(a) Use Formula (1) to obtain the local linear approximation...

(a) Use Formula (1) to obtain the local linear approximation of $1 / x$ at $x_{0}=2$. (b) Use Formula (2) to rewrite the approximation obtained in part (a) in terms of $\Delta x$. (c) Use the result obtained in part (a) to approximate $1 / 2.05$, and confirm that the formula obtained in part (b) produces the same result.

(a) Use Formula (1) to obtain the local linear approximation of $1 / x$ at $x_{0}=2$.
(b) Use Formula (2) to rewrite the approximation obtained in part (a) in terms of $\Delta x$.
(c) Use the result obtained in part (a) to approximate $1 / 2.05$, and confirm that the formula obtained in part
(b) produces the same result.

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Key Concepts

Local Linear Approximation

This is the method of estimating the value of a function near a given point using the equation of the tangent line at that point. It involves using the function value and its derivative at the point, ensuring that the linear approximation accurately reflects the behavior of the function in a small neighborhood around that point.

Derivative

The derivative of a function at a point gives the rate at which the function value changes with respect to changes in the independent variable. In the context of local linear approximation, the derivative represents the slope of the tangent line to the function at the specified point, which is essential for creating the linear model.

Tangent Line

A tangent line touches a curve at one point and has the same slope as the curve at that point. In local linear approximations, the tangent line is used to approximate the function, with the line’s equation formed by the function’s value and derivative at the point of tangency.

Increment (?x)

Representing the change in the independent variable, ?x allows one to express how small shifts in the input value affect the function. In linear approximations, multiplying the derivative (slope) by ?x provides an estimate of the change in the function's value relative to its value at the approximation point.

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