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# Calculus Term Definitions

Some rights reserved.

Calculus I
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
can contact the author at:

jasonu at physics utah edu

File last updated on Sunday 8th 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. if
8.
9. 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

If 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.