Chap 01 Solutions Ex 1 3 Calculus

Notes of Chapter 01
Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.

E-XERCISE 1.3
Discuss the continuity of the following functions at the indicated points
1 - 7):
1 . f(x= x - 3|
)
at x= 3

sets (Problems
I

2.

f(x)=_J'xz-9 x_3 ._

1fx¢3

._0

ifx=3'

fx-4

if-1<xS2 if2<x<5 atx=3.
3.

()=<

fx

Vatx=2

x='ag

fm

4
»

_xz-6

r3_

ifx¢3 ifx=3 `6

atx=3
5.

/6.

'_ f(x)=(4s1n atx=0 f(x) =

sinx forall
2

f(x)=

1fx¢0 ifx=o _o

-

LL

7,

'.

1

0

a

-

R.

A

x E

a

x

if

0 < x < a

ifx=a if x

a

>

at x = a

8./ Determine the points of continuity of the function f (x) = all x e R.
'M @»~’~»
/9. Discuss the continuity of x - xl at x
= 1.
W/
10. Show that the function f R-->
R defined by x if x is irrational f<x> =
1 -x lf x is rational

x

- [rr]

for

|

:

_

is continuous at x = 51

_

_

1

GI. Showthatthefi1nctionf:]0,1]->
"

is continuous on ] 0,

1

f(x)=;
Is

R_ deiinedby

Written by: Prof. Muhammad Farooq
Punjab College, Sargodha
Available at http://www.MathCity.org

f (x) bounded on this interval

'Z

EXPN"-

12. Let

f.(x)

{cos

=

(L)
I

ifxat 0. ifx 2

= 0

f continuous at x = 0?
13.11% f(x)=<(x-a)sm (x_a)1f£f(r
Is

_

0

Discuss the continuity of
1

Y

f

at

.

1fx¢0

14_Letf(x)=<xcos(x)

ifx

.0

f is continuous at
T
§15.utf(x)=(x s1n(x)
_Show that

2

= 0

x = 0

1

f

#ug 'Eh

'
If

x = a

-

.,

1fx¢0
.

‘ifx

to

=`o

f at x =
|x|
1_Letf(x)=`xs1n(--x)1fx¢0_
Discuss the continuity of
'

O.

_

_

.0

if x

f

= 0

Discuss the continuity of at x = 0 the function
17. Findc such that 1N/_E

f

(JC)

x

=

-

1

C

if 0

S x <

if x

=

1

1

is continuous for all x 6 [0, 1 ]
In Problems 18 - 20, find the points of discontinuity of the given function x + 4 if - 6 S x < - 2
_

18.

19;

f(x)=

if-2Sx<2

x-4

if

2$xS ifx x3

gui):

-47-132
6x”+46

f(x)=

x

.

20.

x

x+2

<

6

1”

if1$xS1'0 ifx> 10
-if

0 S x <

1

1$x<2

if x+5 if '2$x’<3
21. Find constants a and b such that the function x’ f(x)=

ax+b
~x’+2

ifx

<

-1