AP Calculus BC Review

Comprehensive vocabulary flashcards summarizing key definitions, theorems, differentiation/integration rules, parametric/polar calculus formulas, and series convergence tests for AP Calculus BC.

Intermediate Value Theorem

If $f$ is continuous on $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there is at least one number $c$ in $[a, b]$ such that $f(c) = k$.

Chain Rule

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Derivative of an Inverse Function

$$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$$

Limit Definition of a Derivative

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

Alternate Definition of a Derivative (at a point 'a')

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Rolle's Theorem

If $f$ is a function that is continuous on $[a, b]$ and differentiable on $(a, b)$, and $f(a) = f(b)$, then there is at least one number $c$ such that $f'(c) = 0$.

Mean Value Theorem

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

Extreme Value Theorem

If $f(x)$ is continuous on $[a, b]$, then $f(x)$ has an absolute maximum and an absolute minimum on that interval.

First Derivative Test

Let $c$ be a critical number of $f$ that is continuous on an open interval $I$ containing $c$ and differentiable except possibly at $c$: 1. If $f'(x)$ changes from negative to positive at $c$, $f(c)$ is a relative minimum. 2. If $f'(x)$ changes from positive to negative at $c$, $f(c)$ is a relative maximum. 3. If $f'(x)$ does not change sign at $c$, $f(c)$ is neither a relative minimum nor relative maximum.

The Second Derivative Test

Let $f$ be a function such that $f'(c) = 0$ and the second derivative exists on an open interval containing $c$: 1. If $f''(c) > 0$, then $f(c)$ is a relative minimum. 2. If $f''(c) < 0$, then $f(c)$ is a relative maximum. 3. If $f''(c) = 0$, the test is inconclusive.

The Fundamental Theorem of Calculus

Let $f$ be continuous on $[a, b]$ and the function $F$ be any antiderivative on $(a, b)$: 1. $\int_a^b f(x)\,dx = F(b) - F(a)$ 2. $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$

Integration by Parts

$$\int u\,dv = uv - \int v\,du$$

Logistic Differential Equation

$\frac{dy}{dt} = ky\left(1 - \frac{y}{L}\right)$ with the general solution $y = \frac{L}{1 + Ae^{-kt}}$

Volume (Disk Method)

$$\pi \int_a^b (R(x))^2\,dx$$

Volume (Washer Method)

$$\pi \int_a^b \left((R(x))^2 - (r(x))^2\right)\,dx$$

Rectangular Arc Length

$$\int_a^b \sqrt{1 + (f'(x))^2}\,dx$$

Average Value Theorem

If $f$ is integrable on $(a, b)$, then the average value on $(a, b)$ is given by $\frac{1}{b - a}\int_a^b f(x)\,dx$.

Parametric Derivatives

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ and $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$

Parametric Arc Length

$$\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$$

Parametric Speed

$$\text{Speed} = |\vec{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2}$$

Polar Area

$$A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta$$

Geometric Series Test

Given a geometric series $\sum_{n=0}^{\infty} ar^n$, the series converges if $|r| < 1$ and diverges if $|r| \ge 1$. If it converges, the sum is $S = \frac{a}{1 - r}$.

Harmonic Series Test

The harmonic series is $\sum_{n=1}^{\infty}\frac{1}{n}$ and diverges.

P-Series Test

A series of the form $\sum_{n=1}^{\infty}\frac{1}{n^p}$ where $p$ is a positive real number converges if $p > 1$ and diverges if $0 < p \le 1$.

Nth-Term Test for Divergence

Given a series $\sum_{n=0}^{\infty} a_n$, if $\lim_{n \to \infty} a_n \neq 0$, then the series diverges.

Direct Comparison Test

Let $a_n \ge 0$ and $b_n \ge 0$ for all $n$ where $a_n \le b_n$: 1. If $\sum b_n$ converges, then $\sum a_n$ converges. 2. If $\sum a_n$ diverges, then $\sum b_n$ diverges.

Limit Comparison Test

Let $a_n > 0$ and $b_n > 0$ for all $n$. If $\lim_{n \to \infty}\frac{a_n}{b_n}$ exists and is both positive and finite, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Integral Test

If $f$ is a positive, continuous, and decreasing function such that $f(n) = a_n$, then $\sum_{n=c}^{\infty} a_n$ and $\int_c^{\infty} f(x)\,dx$ either both converge or both diverge.

Alternating Series Test

An alternating series $\sum (-1)^n a_n$ converges if $|a_{n+1}| < |a_n|$ and $\lim_{n \to \infty} a_n = 0$.

Ratio Test

Let $\sum a_n$ be a series: 1. If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$, then $\sum a_n$ converges. 2. If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| > 1$, then $\sum a_n$ diverges. 3. If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = 1$, the ratio test is inconclusive.

Alternating Series Error Bound

If $\sum a_n$ is a convergent alternating series where $|a_{k+1}| < |a_k|$, then the error in the $n^{\text{th}}$ partial sum is bounded by $R_n = |S - S_n| \le a_{n+1}$.

Taylor Polynomial

$$\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x - a)^n = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \dots + \frac{f^{(n)}(a)}{n!}(x - a)^n$$

Lagrange Form of the Error Bound

$$|R_n(x)| \le \left|\frac{f^{(n+1)}(z)}{(n+1)!}(x - c)^{n+1}\right|$$