AP Calculus BC Core Concepts Flashcards

Vocabulary flashcards reviewing essential AP Calculus BC concepts, theorems, formulas, series expansions, and convergence tests.

Continuity at $c$

A function $f(x)$ is continuous at $c$ if: 1) $f(c)$ is defined, 2) $\lim_{x \to c} f(x)$ exists (meaning $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$), and 3) $\lim_{x \to c} f(x) = f(c)$.

L'Hopitals Rule

A method to evaluate limits of indeterminate forms by showing separately that $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$ (or both are $\pm \infty$).

Fundamental Theorem of Calculus (Integral Derivative)

The derivative of an accumulation function: $\frac{d}{dx}\left[\int_a^{g(x)} f(t)\,dt\right] = f(g(x))g'(x)$.

Average Value of $f(x)$ on $(a,b)$

The average value of $f(x)$ on the interval $(a,b)$ defined as $\frac{1}{b-a} \int_a^b f(x)\,dx$.

Average Rate of Change of $f(x)$ on $(a,b)$

The slope of the secant line on $(a,b)$, calculated as $\frac{f(b)-f(a)}{b-a}$.

Equation of the Tangent Line

The equation of the line tangent to $f(x)$ at $x = a$, given by $y - f(a) = f'(a)(x - a)$.

Tangent Line Approximation Errors

An approximation using a tangent line at $x = a$ that yields an underestimate if $f(x)$ is concave up and an overestimate if $f(x)$ is concave down.

Product Rule

The derivative formula for a product of two functions: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + g'(x)f(x)$.

Quotient Rule

The derivative formula for a quotient of two functions: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - g'(x)f(x)}{(g(x))^2}$.

Extreme Value Theorem

States that if $f$ is continuous on $[a,b]$, then $f$ attains an absolute maximum and an absolute minimum on $[a,b]$.

Rolle's Theorem

States that if $f$ is continuous and differentiable on $[a,b]$ and $f(a) = f(b)$, then there exists a value $c$ in $(a,b)$ such that $f'(c) = 0$.

Intermediate Value Theorem

States that if $f(x)$ is continuous on $[a,b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists a $c$ in $(a,b)$ such that $f(c) = k$.

Mean Value Theorem

States that if $f(x)$ is continuous and differentiable on $[a,b]$, then there is at least one number $c$ in $(a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Critical Number

A number $x$ in the domain of $f$ where $f'(x) = 0$ or $f'(x)$ is undefined.

Point of Inflection

A point on the graph of $f(x)$ where $f''(x)$ changes sign.

Disk Method

A formula for volume of a solid of revolution about a horizontal line where cross sections are circular disks: $V = \pi \int (r(x))^2\,dx$.

Washer Method

A formula for volume of a solid of revolution about a horizontal line: $V = \pi \int \left[(R(x))^2 - (r(x))^2\right]\,dx$, where outer and inner radius functions are subtracted.

Derivative of an Inverse Function

The derivative of $f^{-1}$ at $b$ given by $(f^{-1})'(b) = \frac{1}{f'(a)}$, where $f(a) = b$.