Help finding the equation of a tangent plane

[cherryyosh]

Let [itex]F: R^2 \rightarrow R^4[/itex] be
$$F(x,y) = (x^3y,sin(xy),3,xy^3)$$

i) Find the Jacobian matrix of F at (1, pi)
ii) What is the local linear approximation to F at (1, pi)?
iii) Write the equation of the tangent plane to the graph of F(1,pi)
iv)use ii) to compute (0.99, pi+.001) approximately

Homework Equations

The Attempt at a Solution

for i) I got the jacobian being $$\left(\begin{array} 3\pi & 1 \\ -\pi & -1 \\ 0 & 0 \\ \pi^3 & 3\pi^2 \end{array} \right) $$

With a Linear approximation of
$$\left(\begin{array} 3\pi x + y - 3\pi \\ -\pi x - y + 2\pi \\ 3 \\ \pi^3 x + 3\pi^2 y - 3 \pi^3 \end{array} \right) $$

But I am not sure where to go from here to get the tangent plane equation. Any help would be appreciated.

 

[mighty2000]

For iii) The equation of the tangent plane to the graph of F(1,pi) would be:
$$z = 3\pi + 2\pi^2(y-\pi) $$

For iv) To compute (0.99, pi+.001) approximately, we can use the linear approximation from part ii) by plugging in the values of x=0.99 and y=pi+.001 into the linear approximation equation. This would give us an approximate value for F(0.99,pi+.001).