Question

Use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of \(x\). $$ y=\frac{1}{3} x^{3}, \quad x=1 $$

Short answer

The task involves graphing a function and its corresponding curvature circle at a particular point. To solve it you need to graph the initial function, calculate the curvature, the radius and the circle center at that point, and then, on the same viewing window, graph the circle of curvature.

Step-by-step solution

Graph the Function

Firstly,use the graphing utility to graph the function \(y = \frac{1}{3}x^3\). You have to input the function equation into the dedicated field and draw it out. You will observe the cubic curve of the function on the plane.

Compute for Curvature

The curvature at a given point \(x\) for a function \(y\) is calculated with the formula: \[K(x) = \frac{{|y''(x)|}}{{(1+(y'(x))^2)^{3/2}}}\] To calculate the curvature at \(x=1\), differentiating the function twice. The first derivative \(y'(x)\) is \(\frac{1}{3}(3x^2)\). The second derivative \(y''(x)\) is \(\frac{1}{3}(2x)\). Now, substitute \(x=1\) into these derivative equations to calculate the values needed.

Calculating Radius and Center of Curvature

After finding the curvature, calculate the radius of curvature \$R\$ at point \(x=1\) with formula \[R = \frac{1}{K(1)}\] Then, calculate the center of curvature \(O(x_0, y_0)\) by using formula \[y_0 = y(1) + \frac{1}{K(1)}\] and \[x_0 = x(1) + \frac{1}{K(1) \cdot y'(1)}\]

Graph the Circle of Curvature

Next, graph the circle of curvature on the same viewing window as the initial function. The circle's equation can be written as \[ (x - x_0)^2 + (y - y_0)^2 = R^2 \] Graph this circle using your earlier calculated values for \(x_0\), \(y_0\) and \(R\).