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 \(x\). $$ y=\ln x, \quad x=1 $$

Short answer

The graph of \(y=\ln x\) and its circle of curvature at \(x=1\) can be obtained using a graphing utility. The circle of curvature at \(x=1\) has a radius of 1 and its center is at the point (1,0).

Step-by-step solution

Understand the Function

The function given is \(y=\ln x\), which is the natural logarithm of x. This function is defined only for \(x > 0\). The graph of \(y=\ln x\) is a curve that passes through (1,0), and it increases without bound as \(x\) increases and approaches zero as \(x\) approaches zero from the right.

Graph the Function

To graph this function, use a graphing utility. Input the function and establish a suitable viewing window. It's also possible to mark the point (1,0), the point at which we'll be examining the circle of curvature.

Understand the Circle of Curvature

The circle of curvature (also known as the osculating circle) at a point on a curve is the circle that best approximates the curve at that point. It has radius equal to the reciprocal of the curvature at that point.

Calculate the Curvature

The curvature \(k\) at a particular point is given by the formula \(k = |y''| / (1+(y')^2)^(3/2)\). In this case, \(y' = 1/x\) and \(y'' = -1/x^2\). Substituting \(x = 1\) in the formula gives the curvature at \(x = 1\) is 1.

Draw the Circle of Curvature

The circle of curvature at \(x = 1\) would have radius \(r = 1/k=1\), center at \((1,0)\). Graph this circle on the same viewing window using the graphing utility. The circle should pass through the point (1,0) and touch the curve at this point.