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) = rac{1}{2sqrt{x}}, f'(4) = rac{1}{4}

L(x) = 2 + rac{1}{4}(x - 4)

L(4.1) = 2 + rac{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 $x$ is close to $a$
3. This is simply the tangent line used as an approximation

Linearization & Differentials

Part 2 of 7 — Differentials

The Differential $dy$

$dy = f'(x),dx$

$dx$ is a small change in $x$, $dy$ is the corresponding estimated change in $y$.

Differentials vs Actual Change

• $Delta y = f(x + Delta x) - f(x)$ — exact change
• $dy = f'(x) cdot dx$ — estimated change (using tangent line)

For small $dx$: $Delta y approx dy$

Differentials 🎯

Key Takeaways — Part 2

1. $dy = f'(x),dx$
2. Differentials estimate the change in output for a small change in input
3. Error propagation uses differentials

Linearization & Differentials

Part 3 of 7 — Over/Underestimates

Concavity Determines the Error

This is a common AP exam question!

Over or Under? 🎯

Key Takeaways — Part 3

1. Concave up → tangent line below → underestimate
2. Concave down → tangent line above → overestimate

Linearization & Differentials

Part 4 of 7 — Percentage Error

Relative and Percentage Error

ext{Relative error} = rac{dy}{y} = rac{f'(x),dx}{f(x)}

ext{Percentage error} = rac{dy}{y} imes 100%

Error Estimation 🎯

Key Takeaways — Part 4

1. Relative error = $dy/y$
2. For $V = \frac{4}{3}\pi r^3$: percentage error in $V$ = $3 \times$ percentage error in $r$

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:

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

Table-Based Linearization 🎯

Given: $f(3) = 7$ and $f'(3) = -2$.

Key Takeaways — Part 5

1. Table problems give you $f(a)$ and $f'(a)$ directly
2. Just plug into $L(x) = f(a) + f'(a)(x-a)$

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! ✅