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Seatwork No.4 Basic Calculus:
Continuity and Differentiability of a Function
Justify your answer whether the following functions are
continuous and differentiable at a certain point. Show your
complete solutions. (10pts each)
f ( x )= 3 x + 4 , x = 5
Find if its continuous at
x = 5
Condition 1:
f ( x )= 3 x + 4
f ( 5 )= 3 ( 5 )+ 4
f ( 5 )= 15 + 4
f ( 5 )= 19
Condition 2:
lim ¿ x → 5 f ( x ) = 3 x + 4
lim ¿ x → 5 f ( x ) = 3 ( 5 )+ 4
lim ¿ x → 5 f ( x ) = 15 + 4
lim ¿ x → 5 f ( x ) = 19
Condition 3:
f ( c ) =lim ¿ x →c f ( x )
f ( 5 )=lim ¿ x → 5 3 x + 4
Find if it’s differentiable at
x = 5
f
'
( x )=lim ¿ h → 0
f ( x + h )− f ( x )
h
f
'
( x )=lim ¿ h → 0
3 ( x + h )+ 4 −( 3 x + 4 )
h
f
'
( x )=lim ¿ h → 0
3 x + 3 h + 4 − 3 x − 4 ¿
h
f
'
( x )=lim ¿ h → 0
3 x + 3 h + 4 − 3 x − 4
h
f
'
( x )=lim ¿ h → 0
3 h
h
f
'
x
=lim ¿ h → 0 3
f
'
x
Conclusion:
Therefore the function
f ( x )= 3 x + 4
is differentiable and
continuous at
x = 5
f ( x )=¿
Find if its continuous at
x =
Condition 1:
f ( x )=¿
f
f
1
5
f
1
5
f
'
( x )=lim ¿ h → 0
2 x + 2 h − 3
1
5
2 x − 3
1
5
h
f
'
x
=lim ¿ h → 0
2 x + 2 h − 3
1
5
f
'
x
=lim ¿ h → 0
2 h
1
5
h
f
'
x
=lim ¿ h → 0
2 h
1
5
h
f
'
x
=lim ¿ h → 0
1
5
f
'
x
f ( x )=
4 x + 5
9 − 3 x
, x =− 1
Find if its continuous at
x =− 1
Condition 1:
f ( x )=
4 x + 5
9 − 3 x
f ( x )=
4 x + 5
3 (− x + 3 )
f (− 1 )=
f (− 1 )=
f (− 1 )=
f (− 1 )=
f (− 1 )=
Condition 2:
lim ¿ x → − 1
4 x + 5
9 − 3 x
→ lim ¿ x → − 1
4 x + 5
3 (− x + 3 )
lim ¿ x → − 1 ( 4 x + 5 )
lim ¿ x → − 1
3 (− x + 3 )
lim ¿ x → − 1 ( 4 x + 5 )
lim ¿ x → − 1
3 (− x + 3 )
