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

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Calculus
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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

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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:

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Consider the function, To find the maximum and minimum, changes in concavity, intervals of increasing and decreasing, and asymptotes of the above function. Differentiating with respect to To find maximum and minimum. Concave up intervals: None Concave down intervals: Intervals increasing: Intervals decreasing: Asymptotes: None The curve as follows: Consider the function, To find the maximum and minimum, changes in concavity, intervals of increasing and decreasing, and asymptotes of the above function. Differentiating with respect to To find maximum and minimum. Concave up intervals: Concave down intervals: Intervals increasing: Intervals decreasing: Asymptotes: None The curve as follows: Consider the function, To find the maximum and minimum, changes in concavity, intervals of increasing and decreasing, and asymptotes of the above function. Dif ...
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