Mathematics-1 ,subject code 3110014, Slides of Mathematics

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• UNIT - 1). METHOD – 1: 0/0 TYPE INDETERMINATE FORM - 2). METHOD – 2: ∞/∞ TYPE INDETERMINATE FORM - 3). METHOD – 3: 0 × ∞ FORM - 4). METHOD – 4: ∞ – ∞ FORM - 5). METHOD – 5: - & - FORM ∞ - 6). METHOD – 6: IMPROPER INTEGRAL OF FIRST KIND - 7). METHOD – 7: IMPROPER INTEGRAL OF SECOND KIND - 8). METHOD – 8: CONVERGENCE OF IMPROPER INTEGRAL OF FIRST KIND - 9). METHOD – 9: CONVERGENCE OF IMPROPER INTEGRAL OF SECOND KIND
  • 10). METHOD – 10: EXAMPLE ON GAMMA FUNCTION AND BETA FUNCTION
  • 11). METHOD – 11: VOLUME BY SLICING METHOD...............................................................................
  • 12). METHOD – 12: VOLUME OF SOLID BY ROTATION USING DISK METHOD
  • 13). METHOD – 13: VOLUME OF SOLID BY ROTATION USING WASHER METHOD
  • 14). METHOD – 14: LENGTH OF PLANE CURVES
  • 15). METHOD – 15: AREA OF SURFACES BY REVOLUTION
• UNIT
  • 16). METHOD – 1: CHARACTERISTICS OF SEQUENCE AND LIMIT
  • 17). METHOD – 2: CONVERGENCE OF SEQUENCE
  • 18). METHOD – 3: CONTINUOUS FUNCTION THEOREM
  • 19). METHOD – 4: GEOMETRIC SERIES
  • 20). METHOD – 5: TELESCOPING SERIES
  • 21). METHOD – 6: ZERO TEST
  • 22). METHOD – 7: COMBINING SERIES
  • 23). METHOD – 8: INTEGRAL TEST
  • 24). METHOD – 9: DIRECT COMPARISION TEST
  • 25). METHOD – 10: LIMIT COMPARISION TEST
  • 26). METHOD – 11: RATIO TEST
  • 27). METHOD – 12: RABBE’S TEST - ROOT TEST TH
  • 29). METHOD – 14: LEIBNITZ’S TEST............................................................................................................
  • 30). METHOD – 15: ABSOLUTE CONVERGENT SERIES
  • 31). METHOD – 16: CONDITIONALLY CONVERGENT SERIES I N D E X
  • 32). METHOD – 17: POWER SERIES
  • 33). METHOD – 18:
    • FORM OF TAYLOR’S SERIES.............................................................................. ST
  • 34). METHOD – 19: - FORM OF TAYLOR’S SERIES ND
  • 35). METHOD – 20: MACLAURIN’S SERIES
• UNIT
  • 36). METHOD – 1: FOURIER SERIES IN THE INTERVAL (0, 2L)
  • 37). METHOD – 2: FOURIER SERIES IN THE INTERVAL (−𝐋, 𝐋)
  • 38). METHOD – 3: HALF-RANGE COSINE SERIES IN THE INTERVAL (𝟎, 𝐋)...............................
  • 39). METHOD – 4: HALF-RANGE SINE SERIES IN THE INTERVAL (𝟎, 𝐋)
• UNIT
  • 40). METHOD – 1: LIMIT OF FUNCTION OF TWO VARIABLES..........................................................
  • 41). METHOD – 2: CONTINUITY OF FUNCTION OF TWO VARIABLES
  • 42). METHOD – 3: PARTIAL DERIVATIVES
  • 43). METHOD – 4: CHAIN RULE
  • 44). METHOD – 5: IMPLICIT FUNCTION.......................................................................................................
  • 45). METHOD – 6: TANGENT PLANE AND NORMAL LINE
  • 46). METHOD – 7: LOCAL EXTREME VALUES
  • 47). METHOD – 8: LAGRANGE’S MULTIPLIERS
  • 48). METHOD – 9: GRADIENT
