Calculus 2
Chapter 7
Transcendental Functions
Functions can be categorized into two big groups – algebraic and non-algebraic functions. Algebraic functions: Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division and taking roots). All rational functions are algebraic. Transcendental functions are non-algebraic functions. The following are examples of such functions: i. iii. v. Trigonometric functions Exponential functions Hyperbolic functions ii. iv. vi. Logarithmic functions Inverse trigonometric functions Inverse hyperbolic functions
In this chapter we shall study the properties, the graphs, derivatives and integrals of each of the transcendental function.
Many functions in the field of mathematics and science are inverses of one another. As such, we shall briefly revise the concept of inverse functions before going on to transcendental functions.
7.1
Inverse Functions and Their Derivatives
Objectives
Determine the inverse of a function Obtain the graph of the inverse function from the graph of the function Find the inverse function
What exactly is a function? Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation, a graph, a numerical table or a verbal description. In this section we are going to get familiar with functions and function notation.
MAT133 Calculus with
Analytic
Geometry II
Page 1
An equation is a function if for any x in the domain of the equation, the equation yields exactly one value of y. The set of values that the independent variable is allowed to assume, i.e., all possible input values, is called the domain of the function. The set of all values of f(x) as x varies throughout the domain is called the range of the function.
Example 7.1.1: Given ( ) (a) ( (b) ( )( ) )( )
and ( )
find each of the following.
Notice that in this example (
)( )
(
)( )
. This
Transcendental Functions
Functions can be categorized into two big groups – algebraic and non-algebraic functions. Algebraic functions: Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division and taking roots). All rational functions are algebraic. Transcendental functions are non-algebraic functions. The following are examples of such functions: i. iii. v. Trigonometric functions Exponential functions Hyperbolic functions ii. iv. vi. Logarithmic functions Inverse trigonometric functions Inverse hyperbolic functions
In this chapter we shall study the properties, the graphs, derivatives and integrals of each of the transcendental function.
Many functions in the field of mathematics and science are inverses of one another. As such, we shall briefly revise the concept of inverse functions before going on to transcendental functions.
7.1
Inverse Functions and Their Derivatives
Objectives
Determine the inverse of a function Obtain the graph of the inverse function from the graph of the function Find the inverse function
What exactly is a function? Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation, a graph, a numerical table or a verbal description. In this section we are going to get familiar with functions and function notation.
MAT133 Calculus with
Analytic
Geometry II
Page 1
An equation is a function if for any x in the domain of the equation, the equation yields exactly one value of y. The set of values that the independent variable is allowed to assume, i.e., all possible input values, is called the domain of the function. The set of all values of f(x) as x varies throughout the domain is called the range of the function.
Example 7.1.1: Given ( ) (a) ( (b) ( )( ) )( )
and ( )
find each of the following.
Notice that in this example (
)( )
(
)( )
. This