Author:
Ohio Open Ed Collaborative
Subject:
Calculus
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
  • Cac077
  • Linear Approximation
  • License:
    Creative Commons Attribution Non-Commercial
    Language:
    English
    Media Formats:
    eBook, Interactive

    Linear approximation module

    Overview

    After completing this section, students should be able to do the following.

    • Define linear approximation as an application of the tangent to a curve.
    • Find the linear approximation to a function at a point and use it to approximate the function value.
    • Identify when a linear approximation can be used.
    • Label a graph with the appropriate quantities used in linear approximation.
    • Find the error of a linear approximation.
    • Compute differentials.
    • Use the second derivative to discuss whether the linear approximation over or underestimates the actual function value.
    • Contrast the notation and meaning of dydy versus ΔyΔy.
    • Understand that the error shrinks faster than the displacement in the input.
    • Justify the chain rule via the composition of linear approximations.

    Replacing curves with lines

    Ximera Module

    Two young mathematicians discuss linear approximation.

    Linear approximation

    Ximera Module

    We use a method called ``linear approximation'' to estimate the value of a (complicated) function at a given point.

    Explanation of the product and chain rules

    Ximera Module

    We give explanation for the product rule and chain rule.