Study Materials for 1st Year B.Tech Students.Paper Name: Mathematics

Paper Code : M101Teacher Name: Kakali Ghosh, Rahuldeb Das, Amalendu Singha Mahapatra,

Raicharan DenraChapter – 1

Lecture1.Topic : Limit and continuous of a functionObjective: Historically, Integral Calculus developed much before the Diffential Calculus (though while teaching Differential Calculus proceeds Integral Calculus). The notions of areas/ volumes of objects attracted the attention of the Greek mathematicians like Antiphon (430 B.C.), Euclid (300 B.C.) and Archimedes (987 B.C.- 212 B.C.).

Later in the 17th century, mathematicians were faced with various problems: in mechanics the problem of describing the motion of a particle mathematically; in optics the need to analyze the passage of light through a lens, which gave rise to the problem of defining tangent/ normal to a surface; in astronomy it was important to know when would a planet be at a maximum/ minimum distance from earth; and so on.

The concept of limit is fundamental for the development of calculus. Calculus is built largely upon the idea of a limit and in this present chapter this idea and various related concepts will be studied in a brief manner.

Limit of a Function

Let x be a variable. Let x goes on taking the values 2.9, 2.99, 2.999, 2.9999, …….. Then we see x goes very near to 3 as near as we please. Mathematically we can say the quantity 3-x becomes small as small as we please. In this situation we say x tends to 3 from left. In notation x→3-.

Again let x goes on taking the values 3.1, 3.01, 3.001, 3.0001, …….. Then we see x goes very near to 3 as near as we please. Mathematically we can say the quantity 3-x becomes small as small as we please. In this situation we say x tends to 3 from right. In notation x→3+.

Thus we see that x→3 implies x→3+ and x→3- both.

Formal Definition:

Say that f (x) tends to l as x a iff given > 0, there is some > 0 such that whenever 0 < | x - a| < , then | f (x) - l| < .

Right-handed limit

We say    provided we can make f(x) as close to L as we want for all x sufficiently close to a and x>a without actually letting x be a.

Left-handed limit

We say     provided we can make f(x) as close to L as we want for all x sufficiently close to a and x<a without actually letting x be a.

Properties

First we will assume that  and  exist and that c is any constant.  Then,

1.In other words we can “factor” a multiplicative constant out of a limit. 

2.So to take the limit of a sum or difference all we need to do is take the limit of the individual parts and then put them back together with the appropriate sign.  This is also not limited to two functions.  This fact will work no matter how many functions we’ve got separated by “+” or “-”. 

3.We take the limits of products in the same way that we can take the limit of sums or differences.  Just take the limit of the pieces and then put them back together.  Also, as with sums or differences, this fact is not limited to just two functions. 

4.As noted in the statement we only need to worry about the limit in the denominator being zero when we do the limit of a quotient.  If it were zero we would end up with a division by zero error and we need to avoid that. 

5.In this property n can be any real number (positive, negative, integer, fraction, irrational, zero, etc.).  In the case that n is an integer this rule can be thought of as an extended case of 3.

 For example consider the case of n = 2.

                                          The same can be done for any integer n.

 

6.This is just a special case of the previous example.

                                           

7.In other words, the limit of a constant is just the constant.  You should be able

to convince yourself of this by drawing the graph of . 

8.As with the last one you should be able to convince yourself of this by drawing

the graph of . 

9.

This is really just a special case of property 5 using .

Sandwich TheoremSuppose that for all x on [a, b] (except possibly at  ) we have,

                                                       Also suppose that,

                                                    

for some .  Then,

                                                             

Cauchy necessary and sufficient condition for the existence of a limit.

The necessary and sufficient condition for that the limit exists and is that ,

corresponding to any pre-assigned positive number ε , however small (but not equal to zero), we can find a positive number δ such that and being any two quantities

satisfying

Worked out problems.

Example1: Show that does not exist.

In order that the limit may exist, it must be possible to find a positive number δ such that,

and satisfying

Where ε is any pre-assigned positive quantity.

Now, whatever δ we may choose, if we take and , by taking n

a sufficiently large positive integer, both and will satisfy .

