Question

find radius of curvature for y=x^2 at x=1/2​

Answer 1

Answer:

2/1

Margie did so with a sigh. She was thinking about the old schools they had when her grandfather's grandfather was a little boy. All the kids from the whole neighbourhood came, laughing and shout- ing in the schoolyard, sitting together in the schoolroom, going home together at the end of the day. They learned the same things, so they could help one another on the homework and talk about it. And the teachers were people The mechanical teacher was flashing on the screen: "When we add fractions 'A and 'A - Margie was thinking about how the kids must have loved it in the old days. She was thinking about the fun they had. Questions: (i)'Margie did so with a sigh.' What did Margie do with a sigh? (ii)What was a special feature of the schools in the old times? (iii) What was the feature of learning in the schools of old times? (iv) What topic was the mechanical teacher teaching her?

Answer 2

Concept:

The radius of circle of a particular point on a curve is called the Radius of Curvature for that curve.

If the curve is $y=f(x)$, then the radius of curvature at any point is given by,

$\rho=\frac{(1+y_1^2)^\frac{3}{2}}{y_2}$, where $y_1,y_2$ define the first and second order derivatives of respectively.

Given:

$y=x^2$ is the given equation of the curve whose radius of curvature is to be determined.

To Find:

The radius of the curvature for the above curve at $x=\frac{1}{2}$.

Solution:

The curve is,

$y=x^2$

Now differentiate with respect to x, we get

$y_1=2x\\y_2=2$

Now at point $x=1/2$ the values of first and second order derivatives are,

$y_1|_{x=\frac{1}{2}}=2\times\frac{1}{2}=1\\y_2|_{x=\frac{1}{2}}=2$

So now the radius of curvature at point x=1/2,

$\rho_{x=\frac{1}{2}}=\frac{(1+1^2)^\frac{3}{2}}{2}=\frac{2^\frac{3}{2}}{2}=2^{\frac{3}{2}-1}=2^\frac{1}{2}=\sqrt{2}$

Hence the radius of curvature of the curve $\mathbf{y=x^2}$ at $\mathbf{x=\frac{1}{2}}$ is $\mathbf{\sqrt{2}}$ units.

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