bmm10233 chapter 6 functions

8/11/2019 BMM10233 Chapter 6 Functions

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Chapter 6 | Functions | BMM10233 | 95 of 204

Fuc u p T z = u c H c p p

Graphical representation of function :

F p p uc = c u c p p

Tu c = = p p

T = =; = = ; = = 9 = - = ; = =

Even and odd functions

Even function

I = uc = ; = =

Gp uc c u

Odd Function

I = - - uc

= ; = ; = 5x

x 1+

Graphical representation of odd function :

I c u c p u

Functions

0

0

0y

0y


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Greatest integer function:

I = = []

I qu

F [] = [] = [999] =

[ ] =

Modulus function:

Muuuc x R

T Fuc = < = >H = R R = u

Exponential function:

A uc = x R > p uc D = R = p

u

Inverse of a function:

Supp uc =

N u u Bu c uc uc =

u = 9 u S uc c u

S uc = u cp u T p

uc c u = c c qu

H uc ?

F 1

1 x=

+

1

1 x

=+

+ =1

y

=1

1y

=1 y

y

N = =1 y

y

N q

=1 x

x

N 1 x

x

uc =

1

1 x+


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Concept of maxima and minima:

Supp p uc =

A p p u c u p p c Z

N c p c u c u p c -

F p c p c c z p p c

p c u p p

N c cu uc Z u u

p uc

qu z u u u uc u u

H c p u u S u u c

S cu p

2

2

d y

dx u p

2

2

d y

dx p

Derivatives of some standard function:

d(1) (Constant) 0

dx

d n n 1(2) (x ) nxdxd 1

(3) (log x)edx x

d x x(4) (e ) edx

d x x(5) (a ) e log aedx

=

=

=

=

=

a

b

c

d

e

a

b

c

d

e

a

b

c

d

e

( )y f x=

d y

d x

2

2

d y

d x


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Example:

W u u p + 6 + ?

L = + 6 +

D uc pc

= + 6

N qu

+ 6 = =

D

S = +

Sc + u p uu u u

u u

1. The Algebra of Limits

Lx alim

F =

x alim

= :

x a x a x alim(f g)(x) lim f (x) lim g(x) l m

= =

x a x a x alim (fg)(x) lim f (x). lim g(x) lm

= =

x a

x ax a

lim f (x)f l

lim ( )(x) , Pr ovided m 0g lim f (x) m

= =

x a x alim kf (x) k. lim f (x), Where k is con tan t

=

x a x alim | f (x) | | lim f (x) | | l |

= =

g(x) m

x alim | f (x) | l

=

( )x a x alim fog(x) f lim g(x) f (m)

= =

( )x a x alim fogf (x) log lim f (x) log l

= =

e ex a x aelim f (x) lim .f (x) l

= =

2. Evaluation of Algebraic limits

n nn 1

x a

x alim na , where n Q.

x a

=

p 0, 0a b 0 -z u

m m 10x a x ... a x a1 m 1 m

n n 1x0 1 n 1 n

a

limb x b x ... b x b

+ + + +

+ + + +


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0

0

0

0

a, if m n

b

0, if m n

, if m n and as b 0

, if m n and as b 0

=

>

>


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3 5x xsin x x ...

3! 5= + +

2 4x xcos x 1 ...

2! 4!= + +

3 5x 2tan x x x ...

3 15= + + +

3 5 71 1 x 1 3 x 1 3 5 xsin x x . . . . . ...

2 3 2 4 5 2 4 6 7

= + + + +

1 3 51 1tan x x x x ...

3 5

= + +

2 41 x xsec x 1 5 ...

2! 4!

= + + +

Example 2.

Eu:x 2

1 cos2(x 2)lim

x 2

Sol:

W

x 2 x 2

1 cos2(x 2) 2 | sin(x 2) |lim lim

x 2 x 2

=

x 2 x 2

1 cos 2(x 2) 1 cos 2(x 2)lim lim

x 2 x 2 +

Hcx 2

1 cos2(x 2)lim

x 2

4. Evaluation of Exponential and Logarithmic Limits

x

ex 0

a 1lim log a, a 0

x

= >

x

ex 0

e 1lim log e 1

x

= =

a

ex 0

log (1 x)lim log e

x

+=

Example: 3.

