Estimating ln with Tangent Plane Approximation

Estimate the value of $z = \ln(x-2y)$ at the point (3.1, 0.9) using a tangent plane approximation.

This problem involves the use of the tangent plane approximation to estimate the value of a function at a particular point. The essence of this method lies in using the derivatives of the function, which capture the behavior of the function locally around the point of interest. In this case, the function is the natural logarithm, but more generally, this approach is applicable to a wider array of functions that are differentiable at the point of consideration.

By employing the tangent plane, we create a linear model that approximates the function's behavior near the point. This is especially useful in situations where directly computing the function's value is complex or when one aims to simplify calculations without significant loss of accuracy. When applying the tangent plane approximation, it's crucial to compute the partial derivatives of the function with respect to its variables, evaluate these derivatives at the point of interest, and use these in constructing the equation of the tangent plane.

This technique is a fundamental concept in multivariable calculus, aiding in simplifying complex functions and in understanding the local linear behavior of surfaces.

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