Use Formula to find the curvature. y=x3

Question

Use Formula to find the curvature.  y=x3

Abel Maynard · Accepted Answer

We&#039;re given the curve&amp;nbsp;y=x3&amp;nbsp;and we&#039;re used to find its curvature at an arbitrary point.Remember that the textbook&#039;s Formula states that the curvature of a curve is given byk(x)=|f&amp;nbsp;″(x)|∣1+(f′(x))232}Let&amp;nbsp;f(x)=y=x3&amp;nbsp;Solve for&amp;nbsp;f′(x)&amp;nbsp;by differentiating with respect to x.f′(x)=d&amp;nbsp;dx&amp;nbsp;(x3)=3x2Now solve for&amp;nbsp;f&amp;nbsp;″(x)&amp;nbsp;by differentiating&amp;nbsp;f′(x)=3x2&amp;nbsp;with respect to xf&amp;nbsp;″(x)=d&amp;nbsp;dx&amp;nbsp;(3x2)=6xPlug&amp;nbsp;f′(x)=3x2&amp;nbsp;and&amp;nbsp;f&amp;nbsp;″(x)=6x&amp;nbsp;into k(x) to solve for the curvaturek(x)=|6x||1+(3x2)2|32=6x(1+9x4)32}

user_27qwe · Accepted Answer

Result:κ=6|x|(1+9x4)3/2Solution:Given:κ=|y″|(1+(y′)2)3/2 where y′ represents the first derivative of y with respect to x, and y″ represents the second derivative of y with respect to x.First, let&#039;s find the first derivative y′:y′=dydx=3x2Next, let&#039;s find the second derivative y″:y″=d2ydx2=6xNow, we can substitute these values into the curvature formula:κ=|6x|(1+(3x2)2)3/2Simplifying further:κ=6|x|(1+9x4)3/2So, the curvature of the curve defined by the equation y=x3 is given by κ=6|x|(1+9x4)3/2.

karton · Accepted Answer

To find the curvature of the curve defined by the equation

y = x^{3}

, we need to calculate the second derivative of

y

with respect to

x

, which represents the rate of change of the slope of the curve. The curvature is then given by the formula:

κ = \frac{| \frac{d^{2} y}{d x^{2}} |}{{(1 + {(\frac{d y}{d x})}^{2})}^{\frac{3}{2}}}

Let's begin by finding the first derivative of

y

with respect to

x

:

\frac{d y}{d x} = \frac{d}{d x} (x^{3}) = 3 x^{2}

Now, we can find the second derivative by differentiating

\frac{d y}{d x}

with respect to

x

:

\frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (3 x^{2}) = 6 x

Now that we have the second derivative, we can substitute it into the formula for curvature:

κ = \frac{| 6 x |}{{(1 + (3 x^{2})^{2})}^{\frac{3}{2}}}

Simplifying the denominator, we have:

κ = \frac{6 x}{{(1 + 9 x^{4})}^{\frac{3}{2}}}

So, the curvature of the curve defined by

y = x^{3}

is given by

κ = \frac{6 x}{{(1 + 9 x^{4})}^{\frac{3}{2}}}

.
To summarize, we first found the first derivative of

y

with respect to

x

, which gave us

\frac{d y}{d x} = 3 x^{2}

. Then, we differentiated

\frac{d y}{d x}

with respect to

x

to obtain the second derivative,

\frac{d^{2} y}{d x^{2}} = 6 x

. Finally, we substituted the second derivative into the curvature formula

κ = \frac{| \frac{d^{2} y}{d x^{2}} |}{{(1 + {(\frac{d y}{d x})}^{2})}^{\frac{3}{2}}}

to get the expression

κ = \frac{6 x}{{(1 + 9 x^{4})}^{\frac{3}{2}}}

for the curvature of the curve

y = x^{3}

.

star233 · Accepted Answer

Step 1:Let&#039;s start by finding the first derivative of y with respect to x:y′=ddx(x3)Using the power rule for differentiation, we get:y′=3x2Step 2:Now, we can find the second derivative of y with respect to x:y″=ddx(3x2)Again, applying the power rule, we have:y″=6xTherefore, the curvature of the curve defined by y=x3 is given by y″=6x.

Use Formula to find the curvature. y=x^3

Answered question

Answer & Explanation

New Questions in Algebra II