For function z = f(x,y) where f(x,y) = 4−x^{2}-2y^{2} , we compute that f_{x}(1,1) =−2 and f_{y}(1,1) = −4. We have learned that the partial derivative f_{x}(1, 1) determines a tangent line L_{1} of graph z = f(x, y) at (1, 1, 1). In particular, the line L_{1} can be described by a system of two equations
z − 1 = −2(x − 1), y = 1.
Similarly, f_{y}(1,1) determines one other tangent line L_{2} of graph z = f(x,y) at (1, 1, 1), which can be described by
z − 1 = −4(y − 1), x = 1.
a) Write the vector equations of L_{1} and L_{2}_{,} respectively. Specifically, describe the equation of L_{1} as ⃗r = ⃗r0 + t⃗v1 for some ⃗v1, where ⃗r0 = ⟨1,1,1⟩. Use a similar argument to describe the equation of L_{2} as ⃗r = ⃗r0 + t⃗v2 for some ⃗v2.
b) Use the previous solution, compute the scalar equation of the plane that contains both L_{1} and L_{2}_{.}
The problem concerns vector and scalar equations of lines and planes associated to the given surface
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