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Chapter 11: Problem 35

Compare the osculating circles for \(y=\cos x\) at \(x=0\) and \(x=\pi .\) Compute the concavity of the curve at these points and use this information to help explain why the circles have the same radius.

Short Answer

Expert verified
The radii of the osculating circles at \(x = 0\) and \(x = \pi\) in the function \(y = \cos x\) are the same due to the periodic nature of the function, resulting in the same curvature at these points despite different concavities. The radius inversely proportional to the curvature.

Step by step solution

01

Compute the first and second derivative

For the function \(y=\cos x\), compute the first derivative \(y' = -\sin x\) and the second derivative \(y'' = -\cos x\). The derivatives are crucial for finding the curvature, radius, and concavity at a given point.
02

Find the curvature and radius

The formula for curvature \(k\) at a particular point on the curve is given by \(k = |y''| / (1 + (y')^2)^(3/2)\). For the points \(x = 0\) and \(x = \pi\), calculate the curvature and the radius \(r\), which is given by \(r = 1/k\). This will show that the radius of the osculating circles, despite being at different points, is the same.
03

Determine the concavity

The concavity at a specific point depends on the sign of the second derivative. If \(y'' > 0\), the function is concave up at that point, and if \(y'' < 0\), it is concave down. Determine the concavity at \(x = 0\) and \(x = \pi\). Despite the concavity differing, the fact that the radius remains the same will be easier to understand.
04

Explain why the circles have the same radius

The curvature of a function at a given point represents how quickly the function changes direction at that point. Since the circle that best fits the curve at that point is an osculating circle, a higher curvature leads to a smaller circle. However, since \(y = \cos x\) is a periodic function with a period of \(2\pi\), the curvatures at \(x = 0\) and \(x = \pi\) are the same. This implies the osculating circles at these points have the same radius, regardless of the difference in concavity.

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