Advanced Calculus and Function Analysis: Derivatives, Limits, and Continuity
This assignment provides solutions on derivatives, limits, and function continuity in advanced calculus.
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Advanced Calculus and Function Analysis: Derivatives,
Limits, and Continuity
1)
Find the composite function as( )
( )
( )
( )
( )
( )
3 12
3 12
3
4
4
f g h x f g x
x
f
f x
x
= +
+
=
= +
= +
Hence, correct option isD .
2)
Find the derivative as( ) ( )
( )
h c
h h h
h
h h c h c
h h
2
h
h h c
2
h
c
2
h
Q Qde
dQ dQ Q
Q
Q Q Q Q Q
dQ dQ
Q
Q Q Q
Q
Q
Q
d
dd
−
=
− − −
=
− −
=
=
Hence, correct option isA .
3)
Find derivative of the function as4 2
dy x
dx = −
When4 2 2x − = then1x =
When1x = theny is1
Equation of line having slope 2 and through point( )
1,1 is( )
1 2 1
1 2 2
2 1
y x
y x
y x
− = −
− = −
= −
Hence, correct option isA .
4)
Find the composite function as( )
( )
9
2
9
2 9
2
9 9
x
g f x g
x
x
x
−
=
−
= +
= − +
=
Hence, correct option isA .
5)
For1x the graph is a line with slope2− passing through origin and for1x the graph is
a line with slope 1.( )
2 1
1 1
x x
f x x x
−
= +
Hence, correct option isB .
6)
Find the derivative as
Limits, and Continuity
1)
Find the composite function as( )
( )
( )
( )
( )
( )
3 12
3 12
3
4
4
f g h x f g x
x
f
f x
x
= +
+
=
= +
= +
Hence, correct option isD .
2)
Find the derivative as( ) ( )
( )
h c
h h h
h
h h c h c
h h
2
h
h h c
2
h
c
2
h
Q Qde
dQ dQ Q
Q
Q Q Q Q Q
dQ dQ
Q
Q Q Q
Q
Q
Q
d
dd
−
=
− − −
=
− −
=
=
Hence, correct option isA .
3)
Find derivative of the function as4 2
dy x
dx = −
When4 2 2x − = then1x =
When1x = theny is1
Equation of line having slope 2 and through point( )
1,1 is( )
1 2 1
1 2 2
2 1
y x
y x
y x
− = −
− = −
= −
Hence, correct option isA .
4)
Find the composite function as( )
( )
9
2
9
2 9
2
9 9
x
g f x g
x
x
x
−
=
−
= +
= − +
=
Hence, correct option isA .
5)
For1x the graph is a line with slope2− passing through origin and for1x the graph is
a line with slope 1.( )
2 1
1 1
x x
f x x x
−
= +
Hence, correct option isB .
6)
Find the derivative as
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Subject
Mathematics