[PDF] Exercise 2.2 (Solutions) F.Sc Part 2

View - Download Exercise 2.2 (Solutions) F.Sc Part 2

Previous PDF

Next PDF

MathCity.org

Merging man and maths

Exercise 2.2 (Solutions)Page 53

Calculus and Analytic Geometry, MATHEMATICS 12

Available online @ http://www.mathcity.org, Version: 3.0

Question # 1

Find from first principles, the derivatives of the following expensions w.r.t. their respective independent variables: (i)

3ax b+ (ii) 5(2 3)x+ (iii) ()

23 2t
(iv)

5( )ax b-+ (v) 71

( )az b-

Solution

(i) Let ()

3y ax b= +

3( )y y a x x bδ δ⇒+ = + +

3y ax b a x yδ δ⇒= + + -

33( ) ( )ax b a x ax bδ= + + - +

3 2 2 3 3( ) 3( ) ( ) 3( )( ) ( ) ( )ax b ax b a x ax b a x a x ax bδ δ δ= + + + + + + - +

2 2 2 3 33 ( ) 3 ( )a ax b x a ax b x a xδ δ δ= + + + +

()2 2 3 23 ( ) 3 ( )x a ax b a ax b x a xδ δ δ= + + + +

Dividing by

xδ

2 2 3 23 ( ) 3 ( )ya ax b a ax b x a xx

Taking limit as

0xδ→

2 2 3 2

0 0lim lim 3 ( ) 3 ( )

x x ya ax b a ax b x a xxδ δ

2 2 3 23 ( ) 3 ( )(0) (0)dya ax b a ax b a

dx⇒= + + + +

23 ( ) 0 0dya ax b

dx⇒= + + + 23 ( )dya ax bdx⇒= + (ii) Let

5(2 3)y x= +

52 3y y x xδ δ⇒+ = + +

52 2 3y x x yδ δ⇒= + + -

55(2 3) 2 (2 3)x x xδ= + + - +

http://www.mathcity.org

FSc-II / Ex- 2.2 - 2

Dividing by

xδ

4 3 45 5 52 (2 3) 4 (2 3) .... 321 2 5yx x x xx

Taking limit as

0xδ→

4 3 4 0 0

5 5 5lim lim 2 (2 3) 4 (2 3) .... 321 2 5

x x yx x x xxδ δ

452 (2 3) 0 0 .... 01dyxdx

42(5)(2 3)dyx

dx⇒= + or 410(2 3)dyxdx= + (iii) Let

23 2y t

23( ) 2y y t tδ δ

23 3 2y t t yδ δ

2 2(3 2) 3 3 2y t t tδ δ

2

2233 2 1 3 23 2tt tt

2

233 2 1 13 2ttt

( )( )2

22 2 13 33 2 1 ( 2) .... 13 2 2! 3 2t ttt t

( )()2

22 363 2 1 .... 13 2 2 3 2t ty tt t

2

26 33 2 3 ....3 2 3 2t ttt t

13 33 2 2 3 ....3 2 3 2t ttt t

Dividing by

tδ

2 133 3 2 2 ....3 2y ttt t

Taking limit when

0tδ→, we have

http://www.mathcity.org

FSc-II / Ex- 2.2 - 3

3

0 03lim lim3 3 2 2 ....3 2t t

y ttt t

33 3 2 2 0 0 ....dyt

dx

36 3 2dytdx-⇒= - +

(iv) Let

5( )y ax b-= +

Do yourself

(v) Let 71
( )yaz b=- 7( )az b-= -

7( )y y a z z bδ δ

7 7( )y az b a z az bδ δ

7 7

1 1( )

a zy az baz bδδ

Now do yourself

Error Analyst

Irfan Mehmood 2015-16 Fazaia Degree College Risalpur

Please report us error at www.mathcity.org/error

Book: Exercise 2.2 , page 53

Calculus and Analytic Geometry Mathematic 12

Punjab Textbook Board, Lahore.

Available online at

http://www.MathCity.org in PDF Format (Picture format to view online).

Updated: September,12,2017.

These resources are shared under the licence Attribution-

NonCommercial-NoDerivatives 4.0 International

Under this licence if you remix, transform, or build upon the material, you may not distribute the modified material.quotesdbs_dbs30.pdfusesText_36

[PDF] 3 4 equivalent in cooking

[PDF] 3 as bac

[PDF] 3 commerce private colege in bbsr

[PDF] 3 commerce syllabus

[PDF] 3 ejemplos de indice

[PDF] 3 eme svt genetique

[PDF] 3 exemple qui illustre la puissance mondiale des etat unis

[PDF] 3 membranes de l'oeil

[PDF] 3*500 bac 2015

[PDF] 3*500 bac bareme 2017

[PDF] 3-d maths symbols

[PDF] 3.3.2 maintien opérationnel des postes de travail et aménagement des espaces

[PDF] 31 laminas del tat

[PDF] 31 laminas del tat para imprimir

[PDF] 31 laminas del tat pdf