1. Not finding help here? Sign up for a free 30min 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!

Finding limits in differentiation from first principles

  1. Apr 7, 2012 #1
    1. The problem statement, all variables and given/known data

    Differentiate sin(ax), cos(ax) and tan(ax) from first principles.

    2. Relevant equations

    3. The attempt at a solution

    I have used first principles to differentiate the three expressions and have been successful until I encountered limits of some expressions in the process.

    I need to find the limit as Δx tends to 0 of the following expressions.

    1. sin(aΔx)/Δx
    2. [cos(Δx) - 1]/Δx
    3. tan(Δx)/Δx

    I know some spooky proofs which use the fact that for small Δx, sin(aΔx) ≈ Δx, cos(aΔx) ≈ 1- (aΔx)2/2 and tan(aΔx) ≈ Δx. They do give the right answers and I have been told these methods would give me full marks in the exam (me being a physics student and all that crap!), but I would appreciate it if you give a full rigorous proof of the three limits. (Armed with those, the original problem is just a piece of cake.)
     
  2. jcsd
  3. Apr 7, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi failexam :smile:

    (rsinθ)/rθ = arc-length/chord-length :wink:
     
  4. Apr 7, 2012 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Once you have [tex] \lim_{t \rightarrow 0} \frac{\sin(t)}{t} = 1,[/tex]
    getting
    [tex] \frac{1 - \cos(t)}{t^2} \rightarrow \frac{1}{2} \text{ as } t \rightarrow 0[/tex] follows easily from [tex] \frac{\sin(t)^2}{t^2} = \frac{1 - \cos(t)^2}{t^2}
    = \frac{1-\cos(t)}{t^2} (1+\cos(t)),[/tex] and the limit of [itex] \tan(t)/t[/itex] also follows. So, you need a good proof of [itex] \sin(t)/t \rightarrow 1.[/itex] You can find one in the Khan Academy video .

    RGV
     
    Last edited by a moderator: Sep 25, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding limits in differentiation from first principles
Loading...