For function z = f(x,y) where f(x,y) = 4−x2-2y2 , we compute that fx(1,1) =−2 and fy(1,1) = −4. We have learned that the partial derivative fx(1, 1) determines a tangent line L1 of graph z = f(x, y) at (1, 1, 1). In particular, the line L1 can be described by a system of two equations z − 1 = −2(x − 1), y = 1. Similarly, fy(1,1) determines one other tangent line L2 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 L1 and L2, respectively. Specifically, describe the equation of L1 as ⃗r = ⃗r0 + t⃗v1 for some ⃗v1, where ⃗r0 = ⟨1,1,1⟩. Use a similar argument to describe the equation of L2 as ⃗r = ⃗r0 + t⃗v2 for some ⃗v2. b) Use the previous solution, compute the scalar equation of the plane that contains both L1 and L2.
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For function z = f(x,y) where f(x,y) = 4−x2-2y2 , we compute that fx(1,1) =−2 and fy(1,1) = −4. We have learned that the partial derivative fx(1, 1) determines a tangent line L1 of graph z = f(x, y) at (1, 1, 1). In particular, the line L1 can be described by a system of two equations
z − 1 = −2(x − 1), y = 1.
Similarly, fy(1,1) determines one other tangent line L2 of graph z = f(x,y) at (1, 1, 1), which can be described by
z − 1 = −4(y − 1), x = 1.
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a) Write the
vector equations of L1 and L2, respectively. Specifically, describe the equation of L1 as ⃗r = ⃗r0 + t⃗v1 for some ⃗v1, where ⃗r0 = ⟨1,1,1⟩. Use a similar argument to describe the equation of L2 as ⃗r = ⃗r0 + t⃗v2 for some ⃗v2. -
b) Use the previous solution, compute the scalar equation of the plane that contains both L1 and L2.
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