Learning Outcomes

  1. Find the linear approximation to a function at a point.
  2. Calculate the differential of a given function. 


  • Chapter 4.2  Linear Approximation and Differentials  

Textbook Assignment

  • p. 364:    62, 63, 67-70 all, 72-74 all

WeBWork Assignment

  • Application-Linearization
  • Application-Differentials

Exit problems of the session 

  1. Find the linear approximation to $y=\tan x$ near $a=\dfrac{\pi}{4}$.

  2. Use appropriate linear approximation to estimate $\sqrt[3]{26.95}$.

  3. Find the differential $dy$  for $y=xe^{-x}$ and evaluate it at $x=1$ and $dx=0.02$.


Key Concepts

  • A differentiable function $y=f(x)$ can be approximated at $a$ by the linear function:


  • The differential $dx$ is an independent variable that can be assigned any nonzero real number; the differential $dy$ is define to be


  • For a function $y=f(x)$, if $x$ changes from $a$ to $a+dx$, then $dy=f'(x)dx$ is an approximation for the change in $y$. The actual change in $y$ is $\Delta y=f(a+dx)-f(a)$.  


Videos and Practice Problems of Selected Topics

  1. $\rhd$ Local linearity (9:37) Estimate $\sqrt{4.36}$.
  2. $\rhd$ Linear approximation of a rational function (7:10) Find a linear expression that approximates $\dfrac{y}{x-1}$ around $x=-1$.
  3. $\rhd$ Comparing $\Delta y$ and $dy$ (4:07) Let $y=2x^2$.
    • Find the change in $y$, $\Delta y$, when $x=2$ and $\Delta x=0.3$.
    • Find differential $dy$ when $x=2$ and $dx=0.3$.