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- Math 116 - Chapter 2 - Functions, Limits, and the Derivative

Jake L.

Function

a rule that assigns to each element in set A one and only one element in a set B

Domain

The set A - which input values can be input into the rule.

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Range

The set B - what output values are possible.

Independent Variable

A variable whose value the user sets.

Dependent Variable

A variable that depends on the chosen independent variable

Ordered Pairs

a set of two numbers where the first is the x value and the second is f(x). (x,f(x))

Graph of a funcion

the set of all points (x,y) in the xy-plane such that x is in the domain of ? and y=f(x)

piecewise defined function

A function that is defined by more than one rule.

Graph of an equation

the set of all ordered pairs (x,y) that satisfy the given equation.

Vertical-line test

A curve in the xy-plane is the graph of a function y-f(x) if and only if each vertical line intersects it in at most one point

composite function

Let f and g be functions. The composition of g and f is the function g o f defined by: (g o f)(x) = g(f(x)) The domain of g o f is the set of all x in the domain of f such that f(x) lies in the domain of g.

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Polynomial function

a function of the form f(x) = ax^n + bx^n-1 +...+ cx + d. where a-d are constants and a does not equal 0

linear function

A polynomial function of degree one. Example: f(x) =2x-5

quadratic function

A polynomial function of degree 2. Example: 2x^2+2x+2

cubic function

A polynomial function of degree 3

rational function

The quotient of two polynomials

power function

functions of the form f(x)=x^r where r is any real number.

demand function

p=f(x) where p measures the nit price and x measures the number of units of the commodity in question.

supply function

p=f(x) - generally increasing - relates the unit price and the quantity supplied.

market equilibrium

When the quantity produced is equal to the quantity demanded.

equilibrium quantity

The quantity produced at market equilibrium

equilibrium price

the price at market equilibrium

limit of a function

the function ? has the limit L as x approaches a, if the value f(x) can be made as close to the number L as we please by taking x sufficiently close to (but not equal to) a.

unbounded function

A function where the limit does not exist as x approaches a value

indeterminate form

when something results in an answer of 0/0

limit of a function at infinity

The function ? has the limit L as x increases without bound, if ?(x) can be made arbitrarily close to L by taking x large enough.

right-hand limit of a function

the function ? has this kind of limit as x approaches a from the right, if the values f(x) can be made as close to L as we please by taking x sufiiently close to (but not equal to ) a and to the right of a.

left-hand limit of a function

the function ? has this kind of limit as x approaches a from the left, if the values f(x) can be made as close to L as we please by taking x sufiiently close to (but not equal to ) a and to the left of a.

continuity of a function at a point

A function ? is continuous at the point x=a if the following conditions are satisfied. 1. ?(a) is defined. 2. lim as x approaches a of f(x) exists. and 3. lim as x->a f(x)=f(a)

tangent line to the graph of ?

A line that touches the graph of ? only once and has a slope of lim h->0 (f(x+h) - f(x))/h

differentiable function

a function which has a derivative on the interval [a,b]

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