Grade 12 Mathematics: Derivatives and Their Applications

Vocabulary flashcards covering Grade 12 calculus concepts including definitions, differentiation formulas, monotonicity, extrema, limits, asymptotes, and curve sketching procedures.

Derivative of a function at a point

The finite limit $\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$, denoted as $f'(x_0)$ or $y'(x_0)$, for a function $y = f(x)$ defined on an interval $(a; b)$ containing $x_0$.

Instantaneous velocity

The physical rate of change of a rectilinear motion $s = s(t)$ at time $t_0$, defined as $v(t_0) = s'(t_0)$.

Tangent line equation

The equation of the tangent line to the graph of $y = f(x)$ at point $M(x_0; f(x_0))$, given by $y = f'(x_0)(x - x_0) + f(x_0)$, where $f'(x_0)$ is the slope.

Chain rule for composite functions

If $u = g(x)$ has derivative $u'_x$ and $y = f(u)$ has derivative $y'_u$, the composite function $y = f(g(x))$ has derivative $y'_x = y'_u \cdot u'_x$.

Quotient rule for differentiation

The differentiation formula for the division of two functions: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$, where $g = g(x) \neq 0$.

Derivative of $\tan(u)$ and $\cot(u)$

The differentiation formulas for trigonometric composite functions: $(\tan(u))' = \frac{u'}{\cos^2(u)}$ and $(\cot(u))' = -\frac{u'}{\sin^2(u)}$.

Derivative of $a^u$ and $\log_a(u)$

The differentiation formulas for exponential and logarithmic composite functions: $(a^u)' = u' \cdot a^u \ln(a)$ and $(\log_a(u))' = \frac{u'}{u \ln(a)}$.

Increasing function (đồng biến)

A function $y = f(x)$ defined on $K$ such that for all $x_1, x_2 \in K$, if $x_1 < x_2$, then $f(x_1) < f(x_2)$.

Decreasing function (nghịch biến)

A function $y = f(x)$ defined on $K$ such that for all $x_1, x_2 \in K$, if $x_1 < x_2$, then $f(x_1) > f(x_2)$.

Monotonicity theorem

For a function $f(x)$ differentiable on $K$ with $f'(x) = 0$ only at finitely many points: if $f'(x) > 0$ for all $x \in K$, $f(x)$ is increasing on $K$; if $f'(x) < 0$ for all $x \in K$, $f(x)$ is decreasing on $K$.

Local maximum point of a function

A point $x_0$ for which there exists an interval $(a; b) \subset K$ containing $x_0$ such that $f(x) < f(x_0)$ for all $x \in (a; b)$ with $x \neq x_0$.

Local minimum point of a function

A point $x_0$ for which there exists an interval $(c; d) \subset K$ containing $x_0$ such that $f(x) > f(x_0)$ for all $x \in (c; d)$ with $x \neq x_0$.

Extremum point of a graph

The coordinate point $M(x_0; f(x_0))$ on the Cartesian plane, where $x_0$ is a local extremum point of the function $y = f(x)$.

First derivative test for extrema

A test on continuous $f(x)$: if $f'(x) > 0$ on $(a; x_0)$ and $f'(x) < 0$ on $(x_0; b)$, $x_0$ is a local maximum; if $f'(x) < 0$ on $(a; x_0)$ and $f'(x) > 0$ on $(x_0; b)$, $x_0$ is a local minimum.

Maximum value of a function ($\max_D f(x)$)

A real number $M$ such that $f(x) \le M$ for all $x \in D$ and there exists at least one $x_0 \in D$ such that $f(x_0) = M$.

Minimum value of a function ($\min_D f(x)$)

A real number $m$ such that $f(x) \ge m$ for all $x \in D$ and there exists at least one $x_0 \in D$ such that $f(x_0) = m$.

Horizontal asymptote

The line $y = y_0$ where either $\lim_{x \to +\infty} f(x) = y_0$ or $\lim_{x \to -\infty} f(x) = y_0$.

Vertical asymptote

The line $x = x_0$ where at least one of the one-sided limits $\lim_{x \to x_0^+} f(x)$ or $\lim_{x \to x_0^-} f(x)$ equals $+\infty$ or $-\infty$.

Oblique asymptote

The line $y = ax + b$ ($a \neq 0$) where either $\lim_{x \to +\infty} [f(x) - (ax + b)] = 0$ or $\lim_{x \to -\infty} [f(x) - (ax + b)] = 0$.

Function investigation procedure

A 3-step graphing process: Step 1: Find the domain; Step 2: Examine variation (limits, asymptotes, derivative sign, variation table, extrema); Step 3: Sketch the graph (asymptotes, special points, symmetry).