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Differential Calculus Derivatives of Standard Functions :

. . . . . . . . . . ,

. ,

. √ .

.

.

√

!

. !

Seven Indeterminant Forms (

, , 0x ∞, ∞*∞, 1( , ∞, , 0.

(

.

√ .

.

.

.

.

.

.

.

.

√

" " " " " " " " " " " " " " " √ !

" √

" " .

" .

√

.

" .

√ ! . #$ #$/ ! $#/

# $#/ #$/ . & ' $ $

Maclaurin’s Series // /// //// / / !

!

!

!1 ! ! !

www.MathsHomeWork123.com / Differential Calculus

www.MathsHomeWork123.com

Curvature of Curve The rate of bending of a curve in any interval is called the Curvature of the curve in that interval. Polar Curve r = f(ϴ ϴ)

Cartesian Curve y = f(x) 34 37 9 51 ! 6 8 3* 3*

34 3* 9 51 ! 6 8 37 37 Sin Ψ =

:;

cos Ψ =

:=

tan Ψ =

:;

:<

34 3? 9 5? 9 ! 6 8 3> 3>

34 3> 9 9 5 1!? 6 8 3? 3?

sin @ =

cos @ =

:<

tan @ =

:=

?

:A :<

:B :<

?

:A :B

p= r sin @

Radius of Curvature

Parametric Form

Implicit Form

The reciprocal of the Let x=f(t) and y=g(t) be the Let f(x,y)=o be the implicit Curvature of a curve at parametric equations of form of the given curve. any point is called the the given curve. Radius of Curvature at the point and is denoted by C

D ! 6

E 8 F

/

E

Polar Form Let r = f(ϴ ϴ) be the given curve in polar coordinates. KL ! L M/ C L ! L L

C

G// ! / H

/

C

// // /// /

I/ ! /E J

/

/

/E / E / /E ! /EE /

Centre of Curvature in the Cartesian Form

Circle of Curvature

!

N, E N O ! , E P ! QR

The equation of the circle of curvature is

where

E

,

E

N C N

! E E

Local Maxima and Minima for functions of one variable Given y=f(x), (i) if f/(c)=0 and f//(c)>0, then f has a local minimum at c. (ii) if f/(c)=0 and f//(c) 0 and A0 (or B>0) iii) f(a,b) is not an extremum if AC-B2 < 0 and iv) If AC-B2 > 0, the test is inconclusive.

where λ is called Lagrange Multiplier which is independent of x,y,z, The necessary condition for a maximum or minimum are S/ , S

S/ , SE

S/ ST

Solving the above equations for four unknowns λ, x, y, z, we obtain the point (x,y,z). The point may be a maxima, A function f(x,y) at (a,b) or f(a,b) is said minima or neither which is decided by the to be a Stationary Value of f(x,y) if physical consideration. fx(a,b)=0 and fy(a,b)=0.

Stationary Value

Jacobians

Properties of Jacobian

If u1, u2, u3, …….un are functions of n 1. If u and v are the functions of x and y, S#,$ S ,E variables x1, x2, x3, …xn, then the then

1. S ,E S#,$ Jacobian of the transformation from x1, x2, x3, …xn to u1, u2, u3, …….un is 2. If u,v are the functions of x,y and x,y defined by are themselves functions of r,s then S# W V S V X VS# U S

1 Y

1

S# \ S [ S# , # , … … # ^#,#,……# X [ S## [ S , , … . . S Z

then

S#,$ S ,E

S ,E

SL,

S#,$ SL,

.

3. If u,v,w are functionally dependent function of three independent variables x,y,z then

S#,$,_ S ,E,T

0

www.MathsHomeWork123.com / Differential Calculus

View more...
Differential Calculus Derivatives of Standard Functions :

. . . . . . . . . . ,

. ,

. √ .

.

.

√

!

. !

Seven Indeterminant Forms (

, , 0x ∞, ∞*∞, 1( , ∞, , 0.

(

.

√ .

.

.

.

.

.

.

.

.

√

" " " " " " " " " " " " " " " √ !

" √

" " .

" .

√

.

" .

√ ! . #$ #$/ ! $#/

# $#/ #$/ . & ' $ $

Maclaurin’s Series // /// //// / / !

!

!

!1 ! ! !

www.MathsHomeWork123.com / Differential Calculus

www.MathsHomeWork123.com

Curvature of Curve The rate of bending of a curve in any interval is called the Curvature of the curve in that interval. Polar Curve r = f(ϴ ϴ)

Cartesian Curve y = f(x) 34 37 9 51 ! 6 8 3* 3*

34 3* 9 51 ! 6 8 37 37 Sin Ψ =

:;

cos Ψ =

:=

tan Ψ =

:;

:<

34 3? 9 5? 9 ! 6 8 3> 3>

34 3> 9 9 5 1!? 6 8 3? 3?

sin @ =

cos @ =

:<

tan @ =

:=

?

:A :<

:B :<

?

:A :B

p= r sin @

Radius of Curvature

Parametric Form

Implicit Form

The reciprocal of the Let x=f(t) and y=g(t) be the Let f(x,y)=o be the implicit Curvature of a curve at parametric equations of form of the given curve. any point is called the the given curve. Radius of Curvature at the point and is denoted by C

D ! 6

E 8 F

/

E

Polar Form Let r = f(ϴ ϴ) be the given curve in polar coordinates. KL ! L M/ C L ! L L

C

G// ! / H

/

C

// // /// /

I/ ! /E J

/

/

/E / E / /E ! /EE /

Centre of Curvature in the Cartesian Form

Circle of Curvature

!

N, E N O ! , E P ! QR

The equation of the circle of curvature is

where

E

,

E

N C N

! E E

Local Maxima and Minima for functions of one variable Given y=f(x), (i) if f/(c)=0 and f//(c)>0, then f has a local minimum at c. (ii) if f/(c)=0 and f//(c) 0 and A0 (or B>0) iii) f(a,b) is not an extremum if AC-B2 < 0 and iv) If AC-B2 > 0, the test is inconclusive.

where λ is called Lagrange Multiplier which is independent of x,y,z, The necessary condition for a maximum or minimum are S/ , S

S/ , SE

S/ ST

Solving the above equations for four unknowns λ, x, y, z, we obtain the point (x,y,z). The point may be a maxima, A function f(x,y) at (a,b) or f(a,b) is said minima or neither which is decided by the to be a Stationary Value of f(x,y) if physical consideration. fx(a,b)=0 and fy(a,b)=0.

Stationary Value

Jacobians

Properties of Jacobian

If u1, u2, u3, …….un are functions of n 1. If u and v are the functions of x and y, S#,$ S ,E variables x1, x2, x3, …xn, then the then

1. S ,E S#,$ Jacobian of the transformation from x1, x2, x3, …xn to u1, u2, u3, …….un is 2. If u,v are the functions of x,y and x,y defined by are themselves functions of r,s then S# W V S V X VS# U S

1 Y

1

S# \ S [ S# , # , … … # ^#,#,……# X [ S## [ S , , … . . S Z

then

S#,$ S ,E

S ,E

SL,

S#,$ SL,

.

3. If u,v,w are functionally dependent function of three independent variables x,y,z then

S#,$,_ S ,E,T

0

www.MathsHomeWork123.com / Differential Calculus

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