3. Consider the function g(x, y) = x+y (a) Find the domain of g and draw...

Transcribed Image Text:

3. Consider the function g(x, y) = x+y (a) Find the domain of g and draw it on the graph below. Mark the boundary of the domain and shade the interior. Label the axes and scale. Use the scale 5 boxes equals 1 unit. (b) Find and draw the level curves or contours of g for k = 1, 2, 3, 4, 5, 6 Use the scale 5 boxes=1 unit. o o (b) Find and draw the level curves or contours of g for k = 1, 2, 3, 4, 5, 6 Use the scale 5 boxes=1 unit. (c) The level curves of g appear to be getting closer together as k gets larger. Explain why this is happening. It might be helpful to consider the domain of g. You can also try a 3D grapher (GeoGebra will allow you to rotate the graph) to help you understand this phenomenon. 4. Consider the function h(x, y) = xe 2xy. The domain is the entire xy-plane. This function is also smooth (differentiable everywhere)...no corners, kinks, or asmyptotes. (a) Find h(1, 3) (b) Find hx (x, y) and hy(x, y) (c) Find h (1, 3) and hy (1,3) (d) The quantity hx (1, 3) tells us how much z changes with a 1-unit increase in x and no increase in y. Encapsulate these changes in a vector (hint, the middle component is O). The quantity hy (1, 3) tells us how much z changes with a 1-unit increase in y and no increase in x. Encapsulate these changes in a vector (hint, the first component is O). These vectors lie in the plane tangent to the graph of h at the point (1,3, h(1,3)). Use this fact to find the equation tangent plane at this point. (Hint: you need a normal vector and a point to describe a plane. Can you use the two vectors found at the beginning of this part to create a normal vector? (e) Use your previous answer to approximate h at the point (x, y) = (0.99, 3.01) (round to 2 decimal places). 1. For the following functions. Find the domain of the following functions, and draw the domains in the axes provided with clearly labeled axes and scale, and domain boundaries. The interior of the domain should be shaded in. (a) z = f(x, y) = x - y (b) g(x, y) = y-x 1-x 3. Consider the function g(x, y) = x+y (a) Find the domain of g and draw it on the graph below. Mark the boundary of the domain and shade the interior. Label the axes and scale. Use the scale 5 boxes equals 1 unit. (b) Find and draw the level curves or contours of g for k = 1, 2, 3, 4, 5, 6 Use the scale 5 boxes=1 unit. o o (b) Find and draw the level curves or contours of g for k = 1, 2, 3, 4, 5, 6 Use the scale 5 boxes=1 unit. (c) The level curves of g appear to be getting closer together as k gets larger. Explain why this is happening. It might be helpful to consider the domain of g. You can also try a 3D grapher (GeoGebra will allow you to rotate the graph) to help you understand this phenomenon. 4. Consider the function h(x, y) = xe 2xy. The domain is the entire xy-plane. This function is also smooth (differentiable everywhere)...no corners, kinks, or asmyptotes. (a) Find h(1, 3) (b) Find hx (x, y) and hy(x, y) (c) Find h (1, 3) and hy (1,3) (d) The quantity hx (1, 3) tells us how much z changes with a 1-unit increase in x and no increase in y. Encapsulate these changes in a vector (hint, the middle component is O). The quantity hy (1, 3) tells us how much z changes with a 1-unit increase in y and no increase in x. Encapsulate these changes in a vector (hint, the first component is O). These vectors lie in the plane tangent to the graph of h at the point (1,3, h(1,3)). Use this fact to find the equation tangent plane at this point. (Hint: you need a normal vector and a point to describe a plane. Can you use the two vectors found at the beginning of this part to create a normal vector? (e) Use your previous answer to approximate h at the point (x, y) = (0.99, 3.01) (round to 2 decimal places). 1. For the following functions. Find the domain of the following functions, and draw the domains in the axes provided with clearly labeled axes and scale, and domain boundaries. The interior of the domain should be shaded in. (a) z = f(x, y) = x - y (b) g(x, y) = y-x 1-x