AP Calculus BC Review

Flashcards for AP Calculus BC review, covering integrals, arc length, error bounds, velocity, polar curves, series, L'Hopital's Rule, derivatives, and theorems.

Integral Constant Multiple Rule

(\int k f(u) du = k \int f(u) du)

Integral of du

(\int du = u + C)

Power Rule for Integrals

(\int u^n du = \frac{u^{n+1}}{n+1} + C), where (n \neq -1)

Integral of 1/u

(\int \frac{1}{u} du = \ln|u| + C)

Integral of e^u

(\int e^u du = e^u + C)

Integral of a^u

(\int a^u du = \frac{a^u}{\ln a} + C)

Integral of cos(u)

(\int \cos u du = \sin u + C)

Integral of sin(u)

(\int \sin u du = -\cos u + C)

Integral of tan(u)

(\int \tan u du = -\ln|\cos u| + C)

Integral of cot(u)

(\int \cot u du = \ln|\sin u| + C)

Integral of sec(u)

(\int \sec u du = \ln|\sec u + \tan u| + C)

Integral of csc(u)

(\int \csc u du = -\ln|\csc u + \cot u| + C)

Integral of du/(u sqrt(u^2-a^2))

(\int \frac{du}{u \sqrt{u^2 - a^2}} = \frac{1}{a} \operatorname{arcsec} \frac{u}{a} + C)

Integral of du/sqrt(a^2 - u^2)

(\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin \frac{u}{a} + C)

Integral of du/(a^2 + u^2)

(\int \frac{du}{a^2 + u^2} = \frac{1}{a} \arctan \frac{u}{a} + C)

Integration by Parts

(\int u dv = uv - \int v du)

Arc Length Formula (function)

(L = \int_a^b \sqrt{1 + [f'(x)]^2} dx)

Arc Length Formula (polar)

(L = \int{\theta1}^{\theta_2} \sqrt{[r(\theta)]^2 + [r'(\theta)]^2} d\theta)

Lagrange Error Bound

(|f(x) - P_n(x)| \le \frac{\max|f^{(n+1)}(z)|}{(n+1)!} |x-c|^{n+1})

Velocity

(v(t) = \frac{dx}{dt})

Acceleration

(a(t) = \frac{dv}{dt})

Velocity Vector

( \langle \frac{dx}{dt}, \frac{dy}{dt} \rangle)

Speed

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

Distance Traveled

(\int{ti}^{tf} |v(t)| dt = \int{ti}^{tf} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt)

Position Update Formula (x)

(x(b) = x(a) + \int_a^b x'(t) dt)

Position Update Formula (y)

(y(b) = y(a) + \int_a^b y'(t) dt)

Area inside a polar curve leaf

(\frac{1}{2} \int{\theta1}^{\theta_2} [r(\theta)]^2 d\theta)

Slope of polar curve

(\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{r(\theta) \cos \theta + r'(\theta) \sin \theta}{-r(\theta) \sin \theta + r'(\theta) \cos \theta})

Ratio Test for Convergence

Series converges if (\lim{n \to \infty} \left| \frac{a{n+1}}{a_n} \right| < 1)

Alternating Series Error Bound

(|S - SN| \le a{N+1})

Disk Method

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

Washer Method

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

Shell Method

(V = 2\pi \int_a^b r(x) h(x) dx)

Volume of Known Cross Sections (x-axis)

(V = \int_a^c A(x) dx)

Volume of Known Cross Sections (y-axis)

(V = \int_a^c A(y) dy)

Taylor Series

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

Taylor Series expansion of e^x

(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + …)

Taylor Series expansion of cos(x)

(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + …)

Taylor Series expansion of sin(x)

(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + …)

Taylor Series expansion of 1/(1-x)

(\frac{1}{1-x} = 1 + x + x^2 + x^3 + …)

Taylor Series expansion of ln(x+1)

(\ln(x+1) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + …)

Geometric Series

(\sum_{n=0}^{\infty} Ar^n = \frac{A}{1-r}) if (|r| < 1)

L'Hopital's Rule

If (\lim{x \to a} f(x) = 0) and (\lim{x \to a} g(x) = 0) then (\lim{x \to a} \frac{f(x)}{g(x)} = \lim{x \to a} \frac{f'(x)}{g'(x)})

Average Rate of Change

(\frac{f(b) - f(a)}{b - a})

Definition of Derivative

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

Power Rule

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

Derivative of sin(x)

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

Derivative of cos(x)

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

Derivative of tan(x)

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

Derivative of cot(x)

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

Derivative of sec(x)

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

Derivative of csc(x)

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

Derivative of ln(u)

(\frac{d}{dx} \ln u = \frac{1}{u} \frac{du}{dx})

Derivative of e^u

(\frac{d}{dx} e^u = e^u \frac{du}{dx})

Derivative of log_a(x)

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

Derivative of a^x

(\frac{d}{dx} a^x = a^x (\ln a) \frac{du}{dx})

Intermediate Value Theorem

If f is continuous on [a, b], then for any c between f(a) and f(b), there exists d in (a, b) such that f(d) = c.

Fundamental Theorem of Calculus

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

Second Fundamental Theorem of Calculus

(\frac{d}{dx} \int_a^{g(x)} f(t) dt = f(g(x)) g'(x))

Chain Rule

(\frac{d}{dx} f(u) = f'(u) \frac{du}{dx})

Product Rule

(\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx})

Quotient Rule

(\frac{d}{dx} \frac{u}{v} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2})

Mean Value Theorem & Rolle's Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that (f'(c) = \frac{f(b) - f(a)}{b - a}). If f(a) = f(b), then f'(c) = 0.

Critical Point Condition

(f'(x) = 0) or undefined

Average Value

(f(c) = \frac{1}{b-a} \int_a^b f(x) dx)

Euler's Method

(x{new} = x{old} + \Delta x) and (y{new} = y{old} + \frac{dy}{dx}(x{old}, y{old}) \Delta x)

Local Minimum Conditions

goes (-, 0, +) OR (-, undefined, +) OR (f''(x) > 0)

Local Maximum Conditions

goes (+, 0, -) OR (+, undefined, -) OR (f''(x) < 0)

Absolute Max/Min

Compare local extreme values to endpoints.

Logistics Curves

(P(t) = \frac{L}{1 + Ce^{-(Lk)t}})

Maximum growth rate of logistics curve

(P = \frac{1}{2} L)

Point of Inflection

Concavity changes; goes (+,0,-), (-,0,+), (+,und,-), or (-,und,+)