Julia Pitre jdp8657

Week 8 Problem 5.5.10

url: http:://people.tamu.edu/~jdp8657/homework.html

Problem: Suppose that *x*^{2}+*yz *= 0 and *xyz* +1 = 0, where *x* and *y* are functions of *z*. Find the equation of the tangent line to the curve at any point where *z* = -2. How many such point
s are there?

Solution: To solve this problem, we need to find the values of the implicit derivatives of x and y, as well as the values of x and y at z = -2.

When z = -2, we get :

*x*^{2}+*yz *= 0 *xyz* +1 = 0

*x*^{2}-2y* *= 0 -2*xy* +1 = 0

x = 1, y = 1/2

When we differentiate the equations:

2xy dx/dz + x^{2} dy/dz + y = 0

yz dx/dz + xz dy/dz +xy = 0

if we solve for the values of the derivatives, we get:

dx/dz = (-z+x^{2} ) /(xz)

dy/dz = - [y( 2 +
2x^{2})]/zx

If we plug in the values x=1, y=1/2, z=-2, we get:

dx/dz = -3/2

dy/dz = 1

so the formula for the tangent line is :

(z+2)(-3/2) i + (z+2)(1) j