Visualizing 3D Graphs

Business Calculus · Axiom Academy

Understanding surfaces in three-dimensional space When we have a function of two variables , the graph is no longer a curve but a surface in three-dimensional space. For each point (x, y) in the domain, the function value z = f(x, y) gives the height of the surface above (or below) that point. y = f(x) produces a curve in 2D space (the xy-plane) z = f(x, y) produces a surface in 3D space (xyz-space) Explore how a paraboloid surface looks from different angles. This represents a function like : Just like a topographic map shows elevation with contour lines, we can visualize a 3D surface using level curves . A level curve is the set of all points where f(x, y) = c for some constant c. Business Example: Profit Contours If represents profit based on advertising spending (x) and price (y), then level curves show all combinations of advertising and price that yield the same profit level. A company might want to find all (advertising, price) pairs that give exactly 50,000 profit—this is a level curve! Another way to understand 3D surfaces is through traces —the curves you get when you slice the surface with a plane: Slice parallel to yz-plane. Shows how z changes with y when x is fixed. Slice parallel to xz-plane. Shows how z changes with x when y is fixed. Horizontal slice (level curve). Shows all (x, y) with the same z value. For a cost function C(L, K) depending on labor (L) and capital (K): L = 100: Shows cost as capital varies with labor fixed at 100 workers

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