AP Calculus BC Master Guide Flashcards

Comprehensive vocabulary and formula flashcards for AP Calculus BC, covering limits, derivatives, integrals, differential equations, and series.

Limit Existence Condition

A limit $\lim_{x \to a} f(x)$ exists if and only if the left-sided limit and right-sided limit are equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$

Continuity at a Point

A function is continuous at a point $a$ if 1. $f(a)$ exists, 2. $\lim_{x \to a} f(x)$ exists, and 3. $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuities

These include removable discontinuities (holes), jump discontinuities, and vertical asymptotes (where $\lim_{x \to a} f(x) = \pm \infty$)

Differentiability Breakers

Factors that prevent a function from being differentiable at a point, including discontinuities, corners (e.g., $|x|$), and cusps (e.g., $x^{2/3}$)

Special Trigonometric Limits

Limits derived from the Squeeze (Sandwich) Theorem, such as $\lim_{x \to 0} \frac{\sin(ax)}{bx} = \frac{a}{b}$ and $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$

Intermediate Value Theorem (IVT)

If $f(x)$ is continuous on $[a, b]$, then $f(x)$ takes on every value between $f(a)$ and $f(b)$

Mean Value Theorem (MVT)

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists a value $c$ where a < c < b such that $f'(c) = \frac{f(b)-f(a)}{b-a}$

L'Hôpital's Rule

If $\lim_{x \to a} \frac{f(x)}{g(x)}$ results in an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then the limit equals $\lim_{x \to a} \frac{f'(x)}{g'(x)}$

Formal Derivative Definition

The derivative defined as the limit of the difference quotient: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Product Rule

The rule for differentiating the product of two functions: $\frac{d}{dx} [f(x)g(x)] = f(x)g'(x) + f'(x)g(x)$

Quotient Rule

The rule for differentiating a fraction: $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

Chain Rule

The rule for differentiating composite functions: $\frac{d}{dx} [f(g(x))] = f'(g(x)) \times g'(x)$

Derivative of an Inverse Function

The formula given by $\frac{d}{dx} [f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}$

Critical Points

Points on a function where $f'(x) = 0$ or $f'(x)$ is undefined

Inflection Points

Points where the second derivative $f''(x) = 0$ or is undefined and $f''(x)$ changes sign

Extreme Value Theorem (EVT)

States that if $f(x)$ is continuous on a closed interval $[a, b]$, it must have both an absolute maximum and an absolute minimum

First Derivative Test

A method to find local extrema: if $f'(x)$ changes from positive to negative, it is a local max; if it changes from negative to positive, it is a local min

Concavity Conditions

If f''(x) > 0, the function is concave up ($\cup$); if f''(x) < 0, the function is concave down ($\cap$)

Second Derivative Test

A method to identify extrema using the second derivative: if $f'(c) = 0$ and $f''(c) < 0$, it is a local max; if $f'(c) = 0$ and $f''(c) > 0$, it is a local min

Related Rates Plan

A 4-step process: 1. Find (identify target rate), 2. Given (identify knowns), 3. Equation (relate variables), 4. Derivative (differentiate with respect to $t$)

Linearization

Also known as tangent line approximation, calculated as $f(x) \approx f(a) + f'(a)(x - a)$

Riemann Sum (Trapezoidal Rule)

An approximation of an integral using trapezoids: $T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + … + 2f(x_{n-1}) + f(x_{n})]$

Fundamental Theorem of Calculus (Part 1)

States that $\frac{d}{dx} \int_a^x f(t)\,dt = f(x)$

Integration by Parts

The formula used to integrate products of functions: $\int u\,dv = uv - \int v\,du$

Average Value of a Function

The average value of $f(x)$ on $[a, b]$ is $f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\,dx$

Washer Method (Volume)

Volume formula for revolving a region around the x-axis with a hole: $V = \pi \int_a^b [(f(x))^2 - (g(x))^2]\,dx$

Shell Method (Volume)

Volume of revolution formula (around y-axis): $V = 2\pi \int_a^b xf(x)\,dx$

Arc Length (Rectangular)

The length of a curve from $a$ to $b$: $L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^2}\,dx$

Euler's Method

A recursive numerical method to approximate the solution of a differential equation using the linearization step: $y_{new} = y_{old} + m(\Delta x)$

Logistic Growth Model

A population model defined by the differential equation $\frac{dP}{dt} = kP(1 - \frac{P}{M})$, where $M$ is the carrying capacity

Area of a Polar Curve

The area bounded by a polar function from $\theta_1 = \alpha$ to $\theta_2 = \beta$ is $A = \frac{1}{2} \int_{\alpha}^{\beta} (r(\theta))^2\,d\theta$

Geometric Series Convergence

The series $\sum ar^n$ converges to $\frac{a}{1 - r}$ if |r| < 1 and diverges if $|r| \geq 1$

Taylor Series Formula

An infinite sum centered at $a$: $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$

Maclaurin Series for $\sin(x)$

The expansion: $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + … = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$

Lagrange Error Bound

The remainder of a Taylor polynomial: $|R_n(x)| \leq \frac{M|x - c|^{n+1}}{(n+1)!}$, where $M$ is the maximum value of $|f^{(n+1)}(z)|$ between $c$ and $x$

Ratio Test

A test for convergence where $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$; converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$

Absolute vs. Conditional Convergence

Absolute convergence occurs if $\sum |a_n|$ converges; conditional convergence occurs if $\sum a_n$ converges but $\sum |a_n|$ diverges

Maclaurin Series of $\frac{1}{1-x}$

$$\sum_{n=0}^{a}x^{n}$$

Maclaurin Series of $e^{x}$

$$\sum_{n=0}^{a}\frac{x^{n}}{n!}$$

Maclaurin Series of $\cos x$

$$\sum_{n=0}^{a}\frac{\left(-1\right)^{n}x^{2n}}{\left(2n\right)!}$$

Maclaurin Series of $\tan^{-1}\left(x\right)$

$$\sum_{n=0}^{a-1}\frac{\left(-1\right)^{n}x^{\left(2n+1\right)}}{2n+1}$$

Maclaurin Series for $\ln\left(1+x\right)$

$$\sum_{n=1}^{a}\frac{\left(-1\right)^{\left(n-1\right)}x^{n}}{n}$$