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Chapter 8: Problem 37

Determine the \(t\) intervals on which the curve is concave downward or concave upward. $$ x=2 t+\ln t, \quad y=2 t-\ln t $$

Short Answer

Expert verified
The curve cannot be classified as concave upward or concave downward for any interval of \(t\), because the concavity function \(K(t)\) equals \(0\) for all \(t\).

Step by step solution

01

Compute the first derivatives

Differentiation of the given functions for \(x\) and \(y\) with respect to \(t\) will give us first-order derivatives: \(x'(t) = 2 + \frac{1}{t}\), \(y'(t) = 2 - \frac{1}{t}\)
02

Compute the second derivatives

Differentiation of the first-order derivatives will provide the second-order derivatives: \(x''(t) = -\frac{1}{t^2}\), \(y''(t) = \frac{1}{t^2}\)
03

The Concavity function

The concavity \(K(t)\) can be computed based on \(K(t) = \frac{x'(t)y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^\frac{3}{2}}\). After substituting the values, we will find that \(K(t)\) equals \(0\) for all \(t\). So, it can't be classified as concave upward or concave downward.

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