Find the curvature, the radius of curvature and the center of curvature having a parametric equations at the given point:
y=e^x , (0,1)
Solution;
"y=e^x"
"P(x_0,y_0)=(0,1)"
"y'=e^x"
"y'(0)=1"
"y''=e^x"
"y''(0)=1"
Radius of curvature;
"R=\\frac{(1+y'(x_0)^2)^{\\frac32}}{y''(x_0)}"
By substitution ;
"R=\\frac{(1+1)^{\\frac32}}{1}=\\sqrt{2^3}=2\\sqrt2"
The curvature;
"k=\\frac1R=\\frac{1}{2\\sqrt2}"
The centre of curvature;
Let P be the center of curvature , given as (a,b)
"a=x_o-\\frac{\\frac{dy}{dx}[1+(\\frac{dy}{dx})^2}{\\frac{d^2y}{dx^2}}"
"a=0+\\frac{e^{2x}}{e^x}=e^0=1"
"b=y_o+\\frac{1+(\\frac{dy}{dx})^2}{\\frac{d^2y}{dx^2}}"
"b=1+\\frac{1+e^{2x}}{e^x}=1+2=3"
Therefore;
P=(1,3)
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