Expert: Alon Mandes - 4/4/2009

Question
The function f(x)changes values when x changes from a to a +dx.  What is the approximation error of f(x)= x+x^2, a+3, dx=0.02

(2).  What is and how do I find the linearization  L(x) of f(x) at x=a  F(x)= all of this under a square root 6x+11m a=0 is not under the square root.
(3).  How do I find the implicit differentiation and what is the answer to these particular problems?

(3.) x^3+3x^2y+y^3=8;  
(4). cosxy+x^3=y^3
(5). x/sqrt of 9x+7
(6). 2xy-y^2=1
For some reason I am not getting the same answer the professor is getting with htese problems and need some expert advice on how to solve them and what the answer is?

Answer
I'll start with implicit derivatives & implicit function. To derive
a function which is given in an implicit form, we need to keep in
mind that y is a function of x, say y(x). Thus the derivative of y
will be y'. For example d/dx {y^2}=2yy'. Now, we perform the
derivation operator on both side of the implicit equation. Let's
get started :
(3) x^3+3x^2y+y^3=8 . deriving both sides of the equation yiels:
   3x^2+6xy+3x^2y'+3y^2y'=0
   3x^2+6xy+y'(3x^2+3y^2)=0
   y'=-(3x^2+6xy)/(3x^2+3y^2)
Note that y' is a function of x & y !
(4) cos(xy)+x^3=y^3
   -sin(xy)-sin(xy)y'+3x^2=3y^2y'
   -sin(xy)+3x^2=y'[3y^2+sin(xy)]
   y'=[3x^2-sin(xy)]/[3y^2+sin(xy)]
(5) y=x*(1/sqrt[9x+7])
   y'=(1/sqrt[9x+7])-x*9*[9x+7]^(-3/2)
(6) 2xy-y^2=1
   2y+2xy'-2yy'=0
   y'(2x-2y)=-2y
   y'=(y-x)/y .

Asfor (3) The function f(x)=Sqrt[6x+11] will be linearized as
L(x)=mx+n where m=f'(a), & n can be calculated througth the point
(a,Sqrt[6a+11].

Alon.