How we Define Hyperbolic Functions and their Simple Formulas

Hyperbolic Functions

Define Hyperbolic Function in Simple Way

  • Sin Hyperbolic Function
  • Cos Hyperbolic Function

Simply Define sinh x hyperbolic function

Sinh x = \frac12\left(e^x-\;e^{-x}\right) is called hyperbolic sine function. Its domain and range are the set of all real numbers.

Define sinh x hyperbolic function

Formula for Cosine hyperbolic function in Math with Detail

Cosh x = \frac12\left(e^x+\;e^{-x}\right) is called hyperbolic cosine function. its domain is the set of all real numbers and the range is the set of all numbers in the interval.

[1, +∞)

Formula for Cosine hyperbolic function

Four Hyperbolic Functions and Formulas and Their Details

The remaining four hyperbolic functions are defined in the terms of the hyperbolic sine and the hyperbolic cosine function as follows:

Four Hyperbolic Function Formulas are

Functionsis equal toFormulas
tanh x\frac{\sin h\;x}{\cos h\;x}\frac{e^x\;-\;e^{-x}}{e^x\;+\;e^{-x}}
sech x\frac1{\cos h\;x}\frac2{e^x\;+\;e^{-x}}
coth x\frac{\cos h\;x}{\sin h\;x}\frac{e^x\;+\;e^{-x}}{e^x\;+\;e^{-x}}
csch x\frac1{\sin h\;x}\frac2{e^x\;-\;e^{-x}}

The hyperbolic functions have the same properties that resemble to those of trigonometric functions.

Define Vertical and Horizontal Asymptotes in Calculus

Explain Vertical and Horizontal Asymptotes

What is an Asymptote in Calculus

In mathematical terms, an asymptote is a curve that approaches infinity, asymptotes are lines that appear parallel to it.

What is an Asymptote in Calculus
Asymptote in Math

Types of Asymptotes in Calculus

There are three types of Asymptote in math.

1. Horizontal Asymptote
2. Vertical Asymptote
3. Oblique Asymptote

Difference between Horizontal Vertical and Oblique Asymptotes

Difference between Horizontal Vertical and Oblique Asymptotes

How to find Horizontal Asymptote of Exponential Function?

How to find Horizontal Asymptote of Exponential Function?
It is a Horizontal Asymptote

Horizontal Asymptote: A horizontal asymptote occurs when A curve approaches some constant value b as x approaches infinity (or *infinity)..

How to find Vertical Asymptote of Exponential Function

How to find Vertical Asymptote of Exponential Function
It is a Vertical Asymptote

Vertical Asymptote: A horizontal asymptote occurs when a function of X approaches some constant, c (from the left or right) then the curve continues to infinity (or −infinity).

What is Oblique Asymptote?

What is Oblique Asymptote?
Oblique Asymptote

Oblique Asymptote: A Oblique Asymptote occur when, as x goes to infinity (or −infinity) the curve then becomes a line y=mx+b

Asymptote for a Curve Definition in Math

Definition: A straight line l is called an asymptote for a curve C if the distance between l and C approaches zero as the distance moved along l (from some fixed point on l ) tends to infinity.

The curve C can approach asymptote l as one moves along l in one direction, or in the opposite direction, or in both directions.

How to Find Equation of Horizontal Asymptote?

Suppose the equation y\;=\;f(x) of C is such that y is real and y\rightarrow a as X\rightarrow\infty or X\rightarrow-\infty then y\;=\;a is a Horizontal asymptote. For, the distance between the curve and the straight line y\;=\;a is y\;-\;a and this approaches zero as X\rightarrow\infty or X\rightarrow-\infty.

How to Find Equation of Vertical Asymptote?

If the equation of C is such that y is real and Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a from one side then the straight line x = a is a vertical asymptote.

To see this, observe that (1) x – a is the distance between the curve and the straight line and that this distance is supposed to approach zero (2) Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a , so that

\lim_{y\rightarrow\pm\infty}\;(x-a)\;=\;\lim_{x\rightarrow a}\;(x-a)\;=\;0

Thus, to locate vertical asymptotes we have to find a number a such that \lim_{x\rightarrow a}\;y\;=\;\infty\;or\;-\infty

Similarly, if y\rightarrow mx\;+\;c as x\rightarrow\infty or x\rightarrow-\infty then y\;=\;mx\;+\;c is an asymptotes ( which is neither vertical nor horizontal).

Thus, we inquire for \lim_{x\rightarrow\pm\infty}\;y is studying asymptotes.

Find Horizontal and Vertical asymptote for Algebraic equations

For Algebraic equations, we can find horizontal and vertical asymptotes as follows:

For horizontal asymptotes we write the given equation in the form x\;=\;\frac{\psi(y)}{\theta(y)} and consider those values of y for which \theta(y)\;=\;0.

Similarly, to find a verticle asyptote, we write the given equation in the form y\;=\frac{f(x)}{g(x)} and consider those value of x for which g(x) = 0.

Working Rule for Asymptotes Parallel to the Axes

In an equation of a curve, the coefficient of the highest power of x (respectively of y) equated to zero gives asymptotes (if any) parallel to the x-axis (respectively y-axis)

Method to Draw the Graphs of Algebraic Functions

Defined Graphs of Algebraic Functions

If f is a real-valued function of a real numbers, then the graph of f in the xy-plane is defined to be the graph of equation y = f(x). The graph of a function f is the set of point { (x, y)| y = f(x), x is in thedomain of f } in the cartesian plan for which (x, y) is an ordered pair of f.

The graph provides a visual technique for determining whether the set of points represents a function or not. if a vertical line intersects a graph in more than one point, it is not the graph of a function.

Explain is given in the figure.

Sketch the Graph of a function

A function Graph

Defined Graphs of Algebraic Functions
A Function

Is Also a Function Graph

Sketch the Graph of a function

The Sketch is not a function

The Sketch is not a function
not a function

Best Method to Draw the graph

To draw the graph of y = f(x), we give arbitrary value of our choice to x and find the corresponding values of y. In this way we get ordered pairs \left(x_1\;,\;y_1\right) , \left(x_2\;,\;y_2\right) , \left(x_3\;,\;y_3\right) etc. These orders pairs represent points of the graph in the cartesian. We have these points and join them together to get the graph of the function.