ENGINEERING MATHEMATICS -I SECTION-Bggn.dronacharya.info/APSDept/Downloads/QuestionBank/... · 2016-08-09 · ENGINEERING MATHEMATICS -I SECTION-B. DIFFERENTIAL CALCULUS Successive

ENGINEERING MATHEMATICS-ISECTION-B

DIFFERENTIAL CALCULUS

Successive Differentiation

Leibnitz Theorem and Applications

Taylors and MaclaurinsSeries Curvature

Asymptotes

Curve tracing

Functions of Two or More

Variables

Partial Derivatives of First and

Higher Order

Eulerrsquos Theorem on

Homogeneous Functions

Differentiation of Composite

and Implicit Functions

Jacobians

Taylorrsquos Theorem For A Function of Two Variables

Maxima and Minima of Functions of Two Variables

Lagranges Method of

Undetermined

Multipliers

Differentiation Under

Integral Sign

E-LEARNINGTopic Taylorrsquos seriesE-learning httpnptelacincourses12210401711

Topic Maclaurinrsquos seriesE-learning httpnptelacincourses12210401711

Topic Partial derivatives of first order amp its higher orderE-learning httpnptelacincourses12210100331

Topic Eulerrsquos theorem on homogeneous functions E-learning wwwnptelacincourses122101003downloadsLecture-31pdf

Topic Total differentialDerivatives of composite and implicit functionE-learning httpnptelacincourses12210100332

httpnptelacincourses12210100333

Topic Maxima and minima of function of two variablesE-learning httpnptelacincourses12210401710httpnptelacincourses12210100337httpnptelacincourses12210401726

Topic Lagrangersquos method of undermined multipliersE learning httpnptel ac incourses12210401727

The Process of Differentiating a function again and again is called successive Differentiation

If y be a function of x then its successive derivatives are denoted by

119889119889119889119889119889119889119889119889

11988911988921198891198891198891198891198891198892

11988911988931198891198891198891198891198891198893 hellip hellip hellip hellip hellip hellip hellip

119889119889119899119899119889119889119889119889119889119889119899119899

119889119889111988911988921198891198893 hellip hellip hellip hellip hellip hellip hellip hellip hellip 119889119889119899119899

119889119889prime 119889119889primeprime119889119889primeprimeprime hellip hellip hellip hellip hellip hellip hellip hellip hellip119889119889119899119899

Example 1 Find the fourth derivative of tan x at 119889119889 = 1205871205874

Example 2 119894119894119894119894 119889119889 = 119860119860119890119890119898119898119889119889 + 119861119861119890119890119899119899119889119889 119875119875119875119875119875119875119875119875119890119890 119905119905ℎ119886119886119905119905 1198891198892119889119889

1198891198891198891198892 minus (119898119898 + 119899119899) 119889119889119889119889119889119889119889119889

+119898119898119899119899119889119889 = 0

Example 1 Find the fourth derivative of tan x at

Example 2

SOME STANDARD RESULTS1 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 119889119889119898119898 = 119898119898

(119898119898minus119899119899)119889119889119898119898minus119899119899 119894119894119894119894 119898119898119898119898119898119898119898119898 gt 119899119899

2 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 (119886119886119889119889 + 119887119887)119898119898 = 119898119898(119898119898 minus 1)(119898119898 minus 2) hellip hellip hellip hellip hellip hellip (119898119898 minus 119899119899 +1)(119886119886119889119889 + 119887119887)119898119898minus119899119899 119886119886119899119899 119894119894119894119894 119898119898119898119898119898119898119898119898 gt 119899119899

3 119865119865119894119894119899119899119889119889 119905119905ℎ119890119890 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 1119886119886119889119889+119887119887

= (minus1)119899119899 119899119899 119886119886119899119899

(119886119886119889119889minus119887119887)119899119899+1

4 119865119865119894119894119899119899119889119889 119905119905ℎ119890119890 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894119900119900119875119875119900119900(119886119886119889119889 + 119887119887) = (minus1)119899119899minus1(119899119899minus1)119886119886119899119899

(119886119886119889119889+119887119887)119899119899

5 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 119886119886119898119898119889119889 = 119898119898119899119899119886119886119898119898119889119889 (log119886119886)119899119899 6 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 119890119890119898119898119889119889 = 119898119898119899119899119890119890119898119898119889119889

7 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 sin(119886119886119889119889 + 119887119887) = 119886119886119899119899sin119886119886119889119889 + 119887119887 + 119899119899 1205871205872

8 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 cos(119886119886119889119889 + 119887119887) = 119886119886119899119899cos119886119886119889119889 + 119887119887 + 119899119899 1205871205872

9 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 eax sin(119887119887119889119889 + 119888119888) = (1198861198862 + 1198871198872)1198991198992119890119890119886119886119889119889 sin119887119887119889119889 + 119888119888 +

119899119899119905119905119886119886119899119899minus1 119887119887119886119886

10 119899119899119905119905ℎ 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890 119875119875119894119894 eax cos(119887119887119889119889 + 119888119888) = (1198861198862 + 1198871198872)1198991198992119890119890119886119886119889119889 cos119887119887119889119889 + 119888119888 +

119899119899119905119905119886119886119899119899minus1 119887119887119886119886

Statement- if y=uv where u and v are function of x having derivative of nth order then

119889119889119899119899 = 1198991198991198621198620119906119906119899119899119875119875 + 1198991198991198621198621119906119906119899119899minus11198751198751 + 1198991198991198621198622119906119906119899119899minus21198751198752 + ⋯hellip hellip hellip +119899119899119862119862119875119875119906119906119899119899minus119875119875119875119875119875119875+ ⋯hellip hellip +119899119899119862119862119899119899 119906119906119875119875119899119899

119908119908ℎ119890119890119875119875119890119890 119904119904119906119906119894119894119894119894119894119894119889119889119890119890119904119904 119889119889119890119890119899119899119875119875119905119905119890119890 119905119905ℎ119890119890 119899119899119906119906119898119898119887119887119890119890119875119875 119875119875119894119894 119889119889119890119890119875119875119894119894119875119875119886119886119905119905119894119894119875119875119890119890119904119904

