Learning Outcomes

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

Textbook

  • 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:

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

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

dy=f'(x)dx

  • 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.