Question

If the equation F(x, y, z) = 0 determines z as a differentiable function of x and y, then, at the points where Fz ≠ 0, the following equations are true:

∂z/ ∂x=−Fx / Fz
and ∂z / ∂y=−Fy / Fz
Use these equations to find the values of ∂z/∂x at the given point.
3z³- 5xy + 4yz + 2y³ - 124 = 0, (3,4,2)

Answer 1

To find the value of ∂z/∂x at the given point (3,4,2), we need to apply the equations provided. First, find the partial derivatives of F(x,y,z) with respect to x, y, and z. Then, evaluate these derivatives at the given point. Finally, use the equations to calculate ∂z/∂x.

Step-by-step explanation:

To find the value of ∂z/∂x at the given point (3,4,2), we need to apply the equations provided. First, we need to find the partial derivatives of F(x,y,z) with respect to x, y, and z. Then, we evaluate these derivatives at the given point.

Given equation: 3z³-5xy+4yz+2y³-124 = 0

Partial derivative of F with respect to x: ∂F/∂x = -5y

Partial derivative of F with respect to y: ∂F/∂y = -5x + 4z + 6y²

Partial derivative of F with respect to z: ∂F/∂z = 4y + 3z²

Now, substitute the values x=3, y=4, z=2 into the derivatives:

∂F/∂x = -5(4) = -20

∂F/∂y = -5(3) + 4(2) + 6(4²) = 15 + 8 + 96 = 119

∂F/∂z = 4(4) + 3(2²) = 16 + 12 = 28

Finally, to find ∂z/∂x at the given point:

∂z/∂x = -(∂F/∂x) / (∂F/∂z) = -(-20) / 28 = 20 / 28 = 5/7