Example 1 119868119868119894119894 119889119889 = 119889119889119899119899 log 119889119889 119901119901119875119875119875119875119875119875119890119890 119905119905ℎ119886119886119905119905 119889119889119899119899+1 = 119899119899 119889119889

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 2

119868119868119894119894 119889119889 = cos (119898119898 log119889119889)119901119901119875119875119875119875119875119875119890119890 119905119905ℎ119886119886119905119905 1198891198892119889119889119899119899+2 + (2119899119899 + 1)119889119889119889119889119899119899+1 + (1198981198982 + 1198991198992)119889119889119899119899 = 0

Leibnitzrsquos Theorem

Example 1

LINK FOR REFERENCE

Leibnitzrsquos Theorem for successive differentiation

httpswwwyoutubecomwatchv=67uJGwsZz-Q

The Taylorrsquos series is named after the English mathematician Brook Taylor (1685ndash1731)

The Maclaurinrsquos series is named for the Scottish mathematician Colin Maclaurin(1698ndash1746)

This is despite the fact that the Maclaurinrsquos seriesis really just a special case of the Taylorrsquos series

APPLICATIONS OF TAYLORrsquoS AND MACLAURINrsquoS SERIES

Expressing the complicated functions in terms of simple polynomials

Complicated functions can be made smooth Differentiation of the such functions can be

done as often as we please In the field of Ordinary Differential Equations when

finding series solution to Differential Equations In the study of Partial Differential Equations

GENERAL TAYLORrsquoS SERIES

(I) Expressing f(x + h) in ascending integral powers of h

provided that all derivatives of f(x) are continuous and exist in the interval [x x+h]

(II) Expressing f(x) in ascending integral powers of (x ndash a)

119894119894(119889119889 + ℎ) = 119894119894(119889119889) + ℎ119894119894ʹ(119889119889) + ℎ2

2119894119894ˈʹ(119889119889) +

3119894119894ˈˈˈ(119889119889) + ⋯⋯

119894119894(119889119889) = 119894119894119886119886 + (119889119889 minus 119886119886)

= 119894119894(119886119886) + (119889119889 minus 119886119886)119894119894ʹ(119886119886) + (119889119889 minus 119886119886)2

2119894119894ˈʹ(119886119886) +

(119889119889 minus 119886119886)3

3119894119894ˈˈˈ(119886119886) + ⋯⋯

GUIDELINES FOR FINDING TAYLOR SERIES

Expanding f(x) about x = aDifferentiate f(x) several timesEvaluate each derivative at x = aEvaluate

Substitute the above values in

119894119894(119889119889) = 119894119894(119886119886) + (119889119889 minus 119886119886)119894119894ʹ(119886119886) + (119889119889 minus 119886119886)2

2119894119894ˈʹ(119886119886) +

(119889119889 minus 119886119886)3

3119894119894ˈˈˈ(119886119886) + ⋯⋯

Example

Find the Taylor series for f(x) = sin x at c = π4

f(x) = sin x

f rsquo(x) = cos x

f rsquorsquo(x) = - sin x

f rsquorsquorsquo(x) = - cos x

f(4)(x) = sinx

f π =

f π = minus

(4) 24 2

f π =

Conthellip

( ) 2 2 2 2( ) ( ) ( ) 2 2 4 2 2 4 2 4

f x c x x xn n

π π πinfin

minus= + minus minus minus minus +

sdotsum

2 4 4 41 ( ) 2 4 2 3

x x xx

π π ππ

minus minus minus = + minus minus minus + +

2 3 42 2 2 2 2( ) ( ) ( ) ( ) 2 2 4 2 2 4 2 3 4 2 4 4

x x x xπ π π π= + minus minus minus minus minus + minus

sdot sdot sdot

( ) ( )

( ) ( )( ) ( )( ) ( )

n n nn

f x c f cf c f c x c x cn n

minus= + minus + minus +sum

The Maclaurinrsquos series is simply the Taylorrsquos series about the point x = 0It is given by

119943119943(119961119961) = 119943119943(120782120782) + 119961119961119943119943ʹ(120782120782) + 119961119961120784120784

120784120784 119943119943ˈʹ(120782120782) +

119961119961120785120785

120785120785 119943119943ˈˈˈ(120782120782) + ⋯⋯

MACLAURINrsquoS SERIES

Find the Maclaurinrsquos series for f(x) = ln(x2 + 1)2( ) ln( 1)f x x= + (0) 0f =

(0) 0f =

2 2( )( 1)

xf xxminus

(0) 2f =

4 ( 3)( )( 1)x xf xx

minus=

+(0) 0f =

4 2(4)

12( 6 1)( )( 1)x xf xx

minus + minus=

(4) (0) 12f = minus

4 2(5)

48 ( 10 5)( )( 1)

x x xf xxminus +

(5) (0) 0f =

Conthellip0(0)f =0(0)f =2(0)f =

( 00)f =(4) (0 2) 1f = minus

4 2(5)

48 ( 10 5)( )( 1)

x x xf xxminus +

=+ (5) ( 00)f =

6 4 2(6)

240(5 15 15 1)( )( 1)

x x xf xx

minus minus + minus=

(6) (0 4) 2 0f =

( ) ( )2

(0) (0)(0) (0) 2

n n nn

f x f ff f x x xn n

= + + +sum

2 3 4 5 62 0 12 0 240 2

0 3 4 5 6

0 x x x x x= + + + + +minus

4 62 4

2 3x xx x= minus +

+infin

minus=

Find the Taylor series for f(x) = endash2x at c = 0

f(x) = endash2x f(0) = 1

f rsquo(x) = -2endash2x f rsquo(0) = -2

f rsquorsquo(x) = 4endash2x f rsquorsquo(0) = 4

f rsquorsquorsquo(x) = -8endash2x f rsquorsquorsquo(0) = -8

f(4)(x) = 16endash2x f(4)(0) = 16

( ) ( )2

(0) (0)(0) (0) 2

n n nn

f x f ff f x x xn n

= + + +sum2 34 8 21 2

n nx x xxn

= minus + minus + +

minus= sum

We defined

the nth Maclaurin polynomial for a function as

the nth Taylor polynomial for f about x = x0 as

xfxxxfxxxfxfxxk

)())(()()(

0 minus++minus+minus+=minussum=

fxfxffxk

)0()0()0(

0++++=sum

DERIVATION FOR MACLAURIN SERIES FOR

Derive the Maclaurin series

++++=32

132 xxxex

The Maclaurin series is simply the Taylor series about the point x=0

( ) ( ) ( ) ( ) ( ) ( ) ( ) +primeprimeprimeprimeprime+primeprimeprimeprime+primeprimeprime+primeprime+prime+=+5432

