Advanced Calculus and Mathematical Analysis: Derivatives, Integrals, and Graphical Analysis
This solved assignment explores derivatives, integrals, and graphical analysis in advanced calculus.
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Advanced Calculus and Mathematical Analysis: Derivatives, Integrals,
and Graphical Analysis
1)
Find the slope of curve as( )
( )
( )
( ) ( )
3
3
2
2
9 3
9 3
3 9 0
3 9
d
h t t t
dt
d d d
t t
dt dt dt
t
t
= − −
= − −
= − −
= −
At3t = slope is( ) ( )
2
3 3 3 9
27 9
18
h = −
= −
=
Write the equation of line with slope18m = and point( ) ( )
1 1, 3, 3t y = − as( )
( ) ( )
1 1
3 18 3
3 18 54
18 57
y y m t t
y t
y t
y t
− = −
− − = −
+ = −
= −
Thus, the equation of tangent line is18 57y t= − . Hence, the correct option isB .
2)
The volume of sphere at3.0r = is( )
( )
3
1
3
4 3.0
3
36 3.14
113.04 cm
V
=
=
=
The volume of sphere at3.1r = is( )
( )( )
3
2
3
3
4 3.1
3
4 3.14 3.1
3
124.72 cm
V
=
=
The change in volume is3 3
2 1
3
124.72 cm 113.04 cm
11.68 cm
V V− = −
=
Hence, the correct option isC .
3)
Asx tens to2 from left, the function value is( )
2 2 2 6+ = and asx tens to 2 from right, the
function value is( )
2 4 6+ = , so to remove the discontinuity( )
2f must be equal to 6.
Hence, the correct option isB .
4)
To find the velocity function, differentiate position vector with respect tot as( )
( )
2 2
2
2 2 2
1
2 2
d
v t t
dt
t
t
= +
= +
= +
At1t = ,( )
1
1 2 2 1
1
4
1 m/sec
2
v = +
=
=
Hence, the correct option isB .
5)
Since the slope of line is positive in interval( )
5, 3− − and( )0,3 , so0f in this interval.
Since the slope of line is negative in interval( )
3,0− , so0f in this interval.
and Graphical Analysis
1)
Find the slope of curve as( )
( )
( )
( ) ( )
3
3
2
2
9 3
9 3
3 9 0
3 9
d
h t t t
dt
d d d
t t
dt dt dt
t
t
= − −
= − −
= − −
= −
At3t = slope is( ) ( )
2
3 3 3 9
27 9
18
h = −
= −
=
Write the equation of line with slope18m = and point( ) ( )
1 1, 3, 3t y = − as( )
( ) ( )
1 1
3 18 3
3 18 54
18 57
y y m t t
y t
y t
y t
− = −
− − = −
+ = −
= −
Thus, the equation of tangent line is18 57y t= − . Hence, the correct option isB .
2)
The volume of sphere at3.0r = is( )
( )
3
1
3
4 3.0
3
36 3.14
113.04 cm
V
=
=
=
The volume of sphere at3.1r = is( )
( )( )
3
2
3
3
4 3.1
3
4 3.14 3.1
3
124.72 cm
V
=
=
The change in volume is3 3
2 1
3
124.72 cm 113.04 cm
11.68 cm
V V− = −
=
Hence, the correct option isC .
3)
Asx tens to2 from left, the function value is( )
2 2 2 6+ = and asx tens to 2 from right, the
function value is( )
2 4 6+ = , so to remove the discontinuity( )
2f must be equal to 6.
Hence, the correct option isB .
4)
To find the velocity function, differentiate position vector with respect tot as( )
( )
2 2
2
2 2 2
1
2 2
d
v t t
dt
t
t
= +
= +
= +
At1t = ,( )
1
1 2 2 1
1
4
1 m/sec
2
v = +
=
=
Hence, the correct option isB .
5)
Since the slope of line is positive in interval( )
5, 3− − and( )0,3 , so0f in this interval.
Since the slope of line is negative in interval( )
3,0− , so0f in this interval.
Since the slope of line is constant in interval( )
3,6 , so0f = in this interval.
Hence the correct graph isC .
6)
The derivative of2
0.05d v v= + with respect tov is( ) ( )
( )
( )
2 2
0.05 0.05
0.05 2 1
0.1 1
d d d
v v v v
dv dv dv
v
v
+ = +
= +
= +
At46v = ,( )
0.1 46 1 4.6 1
5.6
+ = +
=
Hence the correct option isA .
7)
Let the length of rectangle isx feet and width of rectangle isy feet.
Since the area is 680 square feet, so680
680
xy
y x
=
=
The cost is given by( ) ( )
7 2 6 2
14 12
680
14 12
8160
14
C x y
x y
x x
x x
= +
= +
= +
= +
Derivative of cost function is2
8160
14C x
= −
SetC equal to zero and solve forx2
2
2
8160
14 0
8160
14
8160
14
8160
14
24.1
x
x
x
x
− =
=
=
=
So, value ofy is680
24.14
28.2
y =
Hence the correct option isC .
8)
Find the composite function as( )
( )
7
2
7
2 7
2
7 7
x
g f x g
x
x
x
−
=
−
= +
= − +
=
Hence, correct option isA .
9)
Integrate the function as
3,6 , so0f = in this interval.
Hence the correct graph isC .
6)
The derivative of2
0.05d v v= + with respect tov is( ) ( )
( )
( )
2 2
0.05 0.05
0.05 2 1
0.1 1
d d d
v v v v
dv dv dv
v
v
+ = +
= +
= +
At46v = ,( )
0.1 46 1 4.6 1
5.6
+ = +
=
Hence the correct option isA .
7)
Let the length of rectangle isx feet and width of rectangle isy feet.
Since the area is 680 square feet, so680
680
xy
y x
=
=
The cost is given by( ) ( )
7 2 6 2
14 12
680
14 12
8160
14
C x y
x y
x x
x x
= +
= +
= +
= +
Derivative of cost function is2
8160
14C x
= −
SetC equal to zero and solve forx2
2
2
8160
14 0
8160
14
8160
14
8160
14
24.1
x
x
x
x
− =
=
=
=
So, value ofy is680
24.14
28.2
y =
Hence the correct option isC .
8)
Find the composite function as( )
( )
7
2
7
2 7
2
7 7
x
g f x g
x
x
x
−
=
−
= +
= − +
=
Hence, correct option isA .
9)
Integrate the function as
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Subject
Mathematics