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.
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.
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.
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.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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