  • 49). METHOD – 10: DIRECTIONAL DERIVATIVE
• UNIT
  • 50). METHOD – 1: DOUBLE INTEGRALS BY DIRECT INTEGRATION
  • 51). METHOD – 2: TRIPLE INTEGRALS BY DIRECT INTEGRATION
  • 52). METHOD – 3: D.I. OVER GENERAL REGION IN CARTESIAN COORDINATES
  • 53). METHOD – 4: DOUBLE INTEGRALS AS VOLUMES
  • 54). METHOD – 5: D.I. BY CHANGE OF ORDER OF INTEGRATION
  • 55). METHOD – 6: D.I. OVER GENERAL REGION IN POLAR COORDINATES............................
  • 56). METHOD – 7: AREA BY DOUBLE INTEGRATION IN CARTESIAN COORDINATES
  • 57). METHOD – 8: AREA BY DOUBLE INTEGRATION IN POLAR COORDINATES..................
  • 58). METHOD – 9: T.I. OVER GENERAL REGION IN CARTESIAN COORDINATES..................
  • 59). METHOD – 10: T.I. OVER GENERAL REGION IN CYLINDRICAL COORDINATES
  • 60). METHOD – 11: T.I. OVER GENERAL REGION IN SPHERICAL COORDINATES
  • 61). METHOD – 12: JACOBIAN........................................................................................................................ I N D E X
  • 62). METHOD – 1 3: D.I. BY CHANGE OF VARIABLE IN CARTESIAN COORDINATES
  • 63). METHOD – 14: D.I. BY CHANGE OF VARIABLE IN POLAR COORDINATES
  • 64). METHOD – 15: T.I. BY CHANGE OF VARIABLE OF INTEGRATION
• UNIT
  • 65). METHOD – 1: ECHELON FORM AND RANK OF MATRIX
  • 66). METHOD – 2: GAUSS ELIMINATION METHOD
  • 67). METHOD – 3: GAUSS - JORDAN METHOD
  • 68). METHOD – 4: INVERSE BY GAUSS - JORDAN METHOD
  • 69). METHOD – 5: EIGEN VALUES, EIGEN VECTORS AND EIGEN SPACE
  • 70). METHOD – 6: ALGEBRAIC AND GEOMETRIC MULTIPLICITY
  • 71). METHOD – 7: DIAGONALIZATION
  • 72). METHOD – 8: THE CAYLEY - HAMILTON THEOREM.................................................................
• FORMULAE, RESULTS & SYMBOLS
  • 73). STANDARD SYMBOLS:
  • 74). BASIC FORMULAE:......................................................................................................................................
  • 75). STANDARD SERIES:
  • 76). TRIGONOMETRIC IDENTITIES AND FORMULAE:
  • 77). VALUES OF CIRCULAR FUNCTIONS:
  • 78). INVERSE TRIGONOMETRIC FUNCTIONS:
  • 79). DIFFERENTIATION:
  • 80). INTEGRATION:
  • 81). RELATION WITH CARTESIAN COORDINATES:
  • 82). HYPERBOLIC FUNCTIONS:
  • 83). FREQUENTLY USED LIMIT:....................................................................................................................
  • 84). LOGARITHM RULES:
  • 85). AREA:.................................................................................................................................................................
  • 86). VOLUME:..........................................................................................................................................................
  • 87). VALUE OF SOME CONSTANTS:
  • 88). TRIGONOMETRIC TABLE:
• LIST OF ASSIGNMENTS...............................................................................

