1. For function z = f (x, y) where f(x, y) = 4 – x² – 2y², we compute that f#(1, 1) = -2 and fy(1,1) = -4. In our class, we have learned that the partial derivative fæ(1, 1) determines a tangent line...


1. For function z =<br>f (x, y) where f(x, y) = 4 – x² – 2y², we compute that f#(1, 1) =<br>-2 and fy(1,1) = -4. In our class, we have learned that the partial derivative<br>fæ(1, 1) determines a tangent line L1 of graph z =<br>the line L1 can be described by a system of two equations<br>f(x, y) at (1,1, 1). In particular,<br>z – 1 = -2(x – 1),<br>1.<br>Similarly, fy(1, 1) determines one other tangent line L2 of graph z =<br>(1, 1, 1), which can be described by<br>f (x, y) at<br>z – 1 = -4(y – 1),<br>x = 1.<br>a) Write the vector equations of L1 and L2, respectively. Specifically, describe<br>the equation of L1 as 7 = ro + tủi for some v1, where ro = (1,1, 1). Use a<br>similar argument to describe the equation of L2 as = ro + ti2 for some 02.<br>b) Use the previous solution, compute the scalar equation of the plane that<br>contains both L1 and L2.<br>

Extracted text: 1. For function z = f (x, y) where f(x, y) = 4 – x² – 2y², we compute that f#(1, 1) = -2 and fy(1,1) = -4. In our class, we have learned that the partial derivative fæ(1, 1) determines a tangent line L1 of graph z = the line L1 can be described by a system of two equations f(x, y) at (1,1, 1). In particular, z – 1 = -2(x – 1), 1. Similarly, fy(1, 1) determines one other tangent line L2 of graph z = (1, 1, 1), which can be described by f (x, y) at z – 1 = -4(y – 1), x = 1. a) Write the vector equations of L1 and L2, respectively. Specifically, describe the equation of L1 as 7 = ro + tủi for some v1, where ro = (1,1, 1). Use a similar argument to describe the equation of L2 as = ro + ti2 for some 02. b) Use the previous solution, compute the scalar equation of the plane that contains both L1 and L2.

Jun 04, 2022
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