2diffcalc

2DIFFCALC
Relations Continuous functions can be graphed with a line or

A relation is a relationship between two sets of smooth curve and contain an infinite number of
values. It is the relation between the x-values and points
y-values of ordered pairs.
Example:
Do the ordered pairs represent a function?
A relation is composed of:
{(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}
● X-values as the domain
No, 3 is repeated in the domain.
● Y-values as the range
Example:
{(2, 6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} {(4, 1), (5, 2), (8, 2), (9, 8)}
Yes.
Domain : {0, 1, 2, 3}
Vertical Line Test
Range : {-6, 0, 4}
A graph of a function exists if a vertical line
Relations can be written through ordered pairs, table, graph, intersects it at exactly one point.
or mapping.
A relation is a function if no vertical line intersects
A mathematical relation is the mapping of inputs the graph of the relation at more than one point.
and outputs.
One Output
If it does, then an input has more than one output
Multiple Outputs
Functions
A function is an equation that has an x and a y value.
We can think of both variables as input values that
we could plug into the equation to get the result.
A function is a relation with the property that for
each input there is exactly one output.
(Not a Function)
In order to be a function, every x-values must have a y value.
No Y’s can share any X’s, No x can have many Y’s.
Set Notations
Discrete functions consists of points that are not D = {x | x ∈ ℝ}
connected Statement: The set of all x such that x belongs to all Real
Numbers.
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R = {y | y ∈ ℝ} Types of Polynomial Functions:

Statement: The set of all y such that y belongs to all Real ⤷ Constant Polynomial Function
Numbers. ⤷ A function where the value does not
change, it does not include any
R = {y | y ≥ -6} variables.
Statement: The set of all y such that y is greater than or
⤷ Graphed like a horizontal line
equal to –6. ⤷ Linear Polynomial Function
⤷ A function that has a degree of 1
Does the graph represent a function? ⤷ Graphed like a diagonal line
⤷ If the diagonal line has a slope that
starts from the top left to the bottom
right, it is a negative slope. If it starts
to the top right and follows through
the bottom left, it is a positive slope.
⤷ Quadratic Polynomial Function
⤷ Graphs as a parabola that open
upward or downward.
⤷ A hyperbola is not considered as a
No, Domain : x ≥ 1/2 quadratic function
Range : all real nos ⤷ Cubic Polynomial Function
⤷ Quartic Polynomial Function
Composition of Functions
(f•g)(x) = f(g(x))
Read as”f composition of g”
Example:
No, Domain : all real nos
f(x) = 2x2 + 3 g(x) = 4x3 + 1
Range : all real nos
(f ⃘ g)(x) = f(g(x))
Polynomial Functions = 2(4x3 + 1)2 + 3
⤷ is a function such as a quadratic, a cubic, a = 2((4x3 + 1) (4x3 + 1)) + 3 FOIL method
quartic, and so on, involving + 16x3 + 2 + 3
non-negative integer powers of x. = 32x6 + 16x3 + 5
⤷ is an expression with a single independent
variable that can appear multiple times with Operations on Functions
different exponent degrees.
Examples:
ADDITION
2
f(x) = 7 - 1.6x - 5x (f + g)(x) = f(x) + g(x)
f(x) = -1.6x2 - 5x + 7 Example:
Degree = 2 f(x) = 2x2 + 3x – 4 and g(x) = 2x + 3
Leading Coefficient = -1.6 (f + g)(x) = (2x2 + 3x – 4) + (2x + 3)
= 2x2 + 5x – 1
p(x) = x + 2x-2 + 9.5
Not a polynomial, negative squared SUBTRACTION
(f – g)(x) = f(x) – g(x)
q(x) = x3 - 6x + 3x4 Example:
q(x) = 3x4 + x3 - 6x f(x) = 2x2 + 3x – 4 and g(x) = 2x + 3
Degree = 4 (f – g)(x) = (2x2 + 3x – 4) – (2x + 3)
Leading Coefficient = 3 = 2x2 + x – 7
MULTIPLICATION
(f•g)(x) = f(x)•g(x)
2DIFFCALC
Example: We can evaluate limits analytically using

f(x) = 3x – 5 and g(x) = x algebraic techniques. These are:
(f•g)(x) = (3x – 5)•(x) ⤷ Substitution
= 3x2 – 5x ⤷ Factoring (simplifying the expressions)
⤷ Rationalizing the numerator or denominator
DIVISION (conjugates)
(f/g)(x) = f(x)/g(x)
Example:
f(x) = 3x2 + 4x – 3 and g(x) = x
(f/g)(x) = (3x2 + 4x – 3)/x
= 3x + 4 – 3/x
Limits of Functions
The limit of a function at a point a in its domain (if it
exists) is the value that the function approaches as
its argument approaches a.
Limits are used to define the derivative, the definite

integral, and analyze the local behaviors of functions
near points of interest.
A function is said to have a limit L at a if it is possible

to make the function arbitrarily close to L by
choosing values closer and closer to a.
The actual value at a is IRRELEVANT to the value of
the limit.
limx⟶a f (x) = L
read as "the limit of f(x) as x approaches a is L.
the limit.
It appears as the x coordinates get closer and closer

to 2, the y coordinates get closer and closer to 3. We
can say that the limit of this function is 3.
2DIFFCALC
2DIFFCALC

2diffcalc - Prelim

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Relations Continuous functions can be graphed with a line or

If it does, then an input has more than one output

R = {y | y ∈ ℝ} Types of Polynomial Functions:

Example: We can evaluate limits analytically using

Limits are used to define the derivative, the definite

A function is said to have a limit L at a if it is possible

It appears as the x coordinates get closer and closer

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