Functions of Two Variables

Business Calculus · Axiom Academy

INTRO Functions of Two Variables Moving beyond single-variable calculus to model real business complexity In the real world, business outcomes rarely depend on just one factor. Profit depends on both price and quantity. Production depends on both labor and capital. Sales depend on both advertising and pricing. In this unit, we extend our calculus toolkit to handle functions of multiple variables - a crucial skill for analyzing complex business scenarios. 1 What is a Function of Two Variables? A function of two variables, written z = f(x, y) , takes two input values and produces one output value. Single variable: y = f(x) → one input, one output (a curve in 2D) Two variables: z = f(x, y) → two inputs, one output (a surface in 3D) Output Q depends on labor L and capital K Profit depends on price p and advertising a Cost depends on quantity of two products x, y Revenue from selling q₁ and q₂ units at prices p₁, p₂ To evaluate f(x, y) at a specific point, substitute values for both x and y. Given f(x, y) = x^2 + 3xy - y^2 , find f(2, 3). Like single-variable functions, we need to identify valid inputs: In business applications, the domain often has natural restrictions: quantities must be non-negative, prices must be positive, labor hours can't exceed available time, etc. Now that we understand functions of two variables, we can ask important questions: How do we graph these functions? (3D surfaces and contour plots)

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