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exponential function

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(1)
( )
1
x
f x
x
=
+
Differentiating with respect to
x
( )
( )
( ) ( ) ( ) ( )
( )
2
d d
g x f x f x g x
f x
d
dx dx
dx g x
g x
=
( )
( ) ( ) ( ) ( )
( )
( )
( )
( )
2
2
2
1 1
1
1
1
1
1
d d
x x x x
dx dx
f x
x
x x
x
x
+ +
=
+
+
=
+
=
+
(2)
( )
3 1f x x= +
Differentiating with respect to
x
( ) ( ) ( )
1
2
1
3 1 3 1
2
d
f x x x
dx
= + +
since
1n n
d
x nx
dx
=
( )
3
2 3 1
f x
x
=
+
(3)
Differentiating with respect to
x
( )
( )
( ) ( ) ( ) ( )
( )
2
d d
g x f x f x g x
f x
d
dx dx
dx g x
g x
=
( )
[ ]
( )
[ ]
[ ]
( ) ( )
[ ]
[ ]
[ ]
2
2
2
2
1 2 1 2 1 1
2 1
1
1
1 2 2 1
1
2 2 2 1
1
3
1
d d
x x x x
d x
dx dx
dx x
x
x x
x
x x
x
x
+ +
+
=
+
=
=
=
(4)
( )
( ) ( )
0
lim
h
F x h F x
F x
h
+
=
Let
( )
sinF x x=
( )
( ) ( )
0
0
0
0
sin sin
lim
2cos sin
2 2
lim
2
2cos sin
2 2
lim
2
2cos sin
2 2
lim
2
2
h
h
h
h
x h x
F x
h
x h x x h x
h
x h h
h
x h h
h
+
=
+ + +
÷ ÷
=
+
÷ ÷
=
+
÷ ÷
=
÷
0
0
2
sin
2
2
lim cos lim
2
2
h
h
h
x h
h
÷
+
=
÷
÷
Since
0 that is 0
2
h
h
( )
( )
0
0
2
lim cos 1
2
2
lim cos
2
2
cos
2
cos
h
h
x h
x h
x
x
+
=
÷
+
=
÷
=
÷
=
(5)
We have
( ) ( ) ( ) ( )
sin cos , cos sin
d d
x x x x
dx dx
= = −

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( )
( )
( ) ( )
( )
2
2
sin
tan
cos
cos sin sin cos
cos
cos cos sin sin
cos
d d x
x
dx dx x
d d
x x x x
dx dx
x
x x x x
x
=
=
=
( )
( )
2 2
2
2
2
cos sin
cos
1
cos
sec
x x
x
x
x
+
=
=
=
(6)
( )
2
16
,1 4f x x x
x
= +
( ) ( )
( )
2
16
1 1
1
1 16
17
f = +
= +
=
( ) ( )
( )
2
16
4 4
4
16 4
20
f = +
= +
=
Therefore, the maximum is 20 and minimum is 17
(7)
( )
3 2
4 3 2, 2 2f x x x x x= +
( ) ( ) ( ) ( )
3 2
2 2 4 2 3 2 2
8 16 6 2
24 8
16
f = +
= − + +
= − +
= −
( ) ( ) ( ) ( )
3 2
2 2 4 2 3 2 2
8 16 6 2
8 4
12
f = +
= +
= −
= −
Therefore, the maximum is
12
and minimum is
16
(8)
( )
2
2 3S t t t= +
The velocity function is the derivative of the position function.
( )
2 2 ,t 2
ds
v t t
dt
= = =
( ) ( )
2 2 2 2
4 2
2
v =
=
=
The acceleration is the derivative of the velocity function.
( )
2
2
2
d s
a t
dt
= =
(9)
( )
2 2
5cos 12sin
3 3
t t
s t
π π
= +
÷ ÷
The velocity function is the derivative of the position function.
( )
( )
( ) ( )
( ) ( )
2 2 2 2
5 sin 12 cos ,t 3
3 3 3 3
2 3 2 3
2 2
5 sin 12 cos
3 3 3 3
2 2
5 sin 2 12 cos 2
3 3
2
12
3
ds t t
v t
dt
v t
π π π π
π π
π π
π π
π π
π
= = − + =
÷ ÷ ÷ ÷
= +
÷ ÷
÷ ÷
= +
÷ ÷
=
÷
The acceleration is the derivative of the velocity function.
( )
( )
( ) ( )
( ) ( )
2
2 2
2
2 2
2 2
2
20 2 16 2
cos sin ,t 3
9 3 3 3
2 3 2 3
20 16
cos sin
9 3 3 3
20 16
cos 2 sin 2
9 3
20
9
d s x x
a t
dt
a t
π π
π π
π π
π π
π π π π
π
= = − =
÷ ÷
= −
÷ ÷
= −
= −
(10)
2
secy x=
Differentiating with respect to
x

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(1)Differentiating with respect to (2)Differentiating with respect to since (3)Differentiating with respect to (4)Let Since (5)We have (6)Therefore, the maximum is 20 and minimum is 17(7)Therefore, the maximum is and minimum is (8)The velocity function is the derivative of the position function.The acceleration is the derivative of the velocity function. (9)The velocity function is the derivative of the position function.The acceleration is the derivative of the velocity function. (10)Differentiating with respect to The tangent equation is at (11)Differentiating with respect to The tangent equation is at (12)Differentiating with respect to And the point is So slope and point form: (13)Therefore, the line is a horizontal asymptote.Since denominator is zero when Therefore, the lines are vertical asymptote.(14)By L'Hospital's rule: (15)By L'Hospital's rule: Again by L'Hospital's rule(16)By L'Hospital's rule: Again by L'Hospital's rule(17)Differentiating with respect (18)Differentiating with respect At the point Point and slope form : (19)Volume of the cone is,Given that,Substitute in (1), we getNow substitute, andin (2), we getTherefore the height of the cone is increasing at the rate of(20)Consider The velocity function is the derivative of the position function.The object is moving to the l ...
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