Linear Approximations and Differentials

Tags: #◇Computer Science #◇CSMATH1 #◇Problem Solving

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◇Overview

Linear approximation uses the tangent line to a function at a specific point to approximate the function's values near that point. Because the tangent line lies very close to the curve near the point of tangency, it provides a simple linear estimate for complex non-linear functions.

◇Important Formulas

The Linearization Equation

Recall the point-slope form of a line: $y - y_1 = m(x - x_1)$.
Consider a function $f(x)$ and a point $(a, f(a))$ on its graph. The slope of the tangent line at this point is $m = f'(a)$.
The equation of the tangent line is:
$y - f(a) = f'(a)(x - a)$
Isolating $y$, we get the linear approximation (or tangent line approximation) of $f$ at $a$:
$L(x) = f(a) + f'(a)(x - a)$
This function $L(x)$ is called the linearization of $f$ at $a$. For $x$ near $a$, $f(x) \approx L(x)$.

Differentials

If $y = f(x)$ is a differentiable function, the differential $dx$ is an independent variable. The differential $dy$ is defined in terms of $dx$ by the equation:
$dy = f'(x) dx$
$dy$ represents the approximate change in $y$ corresponding to a small change $dx$ in $x$.

◇Worked Examples

General Example: Finding a Linear Approximation
Suppose we want to find the linear approximation of a function $f(x)$ at $x = a$ and use it to estimate values near $a$.

1. Compute the function value at $a$, which is $f(a)$.
2. Compute the derivative $f'(x)$ and evaluate it at $a$ to find the slope, $f'(a)$.
3. Construct the linearization: $L(x) = f(a) + f'(a)(x - a)$.
4. To approximate $f(x)$ for an $x$ near $a$, simply plug $x$ into $L(x)$.

(For instance, to approximate values near $x = 1$, we set $a = 1$. The resulting linear equation $L(x)$ provides an easy way to calculate values like $f(0.98)$ or $f(1.05)$ without evaluating the original complex function.)

◇References & Navigation

• Previous Topic: ◇Derivative as Rate of Change and Related Rates Part 1
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• Related Notes: ◇The Tangent Line and the Derivative, ◇Derivative as Rate of Change and Related Rates Part 1