5432 hxfhxfhxfhxfhxfxfhxf

00005432 hfhfhfhfhffhf

Example

DERIVATION (CONT)

Since xnxxx exfexfexfexf ==primeprime=prime= )( )( )( )( and

1)0( 0 == ef n

the Maclaurin series is then

2)()()()( 3

000 heheheehf +++=

11 32 hhh +++=

++++=xxxxf

Find the Maclaurin polynomial for f(x) = x cos x

f(x) = cos x f(0) = 1

f rsquo(x) = -sin x f rsquo(0) = 0

f rsquorsquo(x) = - cos x f rsquorsquo(0) = - 1

f rsquorsquorsquo(x) = sin x f rsquorsquorsquo(0) = 0

f(4)(x) = cos x f(4)(0) = 1( )

2(0) (0)cos (0) (0) 2

nnf fx f f x x x

n= + + +

2 441 0 0 2 4x x

= + minus minus +

( 1)(2 )

+infin

minus= sum

We find the Maclaurin polynomial cos x and multiply by x

1 2 4 6x x x

= minus + minus3 5 7

cos 2 4 6x x xx x x= minus + minus

Find the Maclaurin polynomial for f(x) = sin 3x

f(x) = sin x f(0) = 0

f rsquo(x) = cos x f rsquo(0) = 1

f rsquorsquo(x) = - sin x f rsquorsquo(0) = 0

f rsquorsquorsquo(x) = - cos x f rsquorsquorsquo(0) = -1

f(4)(x) = sin x f(4)(0) = 0( )

2(0) (0)sin (0) (0) 2

nnf fx f f x x x

n= + + +

0 0 03xx= + + minus +

( 1) (3 )(2 1)

+infin

minus=

We find the Maclaurin polynomial sin x and replace x by 3x

3 5 7x x xx= minus + minus

3 5 73 3( ) ( ) ( )sin3 3 3 5 7

x x xxx = minus + minus

Taylorrsquos amp Maclaurinrsquos Theorem for one variable

httpwwwcreativeworld9com201102iit-guest-lecture-mathematics-iii-videohtml

Definition An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity In other words

A Straight line at a finite distance from the origin is said to be an asymptote of an infinite branch of a curve if the perpendicular distance of a point P on that branch from the straight line tends to zero as P tends to infinity along the branch of the curve

Definition An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity In other words

A Straight line at a finite distance from the origin is said to be an asymptote of an infinite branch of a curve if the perpendicular distance of a point P on that branch from the straight line tends to zero as P tends to infinity along the branch of the curve

(Y) (P(xy))

(C) (M)

(N) (O) (A) (X)

A Curve With Finite Branches A Curve With Infinite BranchesEllipse Hyperbola

119961119961120784120784

119938119938120784120784+ 119962119962120784120784

119939119939120784120784= 120783120783 Its two branches are

y= b 120783120783 minus 119961119961120784120784

119938119938120784120784 and y= -b120783120783 minus 119961119961120784120784

119938119938120784120784

(upper half) (lower half) (Both branches lie within x=ax= -a y=b y=-b)

119961119961120784120784

119938119938120784120784minus119962119962120784120784

119939119939120784120784= 120783120783

Its infinite branches are

119962119962 = 119939119939119938119938radic119961119961120784120784 minus 119938119938120784120784 119962119962 = minus119939119939

119938119938radic119961119961120784120784 minus 119938119938120784120784

(Here 119962119962 rarr plusmninfin as 119961119961 rarr plusmninfin)

Its two branches are

y= b and y= -b

(upper half) (lower half)

(Both branches lie within x=ax= -a y=b

y=-b)

Its infinite branches are

(Here as )

KINDS OF ASYMPTOTES

ASYMPTOTE PARALLEL TO AXESAsymptote Parallel to x-axis

Rule to find the asymptote || to X-axis is to equate to zero the real linear factors in the co-efficient of the highest power of x in the equation of the curve

Asymptote Parallel to y-axis

Rule to find the asymptote || to Y-axis is to equate to zero the real linear factors in the co-efficient of the highest power of y in the equation of the curve

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 1119865119865119894119894119899119899119889119889 119905119905ℎ119890119890 11988911988921198891198892 minus 1198861198862(1198891198892 + 1198891198892) minus 1198861198863(119889119889 + 119889119889)+ 1198861198864119860119860119904119904119898119898119901119901119905119905119875119875119905119905119890119890 119875119875119886119886119875119875119886119886119900119900119900119900119890119890119900119900 119905119905119875119875 119886119886119889119889119890119890119904119904

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 6119865119865119894119894119899119899119889119889 119905119905ℎ119890119890 11988911988921198891198892 minus 1198891198892119889119889 minus 1198891198891198891198892 + 119889119889 + 119889119889 + 1= 0 119860119860119904119904119898119898119901119901119905119905119875119875119905119905119890119890 119875119875119886119886119875119875119886119886119900119900119900119900119890119890119900119900 119905119905119875119875 119886119886119889119889119890119890119904119904

The equation of straight line y=mx+c is the oblique asymptote to the given curve

Oblique Asymptote

Asymptote Parallel to x-axis

WORKING RULE FOR FINDING OBLIQUE ASYMPTOTES OF AN ALGEBRAIC CURVE OF THE NTH DEGREE

1 Find the empty119899119899(119898119898) This can be obtained by putting x=1 y=m in the highest degree terms of the given equation of the curve

2 Equate empty119899119899(119898119898) to zero and solve for m Let its roots be m1m2m3helliphelliphellip

3 Find empty119899119899minus1(119898119898) by putting x=1 and y=m in the next lower terms of the equation Similarly empty119899119899minus2(119898119898) may be found out by putting x=1 and y=m in the next lower degree terms in the curve and so on

4 Find the values of c1c2c3helliphelliphelliphelliphelliphellip corresponding to the values m1m2m3helliphelliphellipby using equation 119888119888 = empty119899119899minus1(119898119898)

empty119899119899prime (119898119898)

