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Differentials
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Differentials
3 Activities
Using differentials, estimate the change in the hypotenuse of a right triangle when one of its sides changes from 4 cm to 5 cm and the other changes from 3 cm to 4 cm.
7
5
c
m
\frac{7}{5}\ cm
5
7
c
m
2
c
m
2\ cm\
2
c
m
9
c
m
9\ cm
9
c
m
25
c
m
25\ cm
25
c
m
1
c
m
1\ cm
1
c
m
I don't know
Check Submission
More Differentials Questions:
Volume Differential
The volume
V
V
V
of fluid flowing through a small pipe or tube per unit time at a fixed pressure, is a constant times the fourth power of the tube's radius
r
r
r
, that is,
V
=
k
r
4
V=kr^4
V
=
k
r
4
. How will a 10% increase in
r
r
r
affect
V
V
V
?
Differentials
If the surface area of a sphere increases by 4%, find the percentage change in the volume.
The volume increases by
6
%.
Differentials
Approximate the percentage change in volume of a sphere if its radius is increased by 3%.
Differentials
The kinetic energy of a moving object is equal to
K
=
1
2
m
v
2
K=\frac{1}{2}mv^2
K
=
2
1
m
v
2
where
m
is the mass of the object and
v
is the speed of the object. Estimate the percentage change in the kinetic energy of the object if its velocity is increased by 0.2%.
Differentials
The volume of a cube increases from 125 cubic centimeters to 130 cubic centimeters.
(a) By approximately how many square centimeters does the surface area increase?
(b) By approximately what percentage does the surface area increase?
Differentials
A balloon is spherical in shape and has radius
r
. By approximately what percentage will the volume increase if the radius increases by 1.5%?
Practice: Differential
Practice: Differential
Given that
y
(
x
)
=
2
x
2
−
x
+
3
y\left(x\right)=2x^2-x+3
y
(
x
)
=
2
x
2
−
x
+
3
, find the values of
Δ
x
,
Δ
y
\Delta x,\ \Delta y
Δ
x
,
Δ
y
and
d
x
,
d
y
dx,\ dy
d
x
,
d
y
from
x
=
0
x=0
x
=
0
to
x
=
0.2
x=0.2
x
=
0.2
.
Example: Linear functions
For the function
y
=
4
x
+
5
y = 4x + 5
y
=
4
x
+
5
the ratio of the change in input over the change in output is a constant. This statement is:
Volume Differential
The volume
V
V
V
of fluid flowing through a small pipe or tube per unit time at a fixed pressure, is a constant times the fourth power of the tube's radius
r
r
r
, that is,
V
=
k
r
4
V=kr^4
V
=
k
r
4
. How will a 10% increase in
r
r
r
affect
V
V
V
?
The volume of a cube increases from 125 cubic centimeters to 130 cubic centimeters.
(a) By approximately how many square centimeters does the surface area increase?
(b) By approximately what percentage does the surface area increase?
The kinetic energy of a moving object is equal to
K
=
1
2
m
v
2
K=\frac{1}{2}mv^2
K
=
2
1
m
v
2
where
m
is the mass of the object and
v
is the speed of the object. Estimate the percentage change in the kinetic energy of the object if its velocity is increased by 0.2%.
y
=
C
x
n
y = Cx^n
y
=
C
x
n
for constants
C
C
C
and
n
≠
−
1
n \neq -1
n
=
−
1
. By approximately what percentage will
y
y
y
change if
x
x
x
increases by
r
%
r\%
r
%
?