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Linearization

David Brown

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

  1. What Does Local Linearity Mean?
  2. Using a Tangent to Approximate a Function
  3. Overestimate or Underestimate?
  4. Worked Example
  5. Practice Questions
  6. Glossary
  7. 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=x3−4x+3y = x^3 - 4x + 3. At the point (2, 3), the tangent line y=8x−13y = 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)f(x) at x=ax = a is given by:

y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a)

or

y=f(a)+f′(a)(x−a)y = f(a) + f'(a)(x - a)

Provided that f(x)f(x) is differentiable at aa.

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

For the graph of y=x3−4x+3y = x^3 - 4x + 3 at (2, 3), the tangent is y=8x−13y = 8x - 13. The tangent will be an appro...

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