UNIT 1

❖ INDETERMINATE FORMS:

✓ The following are indeterminate forms which we will study:

, 0 × ∞, ∞ − ∞, 0

0

0

∞

❖ L’ HOSPITAL’S RULE:

If lim

x → a

f

x

g

x

leads to the indeterminate form

or

then

lim

x → a

f

x

g

x

= lim

x → a

f

′

x

g

′

( x

, provided the later limit exists.

✓ Procedure to find the limit using L’ Hospital’s rule:

(1). Differentiate numerator and denominator separately and apply the limit.

(𝟐). If it again reduces to indeterminate form

or

then again

differentiate numerator and denominator separately and apply the limit.

. Continue this process till we get finite or infinite value of the limit.

✓ Remark: lim

x → 0

log x = −∞ & lim

x → ∞

log x = ∞.

METHOD – 1: 0/0 TYPE INDETERMINATE FORM

C 1

Evaluate lim

x → 0

e

x

− 1 − x

x

2

C 2

Evaluate lim

x →

π

2

2x − π

cos x

H 3

Evaluate the examples: (𝟏). lim

x → 0

x − tan x

x

3

) (𝟐). lim

x → 0

sin x − x +

x

3

x

5

T 4

Evaluate lim

x → 0

2x − x cos x − sin x

2 x

3

C 5

Evaluate lim

x → 0

sin x

2

− x

2

x

2

( sin x

2

T 6

Evaluate the examples: (𝟏). lim

x → 0

tan x − x

sin x − x

) (𝟐). lim

x → 0

tan x − sin x

sin x

3

(𝟑). lim

x → 0

x

cos x − 1

sin x − x

) (𝟒). lim

x → 0

1 + x − 2 − x

2 sin

2

x

C 7

Evaluate lim

x → 0

e

x

• e

−x

− x

2

sin x

2

− x

2

H 8

Evaluate lim

x → 0

e

x

− e

sin x

x − sin x

T 9

Evaluate lim

x →

1

2

cos

2

πx

e

2x

− 2ex

𝟐

C 10

Evaluate lim

x → y

x

y

− y

x

x

x

− y

y

H 11

Evaluate lim

x → 0

ln cos √

x

x

H 12

Evaluate lim

x → 0

xe

x

− log

1 + x

x

2

W- 19

❖ 0 × ∞ TYPE INDETERMINATE FORM:

In this case we write f

x

∙ g

x

as

f

x

g

x

or

g

x

f

x

which leads to the form

or

, where L’ Hospital’s rule is applicable.

✓ Procedure to find the limit of indeterminate form 0 × ∞:

. Transform f

x

∙ g

x

into

f(x)

g

x

or

g(x)

f

x

i. e. transform 0 × ∞ into

or

(𝟐). Remember: Don’t put the logarithm function in the denominator in step (1).

. Apply L’ Hospital’s rule to find the value of given limit.

METHOD – 3: 0 × ∞ FORM

C 1

Evaluate lim

x → ∞

{ (a

1

x − 1 ) x }.

H 2

Evaluate lim

x → 1

1 − x

tan (

πx

C 3

Evaluate lim

x → ∞

x + 1 − √

x) log (

x

H 4 Evaluate lim

x → 0

{ (sin x) (ln x) }.

T 5 Evaluate the given examples:

(𝟏). lim

x → a

{ ln ( 2 −

x

a

) cot

x − a

} (𝟐). lim

x →

1

2

{ ln (

− x) cot (x −

❖ ∞ - ∞ TYPE INDETERMINATE FORM:

In this case, we write f

x

− g

x

g

x

f

x

f

x

g

x

which leads to the form

or

where, L’ Hospital’s rule is applicable.

✓ Procedure to find the limit of indeterminate form ∞ − ∞:

. Take L. C. M. in f

x

− g(x) which will transform ∞ − ∞ into

or

(𝟐). Apply L’ Hospital’s rule to find the value of given limit.

METHOD – 4: ∞ – ∞ FORM

C 1

Evaluate lim

x → 0

sin x

x

H 2 Evaluate lim

x → 0

( cosec x − cot x).

T 3

State L

′

Hospital

′

s Rule. Use it to evaluate lim

x → 0

x

2

sin x

2

W- 18

C 4

Evaluate lim

x → 0

x

2

cot x

2

C 5

Use L

′

Hospital

′

s rule to find the limit of lim

x → 1

x

x − 1

log x

S- 19

H 6

Evaluate lim

x → 0

x

e

x

T 7

Evaluate lim

x → 2

x − 2

log(x − 1 )

C 3 Evaluate lim

x → 0

cos x

cot x

T 4 Evaluate given examples: (𝟏). lim

x → 1

x − 1

x− 1

(𝟐). lim

x →

π

2

sin x

tan x

(𝟑). lim

x →

π

2

{ (tan x)

cos x

C 5

Evaluate lim

x → 0

tan x

x

1

x

2

𝟏/𝟑

C 6

Evaluate lim

x → 0

a

x

• b

x

• c

x

1

3x

𝟏/𝟗

H 7 Evaluate the given examples:

(𝟏). lim

x → 0

x

x

x

x

1

x

(𝟐). lim

x → 0

a

x

• b

x

• c

x

• d

x

1

x

𝟏/𝟒

𝟏/𝟒

T 8

Evaluate lim

x → 0

e

x

• e

2x

• e

3x

1

x

𝟐

H 9

Evaluate lim

x → 0

(a

x

• x)

1

x .

H 10

Evaluate lim

x → 1

2 − x

(tan

πx

2

)

𝟐/𝛑

T 11

Evaluate lim

x →

π

2

cos x

(

π

2

− x)

T 12

Evaluate the examples: (𝟏). lim

x →

π

2

cosec x

(tan

2

x)

} (𝟐). lim

x → 0

(cos √

x )

1

x

C 13

Evaluate lim

x → 0

x

(

1

ln(e

x

− 1 )

)

T 14

Evaluate lim

x → 1

( 1 − x

2

(

1

log( 1 −x)

)

❖ IMPROPER INTEGRALS:

✓ If the limit of integral is infinite (one or both) and/or integrand function is discontinuous

for some value(s) on the interval of given integral then such integral is known as an

improper integral.

