Use a tangent line approximation to estimate ((8.06)^(2/3)).  

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You should remember that the linear approximation near a point x = a is given by equation `L(x) = f(a) + f'(a)(x-a).`

Notice that you need to consider the general function`f(x) = x^(2/3) =gt f'(x) = (2/3)*x^(2/3-1) ` => `f'(x) = 2/(3root(3)x).`

`L(x) = f(8.06) + f'(8.06)(x-8.06)`

`L(x) = root(3)(8.06^2) + 2/(3root(3)8.06)(x - 8.06)`

`L(x) = 4.019 + 0.332(x - 8.06)`

`L(x) = 4.019 + 0.332x - 2.680`

`L(x)|_(x=8.06) = 1.338 + 0.332*8.06`

 `L(x)|_(x=8.06) = 4.01392`

Hence, evaluating `root(3)(8.06^2)`  using linear approximation yields `L(x)|_(x=8.06) = 4.01392` . 

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial