# Curve defined by two implicit functions

• Oct 25th 2009, 02:45 PM
My Little Pony
Curve defined by two implicit functions
The curve r(t) passing through the point is defined implicitly by the equations:

x^2y + y^2x + xyz + z^2 = 4
x + y + z = 3

i) Find the equation of the tangent line to this curve at the given point.
ii) Assume that we may choose x = t as the parameter. Find formulae for dy/dt and d^2y/dt^2.

I'm not sure how to tackle these questions. For i), can I use either of the two equations?
• Oct 27th 2009, 06:08 AM
Media_Man
if x=t then dy/dt=dy/dx
Well, if x=t, then what you are really looking for are the first and second derivatives of y with respect to x. Start by simplifying $\displaystyle x^2y + y^2x + xyz + z^2=xy(x+y+z) + z^2=3xy+z^2=4$, and then substituting $\displaystyle z=3-x-y$ to get $\displaystyle x^2-6x+5xy-6y+y^2+5=0$. Next differentiate implicitly to find $\displaystyle \frac{dy}{dx}$: $\displaystyle 2x-6+5xy'+5y-6y'+2yy'=0$. Isolate $\displaystyle y'$ and repeat the process to get $\displaystyle y''$. This should be enough to get you started.
• Oct 27th 2009, 06:14 AM
Jester
Quote:

Originally Posted by My Little Pony
The curve r(t) passing through the point is defined implicitly by the equations:

x^2y + y^2x + xyz + z^2 = 4
x + y + z = 3

i) Find the equation of the tangent line to this curve at the given point.
ii) Assume that we may choose x = t as the parameter. Find formulae for dy/dt and d^2y/dt^2.

I'm not sure how to tackle these questions. For i), can I use either of the two equations?