✓ Types of Improper Integrals:

(1). Improper integral of first kind.

(2). Improper integral of second kind.

(3). Improper integral of third kind (combination of 1

st

nd

kind).

❖ IMPROPER INTEGRAL OF FIRST KIND:

For ∫ f

x

dx

b

a

either a or b or both (a and b) are infinite, then such integral is known as an

improper integral of first kind.

✓ Sub-types of improper integrals of first kind are as follows:

(1). If f is continuous on [a, ∞) then

∫ f(x)dx = lim

b → ∞

∫ f(x)dx

b

a

∞

a

(2). If f is continuous on (−∞, b] then

C 7

Evaluate ∫

1 + x

2

dx

∞

−∞

H 8

Evaluate

x

1 + x

2

2

dx.

∞

−∞

T 9

Evaluate ∫ e

x

dx

∞

−∞

C 10

Evaluate ∫

e

x

• e

−x

dx

∞

−∞

❖ IMPROPER INTEGRAL OF SECOND KIND:

For ∫ f(x)dx

b

a

, f(x) become discontinuous (infinite) at x = a or x = b or at finite

number of points in the interval

a, b

, then such an integral is known as improper

integral of second kind.

✓ Sub-types of improper integrals of second kind are as follows:

(1). If x = a is the point of discontinuity for f(x) then the integral is defined as

∫ f

x

dx = lim

t → a

∫ f

x

dx

b

t

b

a

(2). If x = b is the point of discontinuity for f(x) then the integral is defined as

∫ f

x

dx = lim

t → b

∫ f

x

dx

t

a

b

a

(3). If x = c is the point of discontinuity for f(x) then the integral is defined as

∫ f

x

dx = ∫ f

x

dx + ∫ f

x

dx

b

c

c

a

lim

t → c

∫ f(x)dx

t

a

b

a

• lim

t → c

∫ f

x

dx

b

t

METHOD – 7: IMPROPER INTEGRAL OF SECOND KIND

C 1

Evaluate the integrals ∫

x − 2

dx

5

2

and ∫

(a − x)

2

dx

b

a

H 2

Evaluate the integrals ∫

x

2

dx

1

0

and ∫

x

2 / 3

dx

1

− 1

Answer: ∞ & 6.

H 3

Evaluate

sec x dx

π

2

0

T 4

Evaluate

3 − x

dx.

3

0

T 5

Can we solve the integral ∫

x − 2

2

dx

5

0

directly? Give the reason.

C 6

Evaluate ∫

x − 1

2 / 3

dx.

3

0

𝟏/𝟑

𝟏/𝟑

𝟏/𝟑

T 7

Evaluate ∫

√a

2

− x

2

dx

a

−a

METHOD – 8: CONVERGENCE OF IMPROPER INTEGRAL OF FIRST KIND

C 1

Check the convergence of ∫

x

dx

∞

1

and ∫

x

√ 2

dx

∞

1

C 2

Check the convergence of ∫

dx

1 + x

2

1 + tan

− 1

x

∞

0

H 3

Check the convergence of ∫

x

p

dx

∞

1

T 4

Check the convergence of ∫

sin

2

x

x

2

dx

∞

1

and ∫

sin

3

x

x

3

dx

∞

1

C 5

Check the convergence of ∫

log x

dx

∞

2

H 6

Check the convergence of ∫

x

1 + x

4

dx

∞

2

and ∫

x ( 1 + x)

2

dx

∞

1

T 7

Check the convergence of ∫ e

−x

2

dx

∞

1

C 8

Investigate the convergence of ∫

5x

1 + x

2

)

3

dx.

∞

5

W- 18

H 9

Prove that ∫

e

x

dx

∞

1

is convergent.

H 10

Check the convergence of ∫

3x + 5

x

4

dx

∞

4

T 11

Investigate the convergence of

x

10

1 + x

5

1 + x

27

dx.

∞

0

W- 18

METHOD – 9: CONVERGENCE OF IMPROPER INTEGRAL OF SECOND KIND

C 1

Check the convergence of ∫

x

2

dx.

5

0

H 2

Determine whether the integral ∫

dx

x − 1

3

0

converges or diverges.

S- 19

H 3

Determine

(x − 2 )

2

dx

2

0

and is it convergent or divergent?

T 4

Check the convergence of

dx

√ 1 − x

2

1

0

C 5 Define improper integral of both the kinds and check the convergence of

dx

√ 9 − x

2

3

0

H 6

Check the convergence of ∫

cos 3x

x

5 / 2

dx

3

0