Question

Find the radius of curvature of y=x³ at (1,1)​

Answer 1

Step-by-step explanation:

$\large\mathfrak{\pink{\underline{\underline{❥︎❥︎❥︎ keep\: smiling ........ ♡︎♡︎♡︎}}}}$Solution.

Write the derivatives of the quadratic function:

y

′

=

(

x

2

)

′

=

2

x

;

y

′

′

=

(

2

x

)

′

=

2.

Then the curvature of the parabola is defined by the following formula:

K

=

y

′

′

[

1

+

(

y

′

)

2

]

3

2

=

2

[

1

+

(

2

x

)

2

]

3

2

=

2

(

1

+

4

x

2

)

3

2

.

At the origin (at

x

=

0

), the curvature and radius of curvature, respectively, are

K

(

x

=

0

)

=

2

(

1

+

4

⋅

0

2

)

3

2

=

2

,

R

=

1

K

=

1

2

.