec a \cdot \vec b = 0$

$\overrightarrow{AB} = (x_B - x_A;\, y_B - y_A;\, z_B - z_A)$

$\vec a \cdot \vec b = x_1x_2 + y_1y_2 + z_1z_2$

$|\vec u| = \sqrt{x^2 + y^2 + z^2}$

$\cos(\vec a, \vec b) = \dfrac{\vec a \cdot \vec b}{|\vec a| \cdot |\vec b|}$

$R = a_{k+1} - a_1$

$Q_1 = u_m + \dfrac{\frac{n}{4} - C}{n_m} \cdot (u_{m+1} - u_m)$

$Q_3 = u_j + \dfrac{\frac{3n}{4} - C}{n_j} \cdot (u_{j+1} - u_j)$

$\Delta_Q = Q_3 - Q_1$

$\bar x = \dfrac{1}{n} \sum n_i x_i$

$s^2 = \dfrac{1}{n} \sum n_i (x_i - \bar x)^2 = \dfrac{1}{n} \sum n_i x_i^2 - \bar x^2$

$s = \sqrt{s^2}$

$\int x^\alpha dx = \dfrac{x^{\alpha+1}}{\alpha+1} + C \;(\alpha \ne -1)$, $\quad \int \dfrac{1}{x} dx = \ln|x| + C$

$\int e^x dx = e^x + C$, $\quad \int a^x dx = \dfrac{a^x}{\ln a} + C$

$\int \cos x\, dx = \sin x + C$, $\quad \int \sin x\, dx = -\cos x + C$

$\int \dfrac{dx}{\cos^2 x} = \tan x + C$, $\quad \int \dfrac{dx}{\sin^2 x} = -\cot x + C$

$\int_a^b f(x)\,dx = F(b) - F(a)$

$s = \int_{t_1}^{t_2} v(t)\,dt$

$S = \int_a^b |f(x)|\,dx$, $\quad S = \int_a^b |f(x) - g(x)|\,dx$

$V = \int_a^b S(x)\,dx$

$V = \pi \int_a^b f^2(x)\,dx$

$\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1 \;(abc \ne 0)$

$d(M_0, (P)) = \dfrac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$

$\begin{cases} x = x_0 + at \\ y = y_0 + bt \\ z = z_0 + ct \end{cases}$

$\dfrac{x - x_0}{a} = \dfrac{y - y_0}{b} = \dfrac{z - z_0}{c}$

$\cos(\Delta_1, \Delta_2) = \dfrac{|\vec u_1 \cdot \vec u_2|}{|\vec u_1| \cdot |\vec u_2|}$

$\sin(\Delta, (P)) = \dfrac{|\vec u \cdot \vec n|}{|\vec u| \cdot |\vec n|}$

$\cos((P), (Q)) = \dfrac{|\vec n_1 \cdot \vec n_2|}{|\vec n_1| \cdot |\vec n_2|}$

$R = \sqrt{a^2 + b^2 + c^2 - d}$

$d(I, (P)) = R \iff (P)$ tiếp xúc với mặt cầu

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} \;(P(B) > 0)$

$P(A \cap B) = P(B) \cdot P(A \mid B)$

$P(A) = P(B) \cdot P(A \mid B) + P(\overline{B}) \cdot P(A \mid \overline{B})$

$P(B \mid A) = \dfrac{P(B) \cdot P(A \mid B)}{P(A)}$