5 Then the required asymptotes are y= m1x+c1 y= m2+c2helliphelliphelliphelliphelliphelliphellip

1 Find the This can be obtained by putting x=1 y=m in the highest degree terms of the given equation of the curve

2 Equate to zero and solve for m

Let its roots be m1m2m3helliphelliphellip

3 Find by putting x=1 and y=m in the next lower terms of the equation Similarly may be found out by putting x=1 and y=m in the next lower degree terms in the curve and so on

4 Find the values of c1c2c3helliphelliphelliphelliphelliphellip corresponding to the values m1m2m3helliphelliphellipby using equation

6 If empty119899119899prime (119898119898) = 0 119894119894119875119875119875119875 119904119904119875119875119898119898119890119890 119875119875119886119886119900119900119906119906119890119890 119875119875119894119894 119898119898 119886119886119899119899119889119889 empty119899119899minus1(119898119898) ne 0corressponding to that value then there will be no asymptote corresponding to that value of m

7 If empty119899119899prime (119898119898) = 0 and empty119899119899minus1(119898119898) ne 0 for some value of m the value of c are determined by

1198881198882

2empty119899119899primeprime (119898119898) + 119888119888empty119899119899minus1

prime (119898119898) + empty119899119899minus2(119898119898) = 0

And this will determine two value of c and thus we shall have two parallel asymptotes corresponding to this value of m

6 If corressponding to that value then there will be no asymptote corresponding to that value of m

7 If and for some value of m the value of c are determined by

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 1119865119865119894119894119899119899119889119889 119905119905ℎ119890119890 119886119886119904119904119889119889119898119898119905119905119875119875119905119905119890119890 119875119875119894119894 119905119905ℎ119890119890 119888119888119906119906119875119875119875119875119890119890 (119889119889 minus 119889119889)2(119889119889 + 2119889119889 minus 1)= 3119889119889 + 119889119889 minus 7

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 2119865119865119894119894119899119899119889119889 119886119886119900119900119900119900 119905119905ℎ119890119890 119886119886119904119904119889119889119898119898119905119905119875119875119905119905119890119890 119875119875119894119894 119905119905ℎ119890119890 119894119894119875119875119900119900119900119900119875119875119908119908119894119894119899119899119900119900 119888119888119906119906119875119875119875119875119890119890

(119894119894)1198891198892(119889119889 minus 2119886119886) = 1198891198893 minus 1198861198863

(119894119894119894119894)1198891198893 minus 21198891198891198891198892 minus 1198891198892119889119889 + 21198891198893 + 31198891198892 minus 7119889119889119889119889 + 21198891198892 + 2119889119889 + 2119889119889 + 1 = 0

(119894119894119894119894119894119894)1198891198893 minus 1198891198891198891198892 minus 1198891198892119889119889 + 1198891198893 + 1198891198892 minus 1198891198892 = 0

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 3 119904119904ℎ119875119875119908119908 119905119905ℎ119886119886119905119905 119905119905ℎ119890119890 119886119886119904119904119889119889119898119898119901119901119905119905119875119875119905119905119890119890119904119904 119875119875119894119894 119905119905ℎ119890119890 119888119888119906119906119875119875119875119875119890119890 11988911988921198891198892 minus 1198861198862(1198891198892 + 1198891198892)minus 1198861198863(119889119889 + 119889119889) + 1198861198864 = 0

Form the square through two of whose vertices the curve passes

PICTORIAL EXAMPLE 1graphed on Cartesian coordinates

The x and y-axes are the asymptotes of the curve

PICTORIAL EXAMPLE 2The graph of

y = x is the Asymptote

PICTORIAL EXAMPLE 3The graph of x2 + y2 = (xy)2 with 2 horizontal and 2 vertical asymptotes

The y-axis (x = 0) and the line y = x are both asymptotes

PICTORIAL EXAMPLE 5The graph of x3 + y3 = 3axy

A cubic curve the folium of Descartes (solid) with a single real asymptote (dashed) given by x + y + a = 0

PICTORIAL EXAMPLE 6The graph of Hyperbola

Its asymptotes are y = plusmn

ASYMPTOTE OF THE POLAR CURVES

119868119868119894119894 120572120572 119894119894119904119904 119886119886 119875119875119875119875119875119875119905119905 119875119875119894119894 119905119905ℎ119890119890 119890119890119890119890119906119906119886119886119905119905119894119894119875119875119899119899 119894119894(120579120579)= 0 119905119905ℎ119890119890119899119899 119875119875 sin(120579120579 minus 120572120572)

119894119894prime(120572120572) 119894119894119904119904 119886119886119899119899 119886119886119904119904119889119889119898119898119901119901119875119875119905119905119890119890 119875119875119894119894 119905119905ℎ119890119890 119901119901119875119875119900119900119886119886119875119875 119888119888119906119906119875119875119875119875119890119890

1119875119875

= 119894119894(120579120579)

Working rule for finding the asymptotes of polar curves

1 Write down the given equation as 1119875119875

= 119894119894(120579120579)

2 Equate 119894119894(120579120579) to zero and solve for 120579120579 = 120579120579112057912057921205791205793 hellip hellip hellip hellip 3 Find 119894119894prime(120579120579) and calculate 119894119894prime(120579120579) at 120579120579 = 120579120579112057912057921205791205793 hellip hellip hellip hellip 4 Then write asymptote as 119875119875 sin(120579120579 minus 1205791205791) = 1

119894119894prime (1205791205791) 119875119875 sin(120579120579 minus 1205791205792) =

1119894119894prime (1205791205792)

helliphelliphelliphelliphelliphelliphelliphellip

1 Write down the given equation as

2 Equate to zero and solve for

3 Find and calculate at

4 Then write asymptote as helliphelliphelliphelliphelliphelliphelliphellip

IMPORTANT FORMULAS1 119868119868119894119894 sin(120579120579) = 0 119905119905ℎ119890119890119899119899 120579120579 = 120587120587

2 119868119868119894119894119888119888119875119875119904119904120579120579 = 0 119905119905ℎ119890119890119899119899 120579120579 = (2119899119899 + 1) 1205871205872

