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# calculus and maximum and minimum valuesTo find the maximum and minimum, changes in concavity, interv

Calculus

Homework

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Consider the function,
( )
ln sin 1y x= +
To find the maximum and minimum, changes in concavity, intervals of increasing and
decreasing, and asymptotes of the above function.
( )
ln sin 1y x= +
Differentiating with respect to
x
( )
( )
( )
( )
( )
1
sin 1
sin 1
cos
sin 1
d
y x
x dx
x
x
= +
+
=
+
To find maximum and minimum.
( ) ( ) ( )
( )
( )
, Exact ,Approx , Exact ,Approx Sign of
6.28319 1.
Test value 5 0.144805
Neither
3
4.71239 log 2 0.69 0 local
2
max or min
Test value 0 1 +
1.5708 log 2 0.69 0 Local max
2
Test value 3 0.867562
6.28319 1.
x x f x f x f x f
π
π
+
+
+
Concave up intervals: None
Concave down intervals:
3 3
2 , , , , , 2
2 2 2 2
π π π π
π π
÷
Intervals increasing:
2 ,
2
π
π
Intervals decreasing:
, 2
2
π
π
Asymptotes: None
The curve as follows:
Consider the function,
( )
ln sin 2y x= +
To find the maximum and minimum, changes in concavity, intervals of increasing and
decreasing, and asymptotes of the above function.
( )
ln sin 2y x= +
Differentiating with respect to
x
To find maximum and minimum.
( ) ( ) ( )
( )
( )
, Exact , Approx , Exact , Sign of
6.28319 0.5
Test value 5. 0.0958667
3
4.71239 log 3 1.10 0 Local Max
2
Test value 3. 0.532575
1.5708 0 0 0 Local Min
2
Test value 0 0.5 +
1.5708 log 3 1.10 0 Local Max
2
Test
x x f x f x Approx f x f
π
π
π
+
+
value 3 0.462371
3
4.71239 0 0 0 Local Min
2
Test value 6 0.558049 +
6.28319 0.5 +
π

Concave up intervals:
5 7 11
, ,
6 6 6 6
π π π π
Concave down intervals:
5 7 11
2 , , , , , 2
6 6 6 6
π π π π
π π
÷
Intervals increasing:
3 3
2 , , , , , 2
2 2 2 2
π π π π
π π
÷
Intervals decreasing:
3 3
, , ,
2 2 2 2
π π π π
Asymptotes: None
The curve as follows:
Consider the function,
x
y xe=
To find the maximum and minimum, changes in concavity, intervals of increasing and
decreasing, and asymptotes of the above function.
x
y xe
=
Differentiating with respect to
x
( )
1
x x
x
y xe e
e x
= +
= +
To find maximum and minimum.
( ) ( ) ( )
, Exact , Approx , Exact , Approx Sign of
Test Value 2 0.135335
1
1 1 0.37 0 Local Min
Test Value 0 1
x x f x f x f x f
e
+
Concave up intervals:
[
)
2,
Concave down intervals:
(
]
, 2
−∞
Intervals increasing:
[
)
1,
Intervals decreasing:
(
]
, 1−∞
Asymptotes:
There is one horizontal asymptote is
0y =
Horizontal asymptote : the limit of
x
y xe=
as
x −∞
is 0
The curve as follows:

Consider the function,
2
x
y e
=
To find the maximum and minimum, changes in concavity, intervals of increasing and
decreasing, and asymptotes of the above function.
2
x
y e
=
Differentiating with respect to
x
( )
( )
2
2
2
sin 1
x
d
y e x
dx
x
x
=
=
+
To find maximum and minimum.
( ) ( )
, Exact , Approx Sign of
Test Value 1 0.735759
0 0 1 0 Local Max
Test Value 1 0.735759
x x f x f x f
+
Concave up intervals:
1 1
, , ,
2 2
−∞
÷
Concave down intervals:
1 1
,
2 2
Intervals increasing:
(
]
,0−∞
Intervals decreasing:
[
)
0,
Asymptotes:
There is one horizontal asymptote is
0y
=
Horizontal asymptote: the limit of
2
x
y e
=
as
x
is
0
and the limit of
2
x
y e
=
as
x
is
0
The curve as follows:

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