Calculus Concepts and Formulas

These flashcards cover key concepts and formulas related to calculus, including definitions, rules, and tests that are crucial for understanding the subject.

Formal Definition of a Limit

lim f(x) = L means that for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Simpson's Rule

An approximation for the integral of a function, given by: $rac{b-a}{3n} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + … + f(x_n)]$.

Trig. Substitution

A method used to simplify integrals by substituting trigonometric identities, such as $x = a an( heta)$ or $x = a ext{sin}( heta)$.

Improper Integral

An integral that has either infinite limits of integration or an integrand that approaches infinity at one or more points in the interval.

Geometric Series

A series of the form $S = a + ar + ar^2 + ar^3 + …$ which converges if the common ratio r satisfies |r| < 1.

Alternating Series Test

A test used to determine the convergence of a series where the terms alternate in sign.

Taylor Series Expansion

An expansion of a function $f(x)$ about a point $a$ given by: f(x) = f(a) + f'(a)(x-a) + rac{f''(a)}{2!}(x-a)^2 + ….

L'Hôpital's Rule

A method for finding limits of indeterminate forms by taking the derivative of the numerator and denominator.

Logistic Growth Model

A model describing how a population grows more slowly as it approaches a maximum capacity, described by the equation y = rac{A}{1 + Be^{-kt}}.

Definition of Derivative

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero, given by: f'(x) = rac{f(x+h) - f(x)}{h} as $h o 0$.