MA 105 : Calculus Division 1, Lecture 01 · Note that [x] is the largest integer x and it is...

MA 105 : Calculus

Division 1, Lecture 01

Prof. Sudhir R. GhorpadeIIT Bombay

Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 1, Lecture 01

Generalities about the Course

Instructor: Prof. Sudhir R. Ghorpade, 106B, Maths Dept.

Lecture Hours: Mon, Thu 2.00 – 3.25 PM, in LA 001.

Tutorial: Wed, 2 – 2.55 pm, in LT 001 – 006.

Attendance: Compulsory! (Also, it will be good for you!)

Office Hours: Mondays 11.30 am – 12.30 pm.

Evaluation Plan: Short Quizzes in Tuts (10 %), Commonquizzes (10%× 2), Mid-Sem (30 %), End-Sem (40%).

More Information:

The BookletMoodle page of the courseInstructor’s web page, and especially, the course page:http://www.math.iitb.ac.in/∼srg/autumn2019.html


http://www.math.iitb.ac.in/~srg/

Text, References, and Acknowledgements

The treatment of calculus in these lectures will be based onthe following two books by S. R. Ghorpade and B. V. Limaye,which are published by Springer, New York.

[GL-1] A Course in Calculus and Real Analysis, 2nd Ed., 2018.

[GL-2] A Course in Multivariable Calculus and Analysis, 2010.

Besides these, the other references listed in the booklet,especially the book of Thomas and Finney, may be consulted.For later parts of the course, it is also useful to see the bookBasic Multivariable Calculus by J. E. Marsden, A. J. Trombaand A. Weinstein (Springer, New York, 1993).

Acknowledgement: I shall mainly use the slides of Calculuslectures prepared recently by Prof. B. V. Limaye. These slidesacknowledged the use of the lecture notes of similar coursesgiven by myself and by Prof. Prachi Mahajan in the past.


Notation

N := {1, 2, 3, . . .} [the set of positive integers]

Z := {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .} [the set of integers]

Q := the set of all rational numbers= {m/n : m, n ∈ Z, n 6= 0}

There is no rational number whose square is 2.

Proof: Suppose not! Then (p/q)2 = 2, that is, p2 = 2q2 forsome p, q ∈ Z such that q 6= 0, and p and q have no commonfactor. Now p2 is even, and so p is even. Hence there is aninteger r such that p = 2r . Then 2q2 = p2 = (2r)2 = 4r 2,and so q2 = 2r 2. Thus q2 is even, and so q is also even. Thus2 is a common factor of p and q, which is a contradiction.

Optional Exercise: If d ∈ N is not the square of an integer,then show that there is no rational number whose square is d .


Let R denote the set of all real numbers. We will assume thefollowing things about the set R.

The set Q of all rational numbers is contained in R, and theset R of all real numbers satisfies

Algebraic Properties regarding addition and multiplication.

Order Properties: There is a subset R+ of R such that(i) Given a ∈ R, exactly one of the following holds:

a ∈ R+ or a = 0 or − a ∈ R+

(ii) a, b ∈ R+ =⇒ a + b ∈ R+ and ab ∈ R+.

Define a < b if b− a ∈ R+. Thus R+ = {x ∈ R : 0 < x}.Completeness Property, which we shall state later.

Elements of the set R \Q, that is, those real numbers whichare not rational numbers, are called irrational numbers.


Notation: We write a ≤ b if a < b or a = b.Also, we write a > b if b < a, and a ≥ b if a > b or a = b.

Boundedness of a subset of R

Let E be a subset of R, that is, E ⊂ R.

E is called bounded above if there is α ∈ R such thatx ≤ α for all x ∈ E .Any such α is an upper bound of E .

E is called bounded below if there is β ∈ R such thatx ≥ β for all x ∈ E .Any such β is a lower bound of E .

E is bounded if it is bounded above and bounded below.

We say that M is the maximum of E if M is an upper boundof E and M ∈ E , and we say that m is the minimum of E ifm is a lower bound of E and m ∈ E .


Supremum (sup or lub) and Infimum (inf or glb)

Let E ⊂ R.

A real number α is called a supremum or a least upperbound of E if α is an upper bound of E (that is, x ≤ αfor all x ∈ E ), and α ≤ u for every upper bound u of E .

A real number β is called an infimum or a greatestlower bound of E if β is a lower bound of E (that is,β ≤ x for all x ∈ E ), and v ≤ β for every lower bound vof E .

If E has a supremum, then it is unique, and it is denotedby sup E or lub E . Similarily, if E has an infimum, then itis unique and is denoted by inf E or glb E .

Example: Let E := {x ∈ R : 0 < x ≤ 1}. Then supE = 1 andinf E = 0. Also, maxE = 1, but E has no minimum.


Completeness Property of R:

A nonempty subset of R that is bounded above has asupremum, that is, a least upper bound.

Consequences of the Completeness Property:

A nonempty subset E of R that is bounded below has aninfimum, that is, a greatest lower bound.In fact, the set F := {−x : x ∈ E} is bounded above, andif α is the lub of F , then β := −α is the glb of E .

Archimedean Property: Given x ∈ R, there is n ∈ N suchthat n > x .Proof: Suppose not! Then n ≤ x for all n ∈ N, that is, xis an upper bound of the set N. Let α := supN. Thenα− 1 is not an upper bound of N, that is, there is n0 ∈ Nsuch that α− 1 < n0. But then α < n0 + 1 ≤ α, since(n0 + 1) ∈ N and α is an upper bound of N. Thus weobtain α < α, which is a contradiction.


Let x ∈ R. Applying the Archimedean property to x and−x , we see that there are `, n ∈ N such that −` < x < n.The largest among finitely many integers k satisfying−` ≤ k ≤ n and also k ≤ x is called the integer part ofx , and is denoted by [x ] or by bxc. Note that [x ] is thelargest integer ≤ x and it is characterized by the followingtwo properties: (i) [x ] ∈ Z and (ii) x − 1 ≤ [x ] ≤ x .

Let a ∈ R+ and n ∈ N. Then there is a unique b ∈ R+

such that bn = a. This real number b is called thepositive nth root of a, and we denote it by a1/n.

Example (the positive square root of 2):Let S := {x ∈ R : x2 ≤ 2}. Then S is nonempty since1 ∈ S and S is bounded above by 2. By the completenessproperty of R, let b := sup S . Then b ≥ 1 > 0. Also, weobtain b2 = 2 by showing that both b2 < 2 and b2 > 2lead to contradictions. (Verify!) Thus b :=

√2. Since

b ∈ S , we see that b = max S .Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 1, Lecture 01

Let a < b in R. Then there is a rational number r suchthat a < r < b. In fact, we can consider r := m/n, wheren > 1/(b − a) and m := [na] + 1.

Let a < b in R. Then there is an irrational number s suchthat a < s < b. In fact, since a +

√2 < b +

√2, let

r ∈ Q be such that a +√

2 < r < b +√

2. Thena < r −

√2 < b, where s := r −

√2 is an irrational

number.

Thus we obtain the following important result.

Between any two real numbers, there is a rational number aswell as an irrational number.

Optional Exercise: Write down more detailed versions of the“proofs” sketched above. Consult [GL-1], if desired.


Intervals

Given any a, b ∈ R, we define

(a, b) := {x ∈ R : a < x < b} [open interval][a, b] := {x ∈ R : a ≤ x ≤ b} [closed interval]Semi-open intervals (a, b] and [a, b) are defined similarly.It is also useful to consider symbols ∞ and −∞ anddefine the infinite intervals

(a,∞) := {x ∈ R : x > a}, [a,∞) := {x ∈ R : x ≥ a},(−∞, a) := {x ∈ R : x < a}, (−∞, a] := {x ∈ R : x ≤ a}.

Also, one writes R = (−∞,∞) and refers to this as aninfinite interval, or sometimes, a doubly infinite interval.In general, a subset I of R is an interval if

x , y ∈ I , x < y =⇒ [x , y ] ⊆ I .

One can show that every interval in R is open, closed,semi-open, or infinite interval.


Absolute Value

For x ∈ R, the absolute value or the modulus of x is

|x | :=

{x if x ≥ 0,

−x if x < 0.

Basic Properties: For any x , y ∈ R,|x + y | ≤ |x |+ |y | [Triangle Inequality]||x | − |y || ≤ |x − y |.

Optional Exercises: (i) Show that for any a, b ∈ R witha ≥ 0, b ≥ 0, and n ∈ N,∣∣∣ n

√a − n√b∣∣∣ ≤ n

√|a − b|.

(ii) For any n ∈ N and any nonnegative real numbersa1, . . . , an, prove the AM-GM inequality:

a1 + · · ·+ ann

≥ n√a1 · · · an.


Functions

Given sets D,E , a function f : D → E assigns to eachx ∈ D, a unique element of E , denoted f (x). We refer toD as the domain of f and E as the co-domain of f . Theset {f (x) : x ∈ D} of all values taken by the function iscalled the range of f . [For a formal definition, see [GL-1].]

A function f : D → E is said to be:one-one (or injective) if for any x1, x2 ∈ D,

f (x1) = f (x2) =⇒ x1 = x2.

onto (or surjective) if its range is E .

If f : D → E is bijective, then it has an inverseg : E → D with the property that the composites g ◦ fand f ◦ g are identity functions, i.e.,

g(f (x)) = x for all x ∈ D and f (g(y)) = y for all y ∈ E .


Examples and Types of Functions

Example/Exercise: Consider f1, f2, f3 : R→ R andf4 : R \ {0} → R defined by

f1(x) = 2x+1, f2(x) = x2, f3(x) = x3, and f4(x) =1

x.

Are these one-one/onto/bijective?Functions that mainly arise in Calculus are usually offollowing types.

Polynomial functionsRational functionsAlgebraic functionsTranscendental functions (this includes logarithmic,exponential and trigonometric functions).

Besides these, we can construct functions by pieceingtogether known functions (such as those belonging to theabove classes). A good example is the absolute valuefunction x 7−→ |x |.


Sequences

Definition

Let X be any set. A sequence in X is a function from the setN of natural numbers to the set X .

The value of this function at n ∈ N is denoted by an ∈ X , andan is called the nth term of the sequence.

We shall use the notation (an) to denote a sequence.

Note: {an : n ∈ N} is the set of all terms of the sequence (an).Thus if X := R and an := (−1)n for n ∈ N, then the sequence(an) is given by −1, 1,−1, 1, . . ., but {an : n ∈ N} = {−1, 1}.

Initially, we let X := R, that is, we consider sequences in R.Later, we shall consider sequences in R2 and in R3.


Examples

Examples:

1 an := n for n ∈ N: 1, 2, 3, 4, . . .

2 an := 1/n for n ∈ N: 1, 1/2, 1/3, 1/4, . . .

3 an := n2 for n ∈ N: 1, 4, 9, 16, . . .

4 an :=√

2 for n ∈ N:√

2,√

2, . . . This is an example of aconstant sequence.

5 an := 2n for n ∈ N: 2, 4, 8, 16, . . .

6 an := (−1)n for n ∈ N: −1, 1,−1, 1, . . .

7 a1 := 1, a2 := 1 and an := an−1 + an−2 for n ≥ 3:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . This sequence is known asthe Fibonacci sequence.

8 an := 1/2 + · · ·+ 1/2n for n ∈ N. Check: an = 1− (1/2n).


Bounded sequences

A sequence (an) of real numbers is said to be boundedabove if the set {an : n ∈ N} is bounded above, that is, ifthere is a real number α such that an ≤ α for every n ∈ N.

A sequence (an) of real numbers is said to be bounded belowif the set {an : n ∈ N} is bounded below, that is, if there is areal number β such that β ≤ an for every n ∈ N.

A sequence (an) of real numbers is said to be bounded if it isbounded above as well as bounded below, that is, if there arereal numbers α, β such that β ≤ an ≤ α for every n ∈ N.

If a sequence is not bounded, it is said to be unbounded.

Let us check which of the sequences mentioned earlier arebounded above and/or bounded below.


Examples of bounded and unbounded sequences:

an := n for n ∈ N: β = 1.

an := 1/n for n ∈ N: β = 0, α = 1.

an := n2 for n ∈ N: β = 1.

an :=√

2 for n ∈ N: β =√

2 = α.

an := 2n for n ∈ N: β = 2.

an := (−1)n for n ∈ N: β = −1, α = 1.

a1 := 1, a2 := 1 and an := an−1 + an−2 for n ≥ 3: β = 1.

an := 1/2 + · · ·+ 1/2n for n ∈ N: β = 1/2, α = 1.

Note: The sequence given by an := n for n ∈ N, is notbounded above by the Archimedean property of R.


Toward convergence of a sequence

Let (an) be a sequence in R, and let a be a real number.

Convergence of a sequence (an) to a real number ashould mean that the term an is as close to a as we likefor all sufficiently large n.

Fix any positive number. Construct the sequence

|a1 − a|, |a2 − a|, . . . , |an − a|, . . .

After a certain stage, all the entries from this sequenceshould be smaller than the fixed positive number.

The fixed positive number is often denoted by ε (epsilon).

Let n0 indicate how far one needs to go in the sequenceto ensure that the entries from |an0 − a| onward aresmaller than ε, that is, an0 , an0+1, an0+2, . . . all belong to(a − ε, a + ε).


Definition of convergence of a sequence

Definition

Let (an) be a sequence of real numbers. We say that (an) isconvergent if there is a ∈ R such that the following conditionholds. For every ε > 0, there is n0 ∈ N such that|an − a| < ε for all n ≥ n0.

This is known as the ε-n0 definition of convergence of asequence.

In this case, we say that (an) converges to a, or that a is alimit of (an), and we write

limn→∞

an = a or an → a (as n→∞).

If a sequence does not converge, we say that the sequencediverges or it is divergent.


The convergence of a sequence is unaltered if a finite numberof its terms are replaced by some other terms.

Examples:

(i) Let a ∈ R and an := a for all n ∈ N. Then an → a. Wecan let n0 := 1.

(ii) an := 1/n for all n ∈ N. Then an → 0.

Let ε > 0 be given. We want to find n0 ∈ N such that|(1/n)− 0| < ε for all n ≥ n0.

Choose any n0 ∈ N which is greater than 1/ε. This ispossible because of the Archimedean property of R.For example, we can let n0 := [1/ε] + 1.


MA 105 : Calculus Division 1, Lecture 01 · Note that [x] is the largest integer x and it is...

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