3 119868119868119894119894119904119904119894119894119899119899120579120579 = 119904119904119894119894119899119899120572120572 119905119905ℎ119890119890119899119899 120579120579 = 119899119899120587120587 + (minus1)119899119899120572120572 4 119868119868119894119894119888119888119875119875119904119904120579120579 = 119888119888119875119875119904119904120572120572 119905119905ℎ119890119890119899119899 120579120579 = 2119899119899120587120587 plusmn 120572120572 119899119899 isin I 5 119868119868119894119894119905119905119886119886119899119899120579120579 = 119905119905119886119886119899119899120572120572 119905119905ℎ119890119890119899119899 120579120579 = 119899119899120587120587 + 120572120572 6 sin(119899119899120587120587 + 120579120579) = (minus1)119899119899 sin120579120579 7 cos(119899119899120587120587 + 120579120579) = (minus1)119899119899 cos120579120579 8 tan(119899119899120587120587 + 120579120579) = 119905119905119886119886119899119899 120579120579

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 119894119894119894119894119899119899119889119889 119905119905ℎ119890119890 119886119886119904119904119889119889119898119898119901119901119905119905119875119875119905119905119890119890119904119904 119875119875119894119894 119905119905ℎ119890119890 119894119894119875119875119900119900119900119900119875119875119908119908119894119894119899119899119900119900 119901119901119875119875119900119900119886119886119875119875 119888119888119906119906119875119875119875119875119890119890119904119904 (119894119894)119875119875 = 119886119886 tan120579120579 (119894119894119894119894)119875119875 sin120579120579 = 2 cos 2120579120579

Derivative of a function

Single-Variable Function

Recall how we find the derivative for a Single Variable function f(x)

hxfhxf

)()(lim0

minus+=

Rate of change of f with respect to x (slope)

Two-Variable Function

Rate of change of f with respect to xRate of change of f with respect to y

Partial derivatives of a function

kyxfkyxf

hyxfyhxf

)()(lim

minus+=

partpart

minus+=

partpart

RemarksbullIt is called the Partial Derivativebecause it describes the derivative in one directionbullScripted ldquodrdquo not the regular ldquodrdquo or ldquo2rdquobullWhen differentiate f with respect to x we treat y as if y were a constant and vice versa

Partial Derivative of f with respect to xPartial Derivative of f with respect to y

Ex Given f(xy) = xsup3 - xsup2y + xy + 3ysup2

yxyxyxyx

yxyyxxxx

+minus=

++minus=partpart

+partpart

minuspartpart

++minuspartpart

=partpart

230)1()2(3

sup2)3()(sup2)(sup3)(

sup2)3sup2sup3(

xfpartpart Find HERE we treat ldquoyrdquo as a constant

Assignment

w = x2 minus xy + y2 + 2yz + 2z2 + z

partwpartx

partwparty

and partwpartz

Example A cellular phone company has the following production function for a certain product

where p is the number of units produced with x units of labor and y units of capital

a) Find the number of units produced with 125 units of labor and 64 units of capital

b) Find the marginal productivities of labor and of capital

c) Evaluate the marginal productivities at x = 125 and y = 64

p(x y) = 50x2 3y1 3

Higher-Order DerivativesSingle-Variable

Function

)derivative (3rd

)derivative (2nd

e)(derivativ

dxfdxf

dxdfxf

Multi-Variable Function

wrt x)f of derivative partial (3rd

wrt x)f of derivative partial (2nd

wrt x)f of derivative (partial

partpart

yxyxxff x +minus=partpart

= 23 2 found We

Find xxxxf

partpart Find

( )yxf

partpartpart

=partpart

partpart

partpartpart

=partpart

partpart

so yxxy ff =

Mixed Derivatives

If find the following partial derivatives

Assignment

z = f (x y) = x2y3 + x4 y + xey

ffffff

A Function f(xy) is said to be homogeneous of degree (or order) n in the variables x

and y if it can be expressed in the form 119889119889119899119899 empty119889119889119889119889 119875119875119875119875 119889119889119899119899 empty119889119889

119889119889

An alternative test for a function f(xy) to be homogeneous of degree (or order ) n is that

119894119894(119905119905119889119889 119905119905119889119889) = 119905119905119899119899119894119894(119889119889 119889119889)

For example if 119894119894(119889119889119889119889) = 119889119889+119889119889radic119889119889+radic119889119889

(119894119894) 119894119894(119889119889 119889119889) =119889119889(1 + 119889119889

119889119889)

radic119889119889(1 + 119889119889119889119889

= 11988911988912 empty

119889119889119889119889

f(xy) is a homogeneous function of degree frac12 in x and y

Similarly a function f(xyz) is said to be homogeneous of degree n in the variables xyz if

119894119894(119889119889 119889119889 119911119911) = 119889119889119899119899empty 119889119889119911119911 119911119911119889119889 119875119875119875119875 119889119889119899119899(empty)

119889119889119889119889 119911119911119889119889 119875119875119875119875 119911119911119899119899empty

119889119889119911119911 119889119889119911119911

A Function f(xy) is said to be homogeneous of degree (or order) n in the variables x and y if it can be expressed in the form

For example if then

Alternative test is f(txtytz)= tn f(xyz)

Eulerrsquos Theorem on Homogeneous Functions

If u is a homogeneous function of degree n in x and y then 119889119889 120597120597119906119906120597120597119889119889

+ 119889119889 120597120597119906119906120597120597119889119889

119899119899119906119906

Since u is a homogeneous function of degree n in x and y it can be

expressed as 119906119906 = 119889119889119899119899 119894119894 119889119889119889119889

120597120597119906119906120597120597119889119889

= 119899119899119889119889119899119899minus1 119894119894119889119889119889119889 = 119889119889119899119899119894119894prime 119889119889119889119889 minus 1198891198891198891198892

119889119889 120597120597119906119906120597120597119889119889

= 119899119899119889119889119899119899 119894119894 119889119889119889119889 minus 119889119889119899119899minus1 119889119889 119894119894prime 119889119889

119889119889 (119894119894)

119860119860119900119900119904119904119875119875 120597120597119906119906120597120597119889119889

= 119889119889119899119899 119894119894prime 119889119889119889119889

1119889119889

= 119889119889119899119899minus1119894119894prime 119889119889119889119889

119889119889120597120597119906119906120597120597119889119889

= 119889119889119899119899minus1 119889119889 119894119894prime 119889119889119889119889 (119894119894119894119894)