Eu:x x

x 2

(cos ) (sin ) 1lim , x (0, / 2)

x 2

+


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Sol:

W

x x

x 2

(cos ) (sin ) 1lim , x (0, / 2)

x 2

+

x 2 x 2

x 2

{(cos ) (cos ) } (sin ) (sin ) }

lim x 2

+

x 2 x 22 2

x 2

(cos ) 1 (sin ) 1lim (cos ) (sin )

x 2 x 2

= +

2 2e e(cos ) (log cos ) (sin ) (log sin )= +

5. Evaluation of Limits of the Form1

Ix a x a x a

f(x)lim f (x) lim g(x) 0 such that lim exists,

g(x) = =

ex a x a

f(x)lim{1 f (x)}1/g(x) lim

g(x) + =

Particular Cases

1/ x

x 0lim (1 x) e

+ =

x

x

1lim 1 e

x

+ =

1/ x

x 0lim (1 x) e .

+ =

x

xlim 1 e .

x

+ =

Example 4.

Eu:

1/ xx x x

x 0

a b clim e .

3

+ +=

Sol:

W

1/ xx x x

x 0

a b clim e .

3

+ +=

1/ xx x x

x 0

a b c 3lim 1

3

+ + = +

1/ xx x x

x 0

(a 1) (b 1) c 1lim 1

3

+ + = +

x x x

x 0e

a 1 b 1 c 1lim

3x 3x 3x

= + +


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x x x

ex 0 x 0 x 0

e

1 a 1 b 1 c 1 1lim lim lim {log a log b log c}

3 x x x 3=

=

+ + + +

1/ 3 1/ 3e log(abc) (abc)= =

6. Evaluation of limits by using LHospitals Rule

I uc uc

x a x alim f (x) lim g(x) 0

= =

cuu =

=

cuu p =

x a x a

f (x) f '(x)lim lim ,

g(x) g '(x) =

Example 5.

Eu:

a x

x ax a

x alim

x a

Sol:

W

a x

x ax a

x alim

x a

0form

0

a 1 x

xx a

ax a log alim

x (1 log x) 0

=

+ [U: LHp u]

a a

a

a a log a 1 log a

1 log aa (1 log a)

= =

++

SOLVED EXAMPLES

Example: 1.

I = +

Sol:

= + = 6

Example: 2.

I = /

Sol:

= /=

= /=


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Sol:

F u + 6

u u ccu + = 6

1

x3

= Sc 1

x , (6 2x) (x 5)3

< > +

= + cuu uc uc

F1

x ,3

> + > 6 -

T = 6

Sc 6 p uc 1

x3

= u uc u 1 1

5 53 3

+ =

Example:

9.

I = u u

y

y

xx

f(x)

Sol: = [ p]

= < [ p u ]

T u u

T c j u

Example: 10.

I = + = < < u u

= + =

uc c

Mu ccu =

Mu u = = - - = -

Example:11.

I = [ + ] u u

Sol:


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Example: 12.

I p = [ + ] u u p

Sol:

Example: 13.

F uc 12 2

1y

(x 2x)

=

Sol:

F uc p u qu u u

qu z

2 2x 2x 0, x 2x 0 >

( , ) + cu

Example: 14.

F R | x 3 |

x 3

Sol:

F u c z x 3 0 or x 3 = = u F

u ccp =

A

x 3 0 i.e. x 3 we have | x 3 | x 3,

| x 3 |so that f (x) 1 for all x 3

x 3

and when x 3 0 i.e. x 3we have | x 3 | (x 3), as that

> > =

= = >

<

= < Hc F u c u

Example: 15.

u p p p

F u uc 2FOF F=

F u uc F = F

Sol:

F

3f (x) x=

3 3 3 9(FOF) F(f (x)) F(x ) (x ) x= = =


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2 2 3 2 6F (x) [f (x)] (x ) x= = =

F

2

2

2 2

f (x) sin x

g(x) x

(Fg)x F(x) g(x) x sin x

(Fog)x Fg(x) F(x ) sin x

=

=

= == = =

Example: 16.

