1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Why does differentiation find the approximate value?

  1. Sep 21, 2014 #1
    Like imagine I have to find the cube root of 8.03. So I cube 8 and use the dy=8+dy/dx*(8.03-8.00) formula. But why is this finding the value of cube root of 8.03 and why is this value approximate instead of exact?
     
  2. jcsd
  3. Sep 21, 2014 #2
    What is the derivative at 8, conceptually?
     
  4. Sep 21, 2014 #3
    1/12
     
  5. Sep 21, 2014 #4

    Mark44

    Staff: Mentor

    Look at the graph of ##y = x^{1/3}##. The derivative, dy/dx, gives the slope of the tangent line to this curve. The formula you show gives you the y values on the tangent line, which is close to, but slightly different from the y values on the curve.

    Since the tangent line at (8, 2) lies above the curve ##y = x^{1/3}## , the approximate values will be a little larger than the values on the curve.
     
  6. Sep 21, 2014 #5

    Mark44

    Staff: Mentor

    The value of the derivative isn't what Number Nine was asking. He was asking about the meaning of the derivative at that point.
     
  7. Sep 22, 2014 #6
    Your replies made me realize all the holes in my calculus knowledge. I've been taught calculus in school just through formulas so it looks like my concepts are very weak. I have no idea what to do about it because my textbook also only uses formulas.
     
  8. Sep 22, 2014 #7

    Mark44

    Staff: Mentor

    Most calculus texts contain formulas, but they usually contain explanatory text as well. Are you saying that your textbook doesn't have explanations to go with the formulas? Sometimes students focus on the formulas and ignore the surrounding text.
     
  9. Sep 22, 2014 #8
    Basically it comes from this really simple idea: "The tangent line to ##x_0## resembles the curve near ##x_0##". For instance the curve defined by ##y=\sin x## resembles the tangent line to it at ##0## near zero:

    LGMGc.jpg

    For example it is really hard to determine ##\sin(0.1\,\rm rad)## without using a calculator. But since the tangent line has a simple form, namely ##y=mx+b##, one can easily exploit the fact that the tangent line resembles the curve of ##\sin x## near ##0## since ##0.1## is approximately zero to find a rough estimate for ##\sin(0.1\,\rm rad)##.
     
  10. Sep 24, 2014 #9
    My textbook contains examples of HOW to use, but not WHY
     
  11. Sep 24, 2014 #10

    Mark44

    Staff: Mentor

    Assuming each section of your textbook ends with a set of problems, maybe these problems are the WHY the formulas are used.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Why does differentiation find the approximate value?
  1. Find the values: (Replies: 4)

  2. Why does a = -a? (Replies: 3)

Loading...