midterm 1 calc defs + formulas

function

A relation that assigns exactly one output for each input from a given set.

domain of a function

The set of all possible inputs (or values) for which a function is defined.

codomain of a function

the set of all possible outputs that the function can produce, which may include outputs that are not necessarily achieved by any input.

range of a function

The range of a function is the set of all actual outputs produced by the function for every input in the domain. what actually comes out.

1 to 1 function

each output is produced by exactly one input, meaning no two distinct inputs map to the same output.

lim (fx) = L

The limit of a function $f(x)$ as $x$ approaches a value is the value that $f(x)$ gets arbitrarily close to as $x$ approaches that value, represented as $L$.

1-sided limits

1-sided limits refer to the values that a function approaches as the input approaches a particular point from one side, either the left or the right

horizontal asymptote

horizontal line that a graph approaches as the input values approach infinity or negative infinity, indicating the behavior of the function at the extremes.

vertical asymptote

a vertical line of the form x = k that a function's graph approaches infinitely closely as the output grows or shrinks toward positive or negative infinity (+ ∞ or - ∞), without ever touching or crossing the line

convergent sequence

a sequence of numbers whose terms approach a specific, finite real number limit as the index n goes to infinity

divergent sequence

an infinite sequence of numbers that fails to converge to a finite real-number limit

sum of a series

the total result of adding up the terms of a sequence, either finite or infinite

n-th term arithmetic

aₙ = a₁ + d(n-1)

arithmetic sum of terms

Sₙ = (n/2)(a₁ + aₙ)

arithmetic recursive form

aₙ = aₙ-₁ + d

n-th term geometric

aₙ = a₁rⁿ ⁻ ¹

sum of finite geometric

Sₙ = (a₁(1-n^r))/(1-r)

sum of infinite geometric

S∞ = a₁/(1-r)

recursive form geometric

aₙ = (r)aₙ-₁

when does sum of infinite geometric converge?

when -1≤r≤1