Find an equation of a parabola that has curvature 4 at the origin.

Question

Sculd1987 · Accepted Answer

Let us assume an equation of the form f(x)=ax2,a≠0. Calculate the curvature using the formulak=|f″(x)|[1+(f′(x))2]32f&#039;(x)=2axf&#039;&#039;(x)=2ak=|2a|[1+(2ax)2]32k=2|a|(1+4a2x2)32Now, evaluate the curvature at the origin and set k(0)=4.k(0)=2|a|=4a=2 or a=-2Thus, the equations of the parabola aref(x)=2x2 or f(x)=−2x2Result:The required equation of the parabolas are f(x)=2x2, and f(x)=−2x2.

Jazz Frenia · Accepted Answer

Answer:y=116x2Explanation:To find an equation of a parabola with curvature 4 at the origin, we can start by recalling that the curvature of a parabola is given by the formula κ=14a, where a is the coefficient of the quadratic term in the equation of the parabola.Since we want the curvature to be 4 at the origin, we have κ=4. Plugging this value into the formula, we get 4=14a.To solve for a, we can isolate it by multiplying both sides of the equation by 4, which gives 16a=1. Dividing both sides by 16, we find a=116.Now that we have the value of a, we can write the equation of the parabola in standard form. The general equation of a parabola with vertex at the origin is y=ax2. Substituting a=116, we obtain the equation:y=116x2Therefore, the equation of the parabola with curvature 4 at the origin is y=116x2.

Nick Camelot · Accepted Answer

To find an equation of a parabola with a curvature of 4 at the origin, we can use the general equation of a parabola:

y = a x^{2} + b x + c

.
The curvature of a parabola at a specific point

(x_{0}, y_{0})

is given by the formula:

κ = \frac{2 | a |}{(1 + (2 a x_{0} + b)^{2})^{3 / 2}}

Since we want the curvature to be 4 at the origin

(0, 0)

, we can substitute

x_{0} = 0

and

y_{0} = 0

into the equation. This simplifies the formula to:

κ = \frac{2 | a |}{(1 + b^{2})^{3 / 2}}

Substituting

κ = 4

, we have:

4 = \frac{2 | a |}{(1 + b^{2})^{3 / 2}}

To eliminate the absolute value, we can square both sides of the equation:

16 = \frac{4 a^{2}}{(1 + b^{2})^{3}}

Multiplying both sides by

(1 + b^{2})^{3}

, we obtain:

16 (1 + b^{2})^{3} = 4 a^{2}

Dividing both sides by 4, we have:

(1 + b^{2})^{3} = \frac{a^{2}}{4}

Taking the cube root of both sides:

1 + b^{2} = \sqrt[3]{\frac{a^{2}}{4}}

Simplifying further:

b^{2} = \sqrt[3]{\frac{a^{2}}{4}} - 1

Therefore, the equation of the parabola with a curvature of 4 at the origin is:

y = a x^{2} + (\sqrt[3]{\frac{a^{2}}{4}} - 1) x + c

Note that the constant term

c

can be any real number, as it does not affect the curvature.

Mr Solver · Accepted Answer

Step 1To find an equation of a parabola with curvature 4 at the origin, we can start by using the standard form of a parabola equation:y=ax2+bx+cThe curvature of a parabola at any given point can be expressed as the absolute value of the second derivative of the equation. Since we want the curvature to be 4 at the origin (0,0), we need to find the equation where the second derivative at x=0 is equal to 4.Let&#039;s calculate the second derivative of the equation y=ax2+bx+c:y′=2ax+by″=2aStep 2Now, let&#039;s substitute x=0 into the second derivative and set it equal to 4:2a=4Solving for a, we find that a=2.Thus, the equation of the parabola with curvature 4 at the origin can be either:f(x)=2x2orf(x)=−2x2Both equations represent parabolas with a curvature of 4 at the origin.

Find an equation of a parabola that has curvature 4 at the origin.

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