Linearization

David Brown

6 min read

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Study Guide Overview

This study guide covers approximating function values using local linearity and tangent lines. It explains how to find the equation of a tangent line and use it for approximation. The guide also discusses how a function's concavity determines whether the approximation is an overestimate or underestimate, and provides worked examples and practice questions. Key terms include local linearity, tangent, concavity, overestimate, and underestimate.

#Local Linearity of a Function

#Table of Contents

What Does Local Linearity Mean?
Using a Tangent to Approximate a Function
Overestimate or Underestimate?
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#What Does Local Linearity Mean?

Key Concept

If you 'zoom in' far enough on the graph of a function at a point, a curve can look more like a straight line. This means the tangent to a graph of a function at a point can act as an approximation for the function at that point. This linear approximation of a function is only appropriate very close to the point, hence the term "local linearity."

For instance, consider the function

y = x^3 - 4x + 3

. At the point (2, 3), the tangent line

y = 8x - 13

approximates the curve closely around this point. As you zoom in around (2, 3), the curve looks more like the tangent line.

#Using a Tangent to Approximate a Function

#Equation of the Tangent

The equation of the tangent to $f(x)$ at $x = a$ is given by:

$y - f(a) = f'(a)(x - a)$

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

Provided that $f(x)$ is differentiable at $a$ .

Due to the local linearity of a function, this can be a linear approximation for $f(x)$ at points close to $(a, f(a))$ .

For the graph of

y = x^3 - 4x + 3

at (2, 3), the tangent is

y = 8x - 13

. The tangent will be an appro...

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Question 1 of 10

Zooming in on a function's graph at a point makes it look like a straight line! 🤩 This is the idea behind local linearity. Where is the tangent line approximation most accurate?

Far away from the point of tangency

At the y-intercept

Close to the point of tangency

Where the function's second derivative is zero