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Calculus Term Definitions

Formula

quadratic formula

Calculus I

Definition

absolute value

Calculus I

Theorem

properties of absolute values

Calculus I

Definition

equation of a line in various forms

Calculus I

Definition

equation of a circle

Calculus I

Definition

sin, cos, tan

Calculus I

Definition

sec, csc, tan, cot

Calculus I

Definition

midpoint formula

Calculus I

Definition

function

Calculus I

The solutions or roots of the quadratic equation
ax² + b x + c = 0 are given by

These flashcards and the accompanying

source
code are licensed under a Creative Commons
Attribution–NonCommercial–ShareAlike 2.5 License.
For more information, see creativecommons.org. You
can contact the author at:

jasonu at physics utah edu

File last updated on Sunday 8^th July, 2007,
at 17:15

The equation of a circle centered at (h, k) with radius
r is:
(x − h)² + (y − k)² = r²

Form	Equation
point–slope
slope–intercept	y = m x + b
two point
standard	Ax + By + C = 0

A function is a mapping that associates with each
object x in one set, which we call the domain, a single
value f (x) from a second set which we call the range.

If P(x₁, y₁) and Q(x₂, y₂) are two points, then the midpoint
of the line segment that joins these two points
is given by:

Definition

even and odd functions

Calculus I

Definition

limit

Calculus I

Definition

one–sided limit

Calculus I

Theorem

limit exists iff both the right–handed and
left–handed limits exist and are equal

Calculus I

Theorem

main limit theorem (part 1)

Calculus I

Theorem

main limit theorem (part 2)

Calculus I

Theorem

squeeze theorem

Calculus I

Theorem

two special trigonometric limits

Calculus I

Definition

point-wise continuity

Calculus I

Theorem

composition limit theorem

Calculus I

If a function f (x) is defined on an open interval containing
c, except possibly at c, then the
limit of f (x) as x approaches c equals L is denoted

The above equality holds if and only if for any

there exists

such that

even f(−x) = f (x) for all x e.g. x², cos (x)

odd f(−x) = −f (x) for all x e.g. x, sin (x)

right–handed limit

iff for any

there exists a

such that

Let f, g be functions that have limits at c, and let n
be a positive integer.
7.

provided that

when n is even.

Let k be a constant, and f, g be functions that have limits
at c.

Suppose f, g and h are functions which satisfy the
inequality

for all x near c, (except
possibly at c). Then

and f is continuous at L, then

Let f be defined on an open interval containing c, then
we say that f is point-wise continuous at c if

Definition

continuity on an interval

Calculus I

Definition

derivative

Calculus I

Definition

equivalent form for the derivative

Calculus I

Theorem

differentiability and continuity

Calculus I

Theorem

constant and power rules

Calculus I

Theorem

differentiation rules

Calculus I

Theorem

derivatives of trig functions

Calculus I

Theorem

chain rule

Calculus I

Theorem

generalized power rule

Calculus I

Definition

notation for higher-order derivatives

Calculus I

The derivative of a function f is another function f'
(read “f prime”) whose value at x is

provided the limit exists and is not ∞ or −∞.

function f is said to be continuous on an open
inteval iff f is continuous at every point of the open
interval.

A function f is said to be continuous on a closed
interval [a, b] iff

1. f is continuous on (a, b) and
2.

and
3.

If the function f is differentiable at c, then f is continuous
at c.

Let f and g be functions of x and k a constant.

1. scalar product rule (k f)' = k f'
2. sum rule (f + g)' = f' + g'
3. difference rule (f − g)' = f' − g'
4. product rule (f g)' = f' g + f g'
5. quotient rule

Let u = g (x) and y = f (u). If g is differentiable at x,
and f is differentiable at u = g (x), then the composite
function

is differentiable at x and

In Leibniz notation

Derivative	f' (x)	y'	D	Leibniz
first
second
third
fourth
..
nth

If f is a differentiable function and n is an integer,
then the power of the function

is differentiable and

Theorem

extreme value theorem

Calculus I

Theorem

intermediate value theorem

Calculus I

Definition

critical point
stationary point
singular point

Calculus I

Definition

increasing
decreasing
monotonic

Calculus I

Theorem

monotonicity theorem

Calculus I

Definition

concave up
concave down

Calculus I

Theorem

concavity theorem

Calculus I

Definition

inflection point

Calculus I

definition

local maximum
local minimum
local extremum

Calculus I

Theorem

first derivative test

Calculus I

If the function f is continuous on the closed interval
[a, b] and v is any value between the minimum and
maximum of f on [a, b], then f takes on the value v.

If the function f is continuous on the closed interval
[a, b], then f has a maximum value and a minimum
value on the interval [a, b].

A function f defined on the interval I is

• increasing on I , for every

• decreasing on I , for every

The function f is said to be monotonic on I if f is
either increasing or decreasing on I.

If f is a function defined on an open interval containing
the point c, we call c a critical point of f iff either

• f' (c) = 0 or

• f' (c) does not exist

Furthermore when f' (c) = 0 we call c a stationary
point of f, and when f' (c) does not exist we call c a
singular point of f.

Suppose f is differentiable on an open interval I, then
if f' is increasing on I we say that f is concave up
on I.

If f' is decreasing on I we say that f is concave
down on I.

Suppose f is differentiable on an open interval I, then

• f' (x) > 0 for each

is increasing on I

• f' (x) < 0 for each

is decreasing on I

Let f be continuous at c, then the ordered pair (c, f (c))
is called an inflection point of f if f is concave up
on one side of c and concave down on the other side
of c.

Let f be twice differentiable on the open interval I.

• f'' (x) > 0 for each

is concave up on I

• f'' (x) < 0 for each

is concave down on I

Let f be differentiable on an open interval (a, b) that
contains c.

1.

and

is a local maximum of f.

2.

and

is a local minimum of f.

3. If f' (x) has the same sign on both sides of c,
then f (c) is not a local extremum.

Let the function f be defined on an interval I containing
c. We say f has a local maximum at c iff
there exists an interval (a, b) containing c such that
f (x)≤ f (c) for all x ∈ (a, b).

We say f has a local minimum at c iff there
exists an interval (a, b) containing c such that
f (x) ≥f (c) for all x ∈ (a, b).

A local extremum is either a local maximum
or a local minimum.

Theorem

second derivative test

Calculus I

Theorem

mean value theorem

Calculus

If f is continuous on a closed interval [a, b] and differentiable
on its interior (a, b), then there is at least one
point c in (a, b) such that

or equivalently

f (b) − f (a) = f' (c)(b − a)

Let f be twice differentiable on an open interval containing
c, and suppose f' (c) = 0.

1. If f'' (c) < 0, then f has a local maximum at
c.
2. If f'' (c) > 0, then f has a local minimum at c.

3. If f'' (c) = 0, then the test fails.