Linearization & Differentials

Linearization & Differentials

Part 1 of 7 — The Tangent Line Approximation

Local Linearization

Near a point $x = a$, we can approximate $f(x)$ with its tangent line:

$L(x) = f(a) + f'(a)(x - a)$

This is also called the linear approximation or tangent line approximation.

Why It Works

If $f$ is differentiable at $a$, then for $x$ near $a$: $f(x) approx L(x) = f(a) + f'(a)(x - a)$

Worked Example

Approximate $sqrt{4.1}$ using linearization.

Let $f(x) = sqrt{x}$, $a = 4$.

$f(4) = 2$, $f'(x) = \frac{1}{2sqrt{x}}$,

$L(x) = 2 + \frac{1}{4}(x - 4)$

$L(4.1) = 2 + \frac{1}{4}(0.1) = 2.025$

Actual: $sqrt{4.1} approx 2.02485...$ Very close!

Linearization 🎯

Key Takeaways — Part 1

1. $L(x) = f(a) + f'(a)(x-a)$ is the linearization
2. Works best when is close to

Linearization & Differentials

Part 2 of 7 — Differentials

The Differential $dy$

$dy = f'(x),dx$

$$ is a small change in , is the corresponding estimated change in .

Linearization & Differentials

Part 3 of 7 — Over/Underestimates

Concavity Determines the Error

Linearization & Differentials

Part 4 of 7 — Percentage Error

Relative and Percentage Error

$\text{Relative error} = \frac{dy}{y} = \frac{f'(x),dx}{f(x)}$

Linearization & Differentials

Part 5 of 7 — Linearization with Tables

Using a Table of Values

When given a table of $f(a)$ and $f'(a)$, you can write the linearization immediately:

$L(x)=f(a)+f^{\mathrm{\prime }}(a)$

Linearization & Differentials

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Linearization & Differentials — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Linearization & Differentials — Complete! ✅