Chapter 7Transcendental Functions

Assignment # 1

• Practice Ex for chap 6:

• Ex 6.3 (1 to 12 odd)+ 17 to 22 all

Transcendental Functions

Functions can be classified into two broad classes

• Algebraic

• Transcended

Algebraic functions

Algebraic functions are obtained by division,

multiplication, addition and subtraction of

Polynomials.

Transcendental functions

Functions which are not Algebraic are called polynomial, for example,

• log(x),• ln(x)• sin(x)• ex • sinh(x)

and their inverses

In this section we shall study how to differentiate and integrate these functions.

One to One Functions:

Example

Horizontal Line Test

Example (one to one)

Example (not one to one)

• Only one to one functions can have inverses!

Example

f and f-1 are always symmetric about line y = x

Theorem

Example

Which functions are one to one and which are not ?

Exercise 7.119—28(odd)

• x=[0:.001:10];• y=1-(1./x);• plot(x,y)• x=[0:.01:10];• plot(x,y)• y=1-(1./x);• plot(x,y)• x=[0:.1:10];• plot(x,y)• y=1-(1./x);• plot(x,y)• x=[-10:.1:.99];• plot(x,y)• y=1-(1./x);• plot(x,y)• x=[-10:.1:0];• y=1-(1./x);• plot(x,y)• x=[-10:.1:1];• y=1-(1./x);• plot(x,y)• clc• x=[-5:.01:5];• y=1-(1./x);• plot(x,y)• help grid• GRID ON

7.2- Natural Logarithms

Natural Logarithm

The number “e”

Derivative of ln(u)

Example

Example

Example

Example

Example

Example

Exercise 7.21—68(All even and odd)

(First 4 problems are about properties of Logarithms)

7.3- Exponential Function

An important limit

lim(1 )n

x

n

n= ex

For x = 1

1lim(1 )n n

n

= e

Definition of e

Example

Solving for some Exponent

Example

Generally

Integral of ex

Example

Verification of Solution

The power Rule

Exercise 7.31—66(All even and odd)

(First 16 problems are about properties of Log and ex)

7.4 - ax and logax

Exercise 7.41—74(All even and odd)

Caution

Graphs of Six Basic Trigonometric Functions

Arcsine and Arccosine Functions

Example

Derivatives of Inverse Trigonometric Functions

Integral of Inverse Trigonometric Functions

Example

Example

Example (Completing the Square)

Example

Exercise 7.749—112(All odd)

Hyperbolic Functions

Some Identities for Hyperbolic Functions

Derivatives of Inverse Hyperbolic Functions

Integrals of Inverse Hyperbolic Functions

Example

Note

Exercise 7.8(----)

The End

Thank You !