But in this case ,

,

A finite quantity, and is not less than any chosen ε.Thus, the necessary condition is not sat

Example 2: Show that

Corresponding to a small positive number ε , there exists another small

positive number δ(=ε)>0 such that

< ε whenever

It follows there fore that

Continuity:

Definition: A function f(x) is said to be continuous for x = a, provided exists, is

finite, and is equal to f(a).In other words,

If f(x) be continuous for every value of x in the interval [a,b], it is said to be continuous throughout the interval.A function which is not continuous at a point is said to have a dis continuity at that point.

Corresponding to the analytical definition of limit, provided f(a) exists and given any any pre-assigned positive number ε , however small (but not equal to zero), we can find a

positive number δ such that for all values of x satisfying

Example 3.Let f (x) = for x≠ 2. Show how to define f (2) in order to make f a

continuous function at 2.

Solution. We have

= = (x2 + 2x + 4)

Thus f (x) (22 + 2.2 + 4) = 12 as x 2. So defining f (2) = 12 makes f continuous at 2, (and hence for all values of x).Examples of Discontinuity.

Removable Discontinuity.

Definition

The left hand and right hand limits at a point exist, are equal but?? the function is not defined at this point.???

Example

but the value at x = 0 is undefined.

We remove the problem here by defining the function at point x = 0 to be the limit:

The easiest way to handle removable singularities is to remove them, by defining f to be its limit at any such point.

In the example this means defining at x = 0.

If we always do this, we have for any well defined f(x).

Otherwise the left hand side is undefined at x = a.

Jump Discontinuity:

Definition

The left hand and right hand limits at a point exist, are finite but?they are different. ?

Example

Jump discontinuities only cause trouble is you try to differentiate the function at the

jump. It is possible to get around even with this difficulty by considering ?as

, with and defining its derivative to be

2d(x), .

This object d(x) is called a "delta function". So defined, it is not really a function, since it is 0 except at x = 0, where it is infinite.It is called a "distribution" and is occasionally useful.

Physically it corresponds to density function of a point in one dimension.

Assignment:

1. Show that the function f(x) defined by is continuous at x

= 0 and x = 1.

2. Using Cauchy’s general principal , show that does not exit.

Objective type questions:

1.

(a) 1 (b) -1 (c) 0 (d) none of these

2. is continuous everywhere

(a) Yes (b) no

3. If is equal to

(a) 0 (b) 1 (c) ½ (d) -1.

Lecture 2:

Objective: The limiting process will now be extended to find out the derivative of a function f(x) with respect to x. For this , we begin with the def. of the term increment.

Definition of derivative:

Let y = f(x) be a function defined in an interval , and x be any number in the interior of this interval.

Now let x change to . Then increment of x = . So, y = f(x) changes to

Now consider the increment ratio

When x is fixed, the increment ratio o the difference quotient is a function of .This

expression , takes the form 0/0 which is undefined when we put = 0.If, however,

this increment ratio tends to a definite limit as from either side avoids =

0, then this limit is called the derivative of y with respect to x, and is denoted by or

.

Thus the formal def. of derivative of a function f(x) is

Let y = f(x) be a real – valued function defined in an interval [a,b] containing c is an interior point. Then

If it exists, is called the derivative of f(x) at x = c denoted by at x = c.

Proposition:

If f is differentiable at x0 , then f is continuous at x0.

But continuity does not imply differentiability.

f is differentiable at x0, which implies

To prove:

We have,

Example for converse part.

f(x) = |x| is continuous but not differentiable at x = 0

Worked out problems.

Example 4:Let f (x) = | x|; then f is continuous everywhere, but not differentiable at 0.

Solution: . We compute the Newton quotient directly; recall that | x| = x if x 0, while | x| = - x if x < 0. Thus

= = 1,    while  

   = = - 1.

Thus both of the one-sided limits exist, but are unequal, so the limit of the Newton quotient does not exist.

Assignment:

1. Examine continuity and differentiability of f(x) at x = 0 where

WBUT 2007

And f(0) = 0.

2. Prove that the function f(x) = , 0<x<2 is continuous at x = 1 but not differentiable

there . WBUT 2004

3. Examine the continuity and differentiability of the function

at x = 0. WBUT 2003

Objective type questions:

1. is differentiable at x = 0

(a) Yes (b) no

2. .

(a) derivable in the interval -1<x<1

(b) derivable in the interval -2<x<2

(c) continuous everywhere

(d) discontinuous everywhere