L A u c xth c c L F: A N uc = R u

u F - u

Sol:

Hc F c c / u c u c c Nu

-

A = R F = { } N F c- F

Example: 17.L A B : A B B A = jc

Sol:

Ijc: L

2 2(a ,b ) A B such that

1 1 2 2

1 1 2 2

1 2 1 2

1 1 2 2

1 1 2 2 1 1 2 2

1 1 2 2

f (a , b , ) f (a , b )

(b , a ) (b , a )

b b and a a

(a , b ) (a , b )

then f (a , b ) f (a , b ) (a , b ) (a , b )

for all (a , b ),(a ,b ) A B

=

=

= =

=

= =

F j p

Ijc:

B A

Then b B and a A

(a, b) A B

u B A A B uc =

: A B B A ucHc F jc

Example: 18.

F F: R R g : R R f (x) | x |= g(x) | 5x 2 |=

Sol:

W

= = =


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| 5x 2 |, if x , 0

| 5x 2 |, if x 0

>

=


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Example:22.

I A ={ } B = { 6 } f : A B = F 1F p

Sol:

F = F = F = 6 F =

F = { 6 }

c c jc 1F { 6 }

Example: 23.

p * { } upc

* 1 2 3 4 5

1 1 1 1 1 1

2 1 2 1 2 1

3 1 1 3 1 1

4 1 2 1 4 1

5 1 1 1 1 5

pu * * * *

I * cu?

pu * * *

2 2 3 2 6F (x) [f (x)] (x ) x= = =

Sol:

F cp

* = * =

* * = * = cp c u upp c

c

S * cu F cp

* = * =

* * * = * =

Example: 24.

I z [ 1, 1] uc 1 1 1 3

sin x sin y sin z ,2

+ + = u 2006 2007 2008

2006 2007 2008

9x y z

x y z+ +

+ +

Sol:

W 1

sin x for all x [ 1,1]2 2

Mu u u 1sin x 2

2

pc

N


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1 1 1

1 1 1

1 1 1

3sin x sin y sin z

2

sin x sin y sin z2 2 2

sin x , sin y , sin z2 2 2

x 1, y 1, z 1.

+ + =

+ + = + +

= = = = = =

2006 2007 2008

2006 2007 2008

9x y z

x y z

91 1 1 3 3 0

1 1 1

+ + + +

= + + = =+ +

Example: 25.

F c uc c; c: 3 2f (x) 2x 9x 12x 20= + + +

Sol:W

3 2f (x) 2x 9x 12x 20.= + + +

2 2f '(x) 6x 18x 12x 6(x 3x 2)= + + = + +

F c u >

26(x 3x 2) 0+ + [Q 6 > 26(x 3x 2) 0+ + > 2x 3x 2 0+ + > ]

+ + > [S F ]

< >

x ( , 2) ( 1, )

+ +

2 1

Fig. 1.1

S c ( , 2) ( 1, )

F c u <

26(x 3x 2) 0+ + 26(x 3x 2) 0+ + < 2x 3x 2 0+ + < ]

+ + < [S F]

< <

+ +

2 1

Fig. 1.2

S c


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Example: 26.

F c =4 2x 2x c c

Sol:

4 2f (x) x 2x=

3 2f '(x) 4x 4x 4x(x 1) = = F c u

>

2(x 1) >

2(x 1) > [Q > ]

+ >

< > [S F ]

1, x 0

0, x 0,

1, x 0

+ +

1

10

fig.1

S c ( 1,0) (1, )

F c u

<

2(x 1) <

2(x 1) < [Q > ]

+ < < > [S F ]

< <


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Case I

W < < / 2I c W

< < / 2 c > c > >

Tu > < < / 2 < < / 6

S c / 6 Case II

W / 2 < < 3 / 2I c

/ 2 < < 3 / 2 c < c < <

Tu < / 2 < < 3 / 2 / 6 < < / 2

S c / 6 / 2

Hc c / 6 c / 6 / 2

Example: 28.

uc 2x x 1= + c c ( 1,1)

Sol:W

2x x 1= +

= 2x 1 2(x 1 / 2) =

N < < / 1

2(x )2

< <

/ < < { /} > 1

2(x )2

> >

Tu uu ( 1,1)

Hc c c ( 1,1)

Example: 29.