If u is a homogeneous function of degree n in x and y then

Since u is a homogeneous function of degree n in x and y it can be expressed as

Adding (i) and (ii) we get 119889119889 120597120597119906119906120597120597119889119889

+ 119889119889 120597120597119906119906120597120597119889119889

= 119899119899119889119889119899119899119894119894 119889119889119889119889 = 119899119899119906119906

If u is a Homogeneous function of degree n in x and y then 1198891198892 12059712059721199061199061205971205971198891198892 +

2119889119889119889119889 1205971205972119906119906

120597120597119889119889120597120597119889119889+ 1198891198892 1205971205972119906119906

1205971205971198891198892 = 119899119899(119899119899 minus 1)119906119906

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 1 119894119894119894119894 119906119906 sinminus1 119889119889 + 119889119889radic119889119889 + 119889119889

119901119901119875119875119875119875119875119875119890119890 119905119905ℎ119886119886119905119905 1198891198892 12059712059721199061199061205971205971198891198892 + 2119889119889119889119889

1205971205972119906119906 120597120597119889119889120597120597119889119889

+ 1198891198892 12059712059721199061199061205971205971198891198892 = minus

sin119906119906 cos 2119906119906 4 1198881198881198751198751199041199043119906119906

Composit functions

(119894119894)119894119894119894119894 119906119906 = 119894119894(119889119889119889119889) 119908119908ℎ119890119890119875119875119890119890 119889119889 = empty(119905119905)119889119889 = 120593120593(119905119905)

Then u is called a composit function of t and we can find dudt

(119894119894119894119894) 119894119894119894119894 119911119911 = 119894119894(119889119889119889119889)119908119908ℎ119890119890119875119875119890119890 119889119889 = empty(119906119906 119875119875)119889119889 = 120593120593(119906119906 119875119875)

Then z is called a composite function of u and v so that we can find 120597120597119911119911

120597120597119906119906 119886119886119899119899119889119889 120597120597119911119911 120597120597119875119875

Adding (i) and (ii) we get

If u is a Homogeneous function of degree n in x and y then

Composit functions

Then z is called a composite function of u and v so that we can find

Cor 1 If u=f(xyz) and xyz are function of t then y is a composite function of t and 119889119889119906119906

119889119889119905119905= 120597120597119906119906

120597120597119889119889 119889119889119889119889119889119889119905119905

+ 120597120597119906119906120597120597119889119889

119889119889119889119889119889119889119905119905

+ 120597120597119906119906120597120597119911119911

119889119889119911119911119889119889119905119905

Cor 2 If z = f(xy) and x and y are the functions of u and v then

120597120597119911119911120597120597119906119906

=120597120597119911119911120597120597119889119889

120597120597119889119889120597120597119906119906

+ 120597120597119911119911120597120597119889119889

120597120597119889119889120597120597119906119906

120597120597119911119911120597120597119875119875

=120597120597119911119911120597120597119889119889

120597120597119889119889120597120597119875119875

+ 120597120597119911119911120597120597119889119889

120597120597119889119889120597120597119875119875

Cor 3 If u=f(xy) where y= empty(119889119889) then since x = 120593120593(119889119889) u is a composite function of x

119889119889119906119906119889119889119889119889

= 120597120597119906119906120597120597119889119889

119889119889119889119889119889119889119889119889

+ 120597120597119906119906120597120597119889119889

119889119889119889119889119889119889119889119889

=gt119889119889119906119906119889119889119889119889

= 120597120597119906119906120597120597119889119889

+ 120597120597119906119906120597120597119889119889

119889119889119889119889119889119889119889119889

Cor 1 If u=f(xyz) and xyz are function of t then y is a composite function of t and

Cor 3 If u=f(xy) where y= then since x = u is a composite function of x

Cor 4 If we are given a implicit function f(xy) = c then u=f(xy) where u=c using cor 3 we have

119889119889119906119906119889119889119889119889

= 120597120597119906119906120597120597119889119889

+ 120597120597119906119906120597120597119889119889

119889119889119889119889119889119889119889119889

But dudx=0 120597120597119906119906120597120597119889119889

+ 120597120597119906119906120597120597119889119889

119889119889119889119889119889119889119889119889

= 0 119875119875119875119875 120597120597119894119894120597120597119889119889

+ 120597120597119894119894120597120597119889119889

119889119889119889119889119889119889119889119889

= minus 120597120597119894119894120597120597119889119889120597120597119894119894120597120597119889119889

= minus 119894119894119889119889119894119894119889119889

Hence the differential coefficient of f(xy) wrt x is 120597120597119894119894120597120597119889119889

+ 120597120597119894119894120597120597119889119889

119889119889119889119889119889119889119889119889

But dudx=0

Hence the differential coefficient of f(xy) wrt x is

Cor 5 If f(xy) = c then by cor 4 we have 119889119889119889119889119889119889119889119889

= minus 119894119894119889119889119894119894119889119889

119863119863119894119894119894119894119894119894119890119890119875119875119890119890119899119899119905119905119894119894119886119886119905119905119894119894119899119899119900119900 119886119886119900119900119886119886119894119894119899119899 119908119908 119875119875 119905119905 119889119889119908119908119890119890 119900119900119890119890119905119905

11988911988921199061199061198891198892119889119889

= minus 119894119894119889119889 119889119889119889119889119889119889 (119894119894119889119889) minus 119894119894119889119889

119889119889119889119889119889119889 119894119894119889119889

1198941198941198891198892= minus

119894119894119889119889 120597120597119894119894119889119889120597120597119889119889 + 120597120597119894119894119889119889

120597120597119889119889 119889119889119889119889119889119889119889119889 minus 119894119894119889119889 120597120597119894119894119889119889120597120597119889119889 +

120597120597119894119894119889119889120597120597119889119889 119889119889119889119889119889119889119889119889

1198941198941198891198892

= minus 119894119894119889119889 119894119894119889119889119889119889 minus 119894119894119889119889119889119889 119894119894119889119889119894119894119889119889

minus 119894119894119889119889 119894119894119889119889119889119889 minus 119894119894119889119889119889119889 119894119894119889119889119894119894119889119889

1198941198941198891198892

= minus1198941198941198891198891198891198891198941198941198891198892 minus 119894119894119889119889119894119894119889119889119894119894119889119889119889119889 minus 119894119894119889119889119894119894119889119889119894119894119889119889119889119889 minus 119894119894119889119889119889119889 1198941198941198891198892

1198941198941198891198893

119867119867119890119890119899119899119888119888119890119890 11988911988921198891198891198891198891198891198892 = minus

1198941198941198891198891198891198891198941198941198891198892 minus 2119894119894119889119889119894119894119889119889119894119894119889119889119889119889 minus 119894119894119889119889119889119889 1198941198941198891198892

1198941198941198891198893

Cor 5 If f(xy) = c then by cor 4 we have

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 1 119868119868119894119894 119911119911 = 21198891198891198891198892 minus 31198891198892119889119889 119886119886119899119899119889119889 119894119894 119894119894119899119899119888119888119875119875119890119890119886119886119904119904119890119890119904119904 119886119886119905119905 119905119905ℎ119890119890 119875119875119886119886119905119905119890119890 119875119875119894119894 2 119888119888119898119898 119901119901119890119890119875119875 119904119904119890119890119888119888119875119875119899119899119889119889 119908119908ℎ119890119890119899119899 119894119894119905119905

119901119901119886119886119904119904119904119904119890119890119904119904 119905119905ℎ119875119875119875119875119906119906119900119900ℎ 119905119905ℎ119890119890 119875119875119886119886119900119900119906119906119890119890 119889119889 = 3119888119888119898119898 119904119904ℎ119875119875119908119908 119905119905ℎ119886119886119905119905 119894119894119894119894 119889119889 119894119894119904119904 119901119901119886119886119904119904119904119904119894119894119899119899119900119900 119905119905ℎ119875119875119875119875119906119906119900119900ℎ 119905119905ℎ119890119890 119875119875119886119886119900119900119906119906119890119890 119889119889 = 1 119888119888119898119898

119889119889 119898119898119906119906119904119904119905119905 119887119887119890119890 119889119889119890119890119888119888119875119875119890119890119886119886119904119904119894119894119899119899119900119900 119886119886119905119905 119905119905ℎ119890119890 119875119875119886119886119905119905119890119890 119875119875119894119894 22

15 119888119888119898119898 119901119901119890119890119875119875 119904119904119890119890119888119888119875119875119899119899119889119889 119894119894119899119899 119875119875119875119875119889119889119890119890119875119875 119905119905ℎ119886119886119905119905 119911119911 119904119904ℎ119886119886119900119900119900119900 119875119875119890119890119898119898119886119886119894119894119899119899 119888119888119875119875119899119899119904119904119905119905119886119886119899119899119905119905

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 2 119894119894119894119894 119906119906 119894119894119904119904 119886119886 ℎ119875119875119898119898119875119875119900119900119890119890119899119899119890119890119875119875119906119906119904119904 119894119894119906119906119899119899119888119888119905119905119894119894119875119875119899119899 119875119875119894119894 119899119899119905119905ℎ 119889119889119890119890119900119900119875119875119890119890119890119890 119894119894119899119899 119889119889 119889119889 119911119911 119901119901119875119875119875119875119875119875119890119890 119905119905ℎ119886119886119905119905

119889119889120597120597119906119906120597120597119889119889

+ 119889119889120597120597119906119906120597120597119889119889

+ 119911119911120597120597119906119906120597120597119911119911

= 119899119899119906119906

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 3119865119865119894119894119899119899119889119889119889119889119889119889119889119889119889119889 119908119908ℎ119890119890119899119899

(119894119894) 119889119889119889119889 + 119889119889119889119889 = 119888119888 (119894119894119894119894) (cos 119889119889)119889119889 = (sin119889119889)119889119889

Partial derivatives httpnptelacincourses12210100331

Partial derivatvesand euler th

wwwnptelacincourses122101003downloadsLecture-31pdf

NPTEL LINKS FOR REFERENCE

If u and v are functions of two independent variables x and y then the

determinant

120597120597119906119906120597120597119889119889

120597120597119875119875120597120597119889119889

is called Jacobian of uv with respect to xy and is

denoted by symbol J119906119906 119875119875119889119889 119889119889 119875119875119875119875 120597120597(119906119906 119875119875)

120597120597(119889119889 119889119889)

Simolarly if uvw be the function of xyz then the Jacobian of uvw with respect to xyz is

119869119869 119906119906 119875119875119908119908119889119889119889119889 119911119911

119875119875119875119875 120597120597(119906119906 119875119875119908119908)120597120597(119889119889119889119889 119911119911)

120597120597119906119906120597120597119889119889

120597120597119906119906120597120597119911119911

120597120597119875119875120597120597119889119889

120597120597119875119875120597120597119911119911

120597120597119908119908120597120597119889119889

120597120597119908119908120597120597119911119911

If u and v are functions of two independent variables x and y then the determinant is called Jacobian of uv with respect to xy and is denoted by symbol J

Properties of JACOBIANS

1 If uv are functions of rs where r s are functions of xy then 120597120597(119906119906 119875119875)120597120597(119889119889119889119889)

= 120597120597(119906119906 119875119875)120597120597(119875119875 119904119904)

120597120597(119875119875 119904119904)120597120597(119889119889119889119889)

[ 119888119888ℎ119886119886119894119894119899119899 119877119877119906119906119900119900119890119890 119894119894119875119875119875119875 119869119869119886119886119888119888119875119875119887119887119894119894119886119886119899119899119904119904]

2 If J1 is the Jacobian of uv with respect to xy and J2 is the Jacobian of xy with respect to uv then J1J2 =1 ie 120597120597(119906119906 119875119875)

120597120597(119889119889 119889119889)= 120597120597(119889119889 119889119889)

120597120597(119906119906 119875119875)= 1

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 1 119868119868119894119894 119889119889 = 119875119875 119904119904119894119894119899119899120579120579 119888119888119875119875119904119904empty 119911119911

= 119875119875 cos120579120579 119904119904ℎ119875119875119908119908 119905119905ℎ119886119886119905119905 120597120597(119889119889119889119889 119911119911)120597120597(119875119875120579120579120593120593)

