Answer:

At the displacement  

Δ

s

along the arc of the curve, the point  

M

moves to the point  

M

1

.

The position of the tangent line also changes: the angle of inclination of the tangent to the positive  

x

−

axis

at the point  

M

1

will be  

α

+

Δ

α

.

Thus, as the point moves by the distance  

Δ

s

,

the tangent rotates by the angle  

Δ

α

.

(The angle  

α

is supposed to be increasing when rotating counterclockwise.)

The ABSOLUTE value of the ratio  

Δ

α

Δ

s

is called the mean CURVATURE of the arc  

M

M

1

.

In the limit as  

Δ

s

→

0

,

we obtain the curvature of the curve at the point  

M

:

K

=

lim

Δ

s

→

0

∣

∣

∣

Δ

α

Δ

s

∣

∣

∣

.

From this definition it follows that the curvature at a point of a curve characterizes the speed of rotation of the tangent of the curve at this point.

For a plane curve given by the equation  

y

=

F

(

x

)

,

the curvature at a point  

M

(

x

,

y

)

is expressed in terms of the FIRST and second derivatives of the function  

f

(

x

)

by the formula

K

=

|

y

′

′

(

x

)

|

[

1

+

(

y

′

(

x

)

)

2

]

3

2

.

If a curve is defined in PARAMETRIC form by the equations  

x

=

x

(

t

)

,

y

=

y

(

t

)

,

then its curvature at any point  

M

(

x

,

y

)

is given by

K

=

|

x

′

y

′

′

−

y

′

x

′

′

|

[

(

x

′

)

2

+

(

y

′

)

2

]

3

2

.

If a curve is given by the polar equation  

r

=

r

(

θ

)

,

the curvature is calculated by the formula

K

=

∣

∣

r

2

+

2

(

r

′

)

2

−

r

r

′

′

∣

∣

[

r

2

+

(

r

′

)

2

]

3

2

.

The radius of curvature of a curve at a point  

M

(

x

,

y

)

is called the inverse of the curvature  

K

of the curve at this point:

R

=

1

K

.

Hence for plane curves given by the explicit equation  

y

=

f

(

x

)

,

the radius of curvature at a point  

M

(

x

,

y

)

is given by the following expression:

R

=

[

1

+

(

y

′

(

x

)

)

2

]

3

2

|

y

′

′

(

x

)

|

.

At the displacement  

Δ

s

along the arc of the curve, the point  

M

moves to the point  

M

1

.

The position of the tangent line also changes: the angle of inclination of the tangent to the positive  

x

−

axis

at the point  

M

1

will be  

α

+

Δ

α

.

Thus, as the point moves by the distance  

Δ

s

,

the tangent rotates by the angle  

Δ

α

.

(The angle  

α

is supposed to be increasing when rotating counterclockwise.)

The absolute value of the ratio  

Δ

α

Δ

s

is called the mean curvature of the arc  

M

M

1

.

In the limit as  

Δ

s

→

0

,

we obtain the curvature of the curve at the point  

M

:

K

=

lim

Δ

s

→

0

∣

∣

∣

Δ

α

Δ

s

∣

∣

∣

.

From this definition it follows that the curvature at a point of a curve characterizes the speed of rotation of the tangent of the curve at this point.

For a plane curve given by the equation  

y

=

f

(

x

)

,

the curvature at a point  

M

(

x

,

y

)

is expressed in terms of the first and second derivatives of the function  

f

(

x

)

by the formula

K

=

|

y

′

′

(

x

)

|

[

1

+

(

y

′

(

x

)

)

2

]

3

2

.

If a curve is defined in parametric form by the equations  

x

=

x

(

t

)

,

y

=

y

(

t

)

,

then its curvature at any point  

M

(

x

,

y

)

is given by

K

=

|

x

′

y

′

′

−

y

′

x

′

′

|

[

(

x

′

)

2

+

(

y

′

)

2

]

3

2

.

If a curve is given by the polar equation  

r

=

r

(

θ

)

,

the curvature is calculated by the formula

K

=

∣

∣

r

2

+

2

(

r

′

)

2

−

r

r

′

′

∣

∣

[

r

2

+

(

r

′

)

2

]

3

2

.

The radius of curvature of a curve at a point  

M

(

x

,

y

)

is called the inverse of the curvature  

K

of the curve at this point:

R

=

1

K

.

Hence for plane curves given by the explicit equation  

y

=

f

(

x

)

,

the radius of curvature at a point  

M

(

x

,

y

)

is given by the following expression:

R

=

[

1

+

(

y

′

(

x

)

)

2

]

3

2

|

y

′

′

(

x

)

|

.

Explanation: