Find the radius of curvature at any point on x= e^t cost ,y=e^t sin t
Answers
The radius of curvature at any purpose on the curve given by
R = [1 + cos² t] / (-sin t).
Given,
Point ,
x = e^t cos t ,
y = e^t sin t.
To find,
The radius of curvature
Solution,
To find the radius of curvature at a point on a curve, we can use the formula:
R = [1 + (y'²)] / [y'']
where R is the radius of curvature, y' is the first derivative of y with respect to x, and y'' is the second derivative of y with respect to x.
In this case, the curve is given in parametric form, with x and y given in terms of the parameter t. To find the radius of curvature at any point on the curve, we need to find the first and second derivatives of y with respect to x, and then plug these into the formula above.
To find the first derivative of y with respect to x, we can use the chain rule:
y' = (dy/dt) * (dt/dx)
Similarly, to find the second derivative of y with respect to x, we can use the chain rule again:
y'' = (d²y/dt²) * (dt/dx)² + (dy/dt) * (d²t/dx²)
To find (dy/dt) and (d^2y/dt^2), we can take the first and second derivatives of y with respect to t, using the given equations for x and y:
y = e^t * sin t
y' = e^t * cos t
y'' = e^t * (-sin t)
To find (dt/dx) and (d²t/dx²), we can use the given equation for x and the chain rule:
x = e^t * cos t
x' = e^t * (-sin t)
x'' = e^t * (-cos t)
Substituting these values into the formulas for y', y'', and plugging them into the formula for R, we get:
R = [1 + (e^t * cos t)²] / [e^t * (-sin t)]
R = [1 + cos² t] / (-sin t)
∴The radius of curvature at any point on the curve given by
R = [1 + cos² t] / (-sin t)
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