Respuesta :
Answer:
The equation of a parabola that has a curvature 6 at the origin is [tex]f(x) = 3\cdot x^{2}[/tex].
Step-by-step explanation:
All parabolas are represented by second-order polynomials, whose standard form is:
[tex]f(x) = a\cdot x^{2}+b\cdot x + c[/tex] (Eq. 1)
Where:
[tex]x[/tex] - Independent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
[tex]a[/tex], [tex]b[/tex], [tex]c[/tex] - Coefficient of the parabola, dimensionless.
From Vectorial Calculus we know that curvature ([tex]K[/tex]), dimensionless, is defined by the following expression:
[tex]K = \frac{\left|\frac{d^{2}y}{dx^{2}} \right|}{\left[1+\left(\frac{dy}{dx} \right)^{2}\right]^{\frac{3}{2} }}[/tex] (Eq. 2)
Where [tex]\frac{dy}{dx}[/tex] and [tex]\frac{d^{2}y}{dx^{2}}[/tex] are the first and second derivatives of the function, respectively. Each expression is described below:
[tex]\frac{dy}{dx} = 2\cdot a\cdot x + b[/tex]
[tex]\frac{d^{2}y}{dx^{2}} = 2\cdot a[/tex]
If we know that [tex]a > 0[/tex], [tex]b = c = 0[/tex], [tex]x = 0[/tex] and [tex]K = 6[/tex], then the equation of curvature and polynomial are, respectively:
[tex]\frac{2\cdot a}{1^{\frac{3}{2} }} = 6[/tex]
[tex]a = 3[/tex]
The equation of a parabola that has a curvature 6 at the origin is [tex]f(x) = 3\cdot x^{2}[/tex].
An equation of a parabola that has curvature 6 at the origin is [tex]\rm f(x) = 3x^2[/tex].
We have to determine
An equation of a parabola that has curvature 6 at the origin.
What is the equation of a parabola that has a curvature at the origin?
The equation of a parabola that has a curvature a at the origin is;
[tex]\rm f(x) = ax^2[/tex]
The value of a is given by k(0) = 8;
[tex]\rm k(0) = 2a\\\\6 = 2a\\\\a = \dfrac{6}{2}\\\\a=3[/tex]
Therefore,
The equation of a parabola that has a curvature a at the origin is;
[tex]\rm f(x) = ax^2[/tex]
[tex]\rm f(x) = 3x^2[/tex]
Hence, an equation of a parabola that has curvature 6 at the origin is [tex]\rm f(x) = 3x^2[/tex].
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