AP Calculus AB: Stuff You Must Know Cold

A comprehensive reference set of vocabulary terms, theorems, and formulas adapted from the 'Stuff You Must Know Cold' handout for AP Calculus AB exam preparation.

Power Rule

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Derivative of $\tan(x)$

$$\frac{d}{dx}(\tan(x)) = \sec^2(x)$$

Chain Rule

$\frac{d}{dx}[f(u)] = f'(u)\frac{du}{dx}$ or $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$

Product Rule

$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ or $uv' + vu'$

Quotient Rule

$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ or $\frac{vu' - uv'}{v^2}$

Intermediate Value Theorem (IVT)

If the function $f(x)$ is continuous on $[a, b]$ and $y$ is a number between $f(a)$ and $f(b)$, then there exists at least one number $x = c$ in the open interval $(a, b)$ such that $f(c) = y$.

Mean Value Theorem (MVT)

If the function $f(x)$ is continuous on $[a, b]$ AND the first derivative exists on the interval $(a, b)$, then there is at least one number $x = c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Rolle’s Theorem

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

Extreme Value Theorem (EVT)

If the function $f(x)$ is continuous on $[a, b]$, then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.

Critical Point

A point where $\frac{dy}{dx} = 0$ or is undefined; endpoints must also be considered for absolute extrema.

Point of Inflection

A point where concavity changes and $\frac{d^2y}{dx^2}$ changes sign (goes from positive to negative or vice versa).

Derivative of an Inverse Function

If $f$ has an inverse function $g$, then $g'(x) = \frac{1}{f'(g(x))}$.

Average Rate of Change (ARoC)

$$m_{sec} = \frac{f(b) - f(a)}{b - a}$$

First Derivative Test: Relative Maximum

Occurs if $f'(x) = 0$ or DNE and the sign of $f'(x)$ changes from positive (+) to negative (-).

Horizontal Asymptotes Cases

1. Numerator exponent < denominator: $y = 0$. 2. Numerator exponent > denominator: DNE. 3. Numerator exponent = denominator: $y = \frac{a}{b}$ (leading coefficients quotient).

Fundamental Theorem of Calculus (FTC)

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

Speed

The absolute value of velocity: $|v(t)|$.

Displacement

$$\int_{t_0}^{t_f} v(t)\,dt$$

Total Distance

Integral of $|v(t)|$ over the time interval.

Average Velocity

$$\frac{\Delta x}{\Delta t} = \frac{\text{final position} - \text{initial position}}{\text{total time}}$$

Exponential Growth and Decay

Uses $y = Ce^{kt}$. Occurs when the rate of change of $y$ is proportional to $y$ ($y' = ky$).

Mean Value Theorem for Integrals (Average Value)

If $f(x)$ is continuous on $[a, b]$, then there exists $x = c$ such that $f_{avg} = \frac{1}{b - a} \int_a^b f(x)\,dx$.

Trapezoidal Rule (Even Intervals)

$$\int_a^b f(x)\,dx \approx \frac{b - a}{2n} [y_0 + 2y_1 + 2y_2 + \dots + 2y_{n-1} + y_n]$$

Area Between Two Curves (Slices $\perp$ to x-axis)

$$A = \int_a^b [f(x) - g(x)]\,dx$$

Volume by Disk Method (About x-axis)

$$V = \pi \int_a^b [R(x)]^2\,dx$$

Volume by Washer Method (About x-axis)

$$V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2)\,dx$$

Cross Section Volume: Equilateral Triangles

$$V = \frac{\sqrt{3}}{4} \int_a^b (base)^2\,dx$$

Basic Integral: $\int \frac{du}{u}$

$$\ln|u| + C$$

Derivative of $\sin^{-1}\left(\frac{u}{a}\right)$

$$\frac{d}{dx} \left[ \sin^{-1}\left(\frac{u}{a}\right) \right] = \frac{1}{\sqrt{a^2 - u^2}} \frac{du}{dx}$$

Derivative of $\tan^{-1}\left(\frac{u}{a}\right)$

$$\frac{d}{dx} \left[ \tan^{-1}\left(\frac{u}{a}\right) \right] = \frac{a}{a^2 + u^2} \frac{du}{dx}$$