F u u u uc

= + 6 + x R

= + x R

= + ( )/ 2, / 2

Sol:

W

= + + + = + +

N + x R

Y

y1

X

(1, 5)

x1

(1, 0) 0

F(x) = 3(x + 1) + 52

Fig. 18.7


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+ + x R

x R

Tu u u c x 1= Sc c p T u u c c F

W

= + x RN

x R

x R

+ x R

x R

F(x)

=|x

5|

+5

(1, 5)

X1

Y1

X

Y

S u u

N

=2

7 + = = =

Tu u u =

Sc c p T u u F

W

= + x R

N x R

+ + + x R

+ x R

x R

Tu u u u u

N =

+ = = =2

=

6


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S u u =6

A = + = =

= 2

=

6

u u = 6

Example: 30.

F p c uc 4 3(x 2) (x 1)= +

c c p c

Sol:

W

4 3(x 2) (x 1)= +

3 3 4 2

3 2

2 2

f '(x ) 4(x 2) (x 1) 3(x 2) (x 1)

f '(x) (x 2) (x 1) (7x 2)

f '(x) (x 2) (x 1) (x 2) (7x 1)

= + + +

= +

= +

N

= = 2

7

Sc 2 2(x 2) (x 1) + p S p up (x 2) (7x 2) T c

c u2

7

+ +

2 1

+

27

c p c u2

7

S =2

7 p c u

W c p c u

S = p c u

T c c u

S = p c

F(x)= sing 3x + 4

0

(0, 4)

XX1

Y

Y1

( , 3)6

( , 0)2

( , 0)3

( , 0)6

( , 0)

6

( , 0)3

( , 0)2


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F uc

( )1/ 2

2

1y

4 x

=

Acc - 6

[ 2,0] [ 2,2]

( , 2) (2, ) excluding 2 and 3 (2, ) ( 2, 2)

T uc1/ 2

10

1y (x 2)

log (1 x)= + +

Acc - 6

cu D uc

Wc uc uc? Acc - 6

B N N

If = fff Acc - 6

A H G N

I u = = f = / u u Acc - 6

21

(4t 5)

4

(t 5)

2

45

t +

6 Wc uc u u = ? Acc - 66f = + f = + f = 6 + 6 f = + D uc

Directions for (Que. 7 - 10): D uc:

f z = + z + z

g z = + z + z

h z = z

F u p:

h[f g h/ / ] Acc - 6

6 69 6

gf g h ] Acc - 6

9

9 f[f g h ] Acc - 69 9

f g + h Acc - 6

6 6

Practice Exercise - 1


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W u < < 6 {} + [] = ? Acc - 6

6

I f = + g = / + = f[g] -g[f] = ? Acc - 6

6 6 N

G = + = + I = Acc - 6

6

F uc 1/ 21

y(x | x |)

=

Acc - 6

x < < x 0 < < 0 x< < N N

F uc Acc - 6

= /+ /+ + /

= [ ] [ ]

6 F uc Acc - 66 =

[ / + ] + /

> 6 < < B < > N

I(t 1)

f(t)(t 1)

=

+ qu Acc - 6

1

t

1

t

Wc uc uc? Acc - 6

t t

t t(a a )f(t)(a a )

+=

t

t(a 1)f(t)(a 1)

+=

t

tt (a 1)f(t)

(a 1) =

+ D uc N

9 Wc uc uc? Acc - 69

( )22f (t) log t t 1= + +

( )( )

t t

t t

a af(t)

a a

+=

( )( )

t

t

a 1f(t)

a 1

+=

A

N


22/29


Directions for (Que. 20 - 25): D uc:

A z = M z z

B z = M z z

z = M z z

D z = M z z

M z = Mu z

M z = Mu zAu z c

F c A z qu M z? Acc - 6

W u W u W z u E N

F c B z qu M z? Acc - 6

W u W z u E N N

F c A z qu B z? Acc - 6

> > z > z > z > > < > z N

U c z qu B z Acc - 6

> > z z > > B N N

Wc u? Acc - 6

I A z M z

II B z M z

III A z B z

I III B A N

T u Acc - 6

Max

Min

A

B

C

D N

6 L =

=

T u qu = Acc - 66 R {} { } N

I f : R R = 1f (x) Acc - 6

1

3x 5

x 5

3

+

cu - cu - N

I = = 2(sin x) , Acc - 6

= = x

= = = = x

c

=


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9 T uc : R { R : < } x x

x x

e eis

e e

=

+Acc - 69

1 1 x

log2 1 x

+

1 2 xlog

2 2 x

+

1 1 xlog

2 1 x

+

1 xlog

1 x

+ N

I f : R R = + 1f (x) qu Acc - 6

1/ 3x 3 1/ 3x 3+ 1/ 3(x 3) 1/ 3(x 3)+ 1/ 2(x 3)

L = uc { } T 1f Acc - 6

{ } { } { } { } N

L = + []

1, x 0

0, x 0,

1, x 0

W [] qu T

qu Acc - 6 N

Ix / 2

2xlim

cot x

Acc - 6

N

x

1/ 2x 0

2 1

(1 x) 1

+ Acc - 6

elog 2 e2 log 2 e1

log 22

e5log 2

I 3x 0

x(1 acosx) bsin xlim 1

x

+ = Acc - 6

/ / / / / / 5 3

,7 7

N

6 = + c cu = u p Acc - 66

= = = / = N

x a

cosx cosalimcot x cota

=

Acc - 6

31

sin a2

31

cosec a2

3sin a 3

cosec a N


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3h 0

1 1lim

2hh 8 h

=

+ Acc - 6

/ / 6/ /6 /

9 I c p

c dx

xx1lim 1

a b

+

+ + = Acc - 69

d / be c / ae (c d /a b)e + + N

I / = qu z qu Acc - 6 / / / N

Use HB pencil only. Abide by the time-limit

SORE SHEET

6

9

6

9

6

9

6

9


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26/29


I = x 4

,x 1

+=

Acc - 6

= =

uc c x 1 =

F u { }[(3@ 4)!(3 2)]@[(4 !3)@(2 # 3)] Acc - 6

9 Wc p u = = Acc - 69

(a!b)

(a # b) (a ! b) (a @ b)

(a # b)

(a !b) (a @ b) B c A

uc 1 1

f ( ) f (x) f ( ) and f (4) 65, what is the value of f (6)x x

= + = Acc - 6

6 N

L + = X, Y R.

= 1 1f (0) 11, then f (3) is given by= Acc - 6

2 2 2n

1 2 nlim ...

1 n 1 n 1 n

+ + +

Acc - 6

1

2

1

2

3

2 N

I =

1

x sin , x 0x

k, x 0= cu = u Acc - 6

1 1

2 2

I = = + ; upp = 1f (0) 3= 1f (5) qu Acc - 6

6 N

L uc + = + 2x g(x)=

W cuu uc 1f (x) qu Acc - 6

1g (x) +


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6 =

2

1 cos4x, x 0

x

a, x 0

x , x 0(16 x ) 4

+

I uc cu = = p Acc - 66

6

uc = []2x 1

cos ,2

[ ] uc cuu : Acc -

6

p c N

T uc

=

x a 2 sin x, 0 x1

2x cot x b, x4 2

a cos 2x b sin x, x2

+


28/29


log(1 ax) log(1 bx)f(x)x

+ = =

T u c u = cuu = Acc - 66

+ + (a b)

2

+ N

SORE SHEET

6

9

6

9

Use HB pencil only. Abide by the time-limit


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bmm10233 chapter 6 functions

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