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!

Homework Help: Calculus II - closed function question

  1. Jan 31, 2012 #1
    there is a function: F ( x, y, z) = 2ln (xz) + sin ( xyz) − y^2 = 0.
    the func is defined by the closed function z=f(x,y) and provides : f(1,0)=1
    we define: g(t)=f(t,1-t^6) . where t is very close to 1.
    I have to find g'(1)

    2. Relevant equations

    I tried to to do like that: find F'x and F'z and did z'x =-(F'x/ F'z) and got -1. but from here I dont know what to do.
    The answer is g'(1)=2.
    Thanks for your help!
    Last edited: Jan 31, 2012
  2. jcsd
  3. Jan 31, 2012 #2


    User Avatar
    Homework Helper

    sorry, can't really follow this - can you explain the question in a little more detail and more clearly? for example what is g in relation to F and f and x in relation to t?
  4. Jan 31, 2012 #3
    sorry about the wrong copying: g(t)=f(t,1-t^6)
  5. Jan 31, 2012 #4


    User Avatar
    Science Advisor

    Are you saying there exist f(x,y) such that z= f(x,y), in the neighborhood of (1, 0, 1), is the same as [itex]2ln(xz)+ sin(xyz)- y^2= 0[/itex]?

    If so, then F(x, y, g)= 2ln(xg)+ sin(xyg)- y^2= 0. And with x= t, y= 1- t^6, that is 2ln(tg)+ sin(t(1- t^6)g)- (1- t^6)^2.

    Differentiating with respect to t,
    [tex]\frac{2}{tg}(1+ tg')+ cos(t(1- t^6)g)((1- t^6)g- 6t^6g+ t(1-t^6)g')- 12(1- t^6)t^5= 0[/tex]
    Set t= 1, so that x= 1, y= 0, and F(1, 0, g(1))= 2ln(g(1))= 0. What is g(1)? Put x= 1, y= 0, and that value of g(1) into the equation and solve for g'(1).

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook