Calc 2 Memorization Sheet Flashcards

Vocabulary flashcards covering key derivatives, integrals, series, theorems, and formulas for AP Calculus BC exam preparation.

Product Rule

$$\frac{d}{dx}(uv) = u'v + uv'$$

Quotient Rule

$$\frac{d}{dx}\left(\frac{t}{b}\right) = \frac{t'b - tb'}{b^2}$$

Chain Rule

$\frac{d}{dx}f(u) = f'(u) \cdot u'$ or $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

Power Rule for Derivatives

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

Derivative of Natural Logarithm

$\frac{d}{dx}\ln(x) = \frac{1}{x}$ or $\frac{d}{dx}\ln(u) = \frac{u'}{u}$

Derivative of General Logarithm

$$\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$$

Derivative of Exponential Functions

$\frac{d}{dx}e^x = e^x$ and $\frac{d}{dx}a^x = a^x\ln(a)$

Derivative of Sine

$$\frac{d}{dx}\sin(x) = \cos(x)$$

Derivative of Cosine

$$\frac{d}{dx}\cos(x) = -\sin(x)$$

Derivative of Tangent

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

Derivative of Cotangent

$$\frac{d}{dx}\cot(x) = -\csc^2(x)$$

Derivative of Secant

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

Derivative of Cosecant

$$\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)$$

Derivative of Arcsine

$$\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}$$

Derivative of Arctangent

$$\frac{d}{dx}\arctan(x) = \frac{1}{1 + x^2}$$

Definition of Derivative

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

Alternative Forms of the Derivative

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

Power Rule for Integrals

$\int x^n\,dx = \frac{x^{n+1}}{n + 1} + C$, where $n \neq -1$

Integral of 1/x

$\int \frac{1}{x}\,dx = \ln|x| + C$ or $\int \frac{u'}{u}\,dx = \ln|u| + C$

Integral of Exponential Functions

$\int e^x\,dx = e^x + C$ and $\int a^x\,dx = \frac{a^x}{\ln(a)} + C$

Integral Resulting in Arctangent

$$\int \frac{u'}{a^2 + u^2}\,du = \frac{1}{a}\arctan\left(\frac{u}{a}\right) + C$$

Integral Resulting in Arcsine

$$\int \frac{u'}{\sqrt{a^2 - u^2}}\,du = \arcsin\left(\frac{u}{a}\right) + C$$

Integration by Parts Formula

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

Maclaurin Series for e^x

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!} + \dots$$

Maclaurin Series for sin(x)

$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + \dots$$

Maclaurin Series for cos(x)

$$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots + (-1)^n \frac{x^{2n}}{(2n)!} + \dots$$

Taylor Series

$$f(x) = f(c) + f'(c)(x - c) + \frac{f''(c)(x - c)^2}{2!} + \frac{f'''(c)(x - c)^3}{3!} + \dots + \frac{f^{(n)}(c)(x - c)^n}{n!} + \dots$$

Maclaurin Series

A Taylor series centered at $c = 0$.

Logistic Growth Model

Differential equation $\frac{dP}{dt} = kP(M - P)$, where $M$ is the carrying capacity; fastest growth occurs when $P = \frac{M}{2}$.

Euler's Method

An approximation technique using tangent lines where $\frac{dy}{dx} \approx \frac{\Delta y}{\Delta x}$.

Second Fundamental Theorem of Calculus

$$\frac{d}{dx}\int_u^v f(t)\,dt = f(v) \cdot v' - f(u) \cdot u'$$

Area of a Trapezoid

$$A = w\left(\frac{h_1 + h_2}{2}\right)$$

Alternating Series Error Bound

$|\text{error}| \le a_{n+1}$ (the absolute value of the first omitted term).

Lagrange Error Bound

$|\text{error}| \le \frac{f^{(n+1)}(z)(x - c)^{n+1}}{(n + 1)!}$, where $f^{(n+1)}(z)$ is the maximum value of the $(n+1)$-th derivative between $x$ and $c$.

Disc Method

Volume formula $V = \pi \int_a^b r^2\,dx$

Washer Method

Volume formula $V = \pi \int_a^b (R^2 - r^2)\,dx$

Cross-Sectional Area Volume Formula

Volume formula $V = \int_a^b A(x)\,dx$

Start plus Accumulation

$$f(b) = f(a) + \int_a^b f'(x)\,dx$$

Fundamental Theorem of Calculus

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

nth Term Test

If $\lim_{n \to \infty} a_n \neq 0$, then the series diverges (cannot be used to show convergence).

Geometric Series Test

The series $\sum_{n=1}^{\infty} a r^{n-1}$ converges if $|r| < 1$ with sum $S = \frac{a}{1 - r}$, and diverges if $|r| \ge 1$.

p-Series Test

The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$.

Ratio Test

If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$, the series converges; if $> 1$, it diverges; if $= 1$, the test is inconclusive.

Average Rate of Change (AROC)

$\frac{f(b) - f(a)}{b - a}$ (slope between two points).

Instantaneous Rate of Change (IROC)

$f'(c)$ (slope at a single point).

Mean Value Theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then $\frac{f(b) - f(a)}{b - a} = f'(c)$ for some $c \in (a, b)$.

Average Value of a Function

$$f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\,dx$$

Intermediate Value Theorem

If a function $f$ is continuous on $[a, b]$, it takes on every $y$-value between $f(a)$ and $f(b)$.

Extreme Value Theorem

If a function $f$ is continuous on $[a, b]$, it has both an absolute minimum and an absolute maximum on the interval.

Definition of Continuity

A function $f$ is continuous at $x = c$ if and only if $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$.

Squeeze Theorem

If $f(x) \le g(x) \le h(x)$ for all $x \neq c$ near $c$, and $\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} g(x) = L$.

Arc Length Formulas

Rectangular: $L = \int_a^b \sqrt{1 + (f'(x))^2}\,dx$; Parametric: $L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$

Speed Formula (Parametric)

$$v(t) = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$$

Total Distance Formula (Parametric)

$$\int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$$

Polar Area Formula

$$A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2\,d\theta$$

Parametric Derivatives

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

Polar Conversions

$r^2 = x^2 + y^2$, $x = r\cos(\theta)$, $y = r\sin(\theta)$, and $\theta = \arctan\left(\frac{y}{x}\right)$