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Lesson 1.2
Calculus
Mathematical
model:
Mathematical
model:
A
mathematical
description of a
real world
situation.
Mathematical
models are
often
represented as
functions.
These
functions may
be linear,
quadratic,
cubic, etc.
A linear function of x
means the graph of
the function
is a line.
A linear function of x
means the graph of
the function
is a line.
Also the equation
could be put in the
form of f(x) = mx + b
where m = slope
and b = y-intercept.
A characteristic of
linear functions is that
they grow at
a constant rate.
For example, the linear function:
f(x) = 3x – 2
For example, the linear function:
f(x) = 3x – 2
For example, the linear function:
f(x) = 3x – 2
Notice from the table, as x increases by 0.1, the
value of f(x) increases by 0.3. We already know that
the slope of the graph is 3, but that is also
interpreted as the constant rate of change.
Example:
Example:
(a) As dry air moves upward, it expands and cools.
If the ground temperature is 200C and the
temperature at a height of I km is 100C, express the
temperature T(in 0C) as a function of the height h
(in km), assuming that a linear model is
appropriate.
Example:
(b) Draw the graph of the function in part (a).
What does the slope represent?
Example:
(b) Draw the graph of the function in part (a).
What does the slope represent?
Example:
(b) Draw the graph of the function in part (a).
What does the slope represent?
(c) What is the temperature at a height of 2.5 km?
Polynomials:
Polynomials:
A function P is called a polynomial if
where n is a nonnegative
integer and the numbers a0,
a1, a2, etc are constants,
called the coefficients.
Polynomials:
The degree of a polynomial is the
term with the largest exponent (or
sum of exponents if more than one
variable exists within the term).
Polynomials:
The degree of a polynomial is the
term with the largest exponent (or
sum of exponents if more than one
variable exists within the term).
Polynomials:
The degree of a polynomial is the
term with the largest exponent (or
sum of exponents if more than one
variable exists within the term).
The above polynomial has
degree 6.
Examples of quadratic functions:
P(x) = ax2 + bx + c
Examples of quadratic functions:
P(x) = ax2 + bx + c
Examples of cubic functions:
P(x) = ax3 + bx2 + cx + d
Examples of cubic functions:
P(x) = ax3 + bx2 + cx + d
Power Functions:
Power Functions:
A function in the form of f(x) = xa.
Power Functions:
A function in the form of f(x) = xa.
If a = n is a positive integer then,
Power Functions:
A function in the form of f(x) = xa.
If a = n is a positive integer then,
Power Functions:
A function in the form of f(x) = xa.
If a = n is a even then,
Power Functions:
A function in the form of f(x) = xa.
If a = n is a even then,
Power Functions:
A function in the form of f(x) = xa.
If a = n is a odd then,
Power Functions:
A function in the form of f(x) = xa.
If a = n is a odd then,
Power Functions:
If a = 1/n, where n is a positive integer:
f(x) = x1/n = ??
(This is called the root function)
Power Functions:
If a = 1/n, where n is a positive integer:
f(x) = x1/n = ??
(This is called the root function)
Power Functions:
If a = -1, we get what is called the
reciprocal function
xy = 1  y = 1/x
Power Functions:
If a = -1, we get what is called the
reciprocal function
xy = 1  y = 1/x
Rational Functions:
Rational Functions:
A function that is a ratio of 2 polynomials.
Rational Functions:
A function that is a ratio of 2 polynomials.
P(x)
f (x) =
Q(x)
Rational Functions:
A function that is a ratio of 2 polynomials.
P(x)
f (x) =
Q(x)
Algebraic Functions:
Any function that can be constructed using
algebraic operations (addition, subtraction,
multiplication, division, and taking roots).
Algebraic Functions:
Any function that can be constructed using
algebraic operations (addition, subtraction,
multiplication, division, and taking roots).
Any rational function is automatically an
algebraic function.
Algebraic Functions:
Here are some examples:
Algebraic Functions:
Here are some examples:
Trigonometric Functions:
Trigonometric Functions:
In Calculus, it is always assumed that radian
measure is always used.
Trigonometric Functions:
In Calculus, it is always assumed that radian
measure is always used.
We should recognize the following graphs:
Trigonometric Functions:
In Calculus, it is always assumed that radian
measure is always used.
We should recognize the following graphs:
Trigonometric Functions:
In Calculus, it is always assumed that radian
measure is always used.
We should recognize the following graphs:
Trigonometric Functions:
The tangent function is related to the sine and
cosine functions as:
Trigonometric Functions:
The tangent function is related to the sine and
cosine functions as:
sin x
tan x =
cos x
Trigonometric Functions:
The tangent function is related to the sine and
cosine functions as:
sin x
tan x =
cos x
Trigonometric Functions:
The tangent function is related to the sine and
cosine functions as:
sin x
tan x =
cos x
Its Range is:
Trigonometric Functions:
The tangent function is related to the sine and
cosine functions as:
sin x
tan x =
cos x
Its Range is:
Its Domain is:
Exponential Functions:
Exponential Functions:
f(x) = ax
where the base a is a positive constant.
Exponential Functions:
f(x) = ax
where the base a is a positive constant.
Exponential Functions:
f(x) = ax
where the base a is a positive constant.
Its Range is:
Exponential Functions:
f(x) = ax
where the base a is a positive constant.
Its Range is:
Its Domain is:
Logarithmic Functions:
Logarithmic Functions:
f(x) = logax
Logarithmic Functions:
f(x) = logax
Logarithmic Functions:
f(x) = logax
Its Range is:
Logarithmic Functions:
f(x) = logax
Its Range is:
Its Domain is: