(a) Find the linear approximation to f(x)=lnx near 1 (b)...

(a) Find the linear approximation to $f(x)=\ln x$ near 1
(b) Illustrate part (a) by graphing $f$ and its linearization.
(c) For what values of $x$ is the linear approximation accurate to within $0.1 ?$

Key Concepts

Linear Approximation

Linear approximation is a method used to approximate the value of a function near a known point using the tangent line at that point. This method relies on the fact that, for functions that are differentiable at a point, the tangent line provides the best linear estimate of the function's behavior close to that point. It is a fundamental concept in calculus that simplifies complex functions to linear forms for ease of computation and analysis.

Derivative and Tangent Line

The derivative of a function at a particular point gives the slope of the tangent line at that point. This tangent line is used as the linear approximation of the function around that point. The concept is central because it connects instantaneous rate of change with the geometric interpretation of the function through its slope. Calculating the derivative is thus essential for forming the linearization formula, f(x) ? f(a) + f’(a)(x – a), where a is the point of approximation.

Graphical Visualization

Graphical visualization involves plotting both the original function and its linear approximation to visually compare their behavior near the point of interest. This comparison helps in understanding how well the linearization approximates the function over a range of values and highlights regions where the approximation is particularly accurate or less reliable.

Error Analysis

Error analysis in the context of linear approximation involves evaluating and quantifying the difference between the actual function and its linear approximation. This analysis is crucial for determining the interval around the point of approximation over which the linear model maintains a desired level of accuracy. Techniques such as estimating the remainder term in Taylor series or using bounds on the higher-order derivatives are typically used to assess and guarantee accuracy within a pre-specified tolerance.

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