= 1198751198752 sin120579120579

1 If uv are functions of rs where r s are functions of xy then

2 If J1 is the Jacobian of uv with respect to xy and J2 is the Jacobian of xy with respect to uv then J1J2 =1 ie

MAXIMA AND MINIMA OF FUNCTIONS OF TWO VARIABLES

A function f(xy) is said to have a maximum value at x = a y = b if f(ab) f(a+hb+k) for small and independent values of h and k positive or negative

A function f(xy) is said to have a minimum value at x = a y = b if f(ab) f(a+hb+k) for small and independent values of h and k positive or negative

RULE TO FIND THE EXTREME VALUES OF A FUNCTION

Let z = f(xy) be a function of two variables(i) Find and(ii) Solve =0 and =0 simultaneouslyLet (ab) (cd)hellip Be the solutions of theseequations(iii) For each solution in step (ii) find r =s = t =

(iv) (a) If rt - 0 and r 0 for a particular solution (ab) of step (ii)then z has a maximum value at (ab)

(b) ) If rt - 0 and r 0 for a particular solution (ab) of step (ii)then z has a minimum value at (ab)

(c) If rt - 0 for a particular solution (ab) of step (ii)then z has no extreme value at (ab)

(d) If rt - =0 the case is doubtful and requires further investigation

ASSIGNMENT

1Examine the extreme values of + + 6 x + 12 2 Find the points on the surface = xy + 1 nearest to the

origin3 A rectangular box open at the top is to have a volume of

32 cc Find the dimensions of the box requiring least material for its construction

4 Divide 24 into three parts such that the continued product of the first square of the second and the cube of the third may be maximum

Differentiation Under Integral SIGN

If a function f(xα) of the two variables x and α α being called parameter be integrated wrt x between limits a and bint 119894119894(119889119889α)119889119889119889119889 119894119894119904119904 119886119886 119894119894119906119906119899119899119888119888119905119905119894119894119875119875119899119899 119875119875119894119894 α for example119887119887

119886119886

sinα dx = minuscosαα

1205871205872

= minus1αcos

1205871205872

α minus 11205871205872

1 minus 1198881198881198751198751199041199041205871205872

119905119905ℎ119906119906119904119904 119894119894119899119899 119900119900119890119890119899119899119890119890119875119875119886119886119900119900 119894119894(119889119889α)119889119889119889119889 = 119865119865(α)119887119887

119886119886

If a function f(xα) of the two variables x and α α being called parameter be integrated wrt x between limits a and b

119923119923119923119923119923119923119939119939119923119923119923119923119923119923119963119963prime119956119956 119929119929119929119929119929119929119923119923

119868119868119894119894 119894119894(119889119889α)119886119886119899119899119889119889 120597120597120597120597119889119889

[119894119894(119889119889α)]119887119887119890119890 119888119888119875119875119899119899119905119905119894119894119899119899119875119875119906119906119904119904 119894119894119906119906119899119899119888119888119905119905119894119894119875119875119899119899119904119904 119875119875119894119894 119889119889 119886119886119899119899119889119889 α then 119889119889119889119889α

119894119894(119889119889α)119889119889119889119889119887119887

119886119886

= 120597120597120597120597119889119889

[119894119894(119889119889α)] 119889119889119889119889 119908119908ℎ119890119890119875119875119890119890 119886119886 119886119886119899119899119889119889 119887119887 119886119886119875119875119890119890 119888119888119875119875119899119899119904119904119905119905119886119886119899119899119905119905119904119904 119894119894119899119899119889119889119890119890119901119901119890119890119899119899119889119889119890119890119899119899119905119905 119875119875119894119894 α119887119887

119886119886

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 1119864119864119875119875119886119886119900119900119906119906119886119886119905119905119890119890 tanminus1 119886119886119889119889119889119889(1 + 1198891198892) 119889119889119889119889 (119886119886 ge 0)119887119887119889119889 119886119886119901119901119901119901119900119900119889119889119894119894119899119899119900119900 119889119889119894119894119894119894119894119894119890119890119875119875119890119890119899119899119905119905119894119894119886119886119905119905119894119894119875119875119899119899 119906119906119899119899119889119889119890119890119875119875 119905119905ℎ119890119890 119868119868119899119899119905119905119890119890119900119900119875119875119886119886119900119900 119904119904119894119894119900119900119899119899infin

119864119864119889119889119886119886119898119898119901119901119900119900119890119890 2 119890119890119875119875119886119886119900119900119906119906119886119886119905119905119890119890 log(1 + 119886119886119889119889)

1 + 1198891198892 119889119889119889119889 119886119886119899119899119889119889 ℎ119890119890119899119899119888119888119890119890 119904119904ℎ119875119875119908119908 119905119905ℎ119886119886119905119905 log(1 + 119889119889)

1 + 1198891198892 1198891198891198891198891

119886119886

= 1205871205878

Engineering Mathematics-ISECTION-B

Differential Calculus

Slide Number 3

E-learning

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SOME STANDARD RESULTS

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Taylor and Maclaurinrsquos Series

Applications of Taylorrsquos and Maclaurinrsquos Series

General TAYLORrsquos SERIES

Guidelines for Finding Taylor Series

Slide Number 14

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Maclaurinrsquos Series

Derivation for Maclaurin Series for

Derivation (cont)

Slide Number 23

Slide Number 24

Slide Number 25

AsymptoteS

Slide Number 27

Kinds of Asymptotes

Asymptote Parallel to Axes

Working rule for finding oblique Asymptotes of an algebraic curve of the nth Degree

Slide Number 31

Slide Number 32

Slide Number 33

Slide Number 34

Slide Number 35

Slide Number 36

Slide Number 37

Slide Number 38

Slide Number 39

Asymptote of the polar curves

Important Formulas

Slide Number 42

Slide Number 43

Slide Number 44

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Slide Number 46

Slide Number 47

Slide Number 48

Slide Number 49

Slide Number 50

Slide Number 51

Slide Number 52

Slide Number 53

Slide Number 54

Slide Number 55

Slide Number 56

Slide Number 57

Slide Number 58

Slide Number 59

Slide Number 60

Maxima and minima of functions of two variables

Rule to find the extreme values of a function

Slide Number 63

assignment

Slide Number 65

Slide Number 66

Slide Number 67