Multivariable Functions

Example 1 — Gradient and Directional Derivative of f(x,y) = 3x²y − y³

Example 2 — Find and Classify All Critical Points of f(x,y) = x² + y² − 2x − 4y + 1

Example 3 — Lagrange: Minimize x² + y² Subject to x + y = 1

Example 4 — Level Curve Tangent Line Using the Gradient

What is a function of two variables?

A function of two variables f(x, y) assigns a single real number to each ordered pair (x, y) in its domain. You can think of it as a surface in 3D space: for every (x, y) point on the xy-plane, f(x, y) gives the height z above that point. Example: f(x, y) = x² + y² is a paraboloid. Its domain is all of ℝ² (any real x and y), and its range is [0, ∞) because x² + y² ≥ 0 always.

What are level curves and how do you find them?

Level curves (also called contour lines) of f(x, y) are the curves in the xy-plane where f has a constant value c: f(x, y) = c. They show the 'shape' of the surface viewed from above. To find level curves: set f(x, y) = c for various values of c and solve for the relationship between x and y. Example: for f(x, y) = x² + y², the level curve at c = 4 is x² + y² = 4, which is a circle of radius 2. Level curves that are close together indicate a steep surface; widely spaced level curves indicate a gentle slope.

What is a partial derivative and how do you compute it?

The partial derivative of f(x, y) with respect to x, written ∂f/∂x or fₓ, measures how f changes as x changes while y is held constant. To compute it, treat y as a constant and differentiate with respect to x using ordinary single-variable rules. Similarly, ∂f/∂y treats x as a constant and differentiates with respect to y. Example: if f(x, y) = 3x²y + sin(y), then ∂f/∂x = 6xy (y is constant, so sin(y) contributes 0) and ∂f/∂y = 3x² + cos(y) (x² is constant with respect to y).

What is the gradient vector and what does it mean geometrically?

The gradient of f(x, y) is the vector ∇f = (∂f/∂x, ∂f/∂y). It points in the direction of steepest ascent on the surface — the direction you would travel in the xy-plane to climb the surface as steeply as possible. Its magnitude |∇f| equals the rate of that maximum increase. Geometrically, ∇f at a point (a, b) is always perpendicular (normal) to the level curve f(x, y) = f(a, b) passing through that point. The gradient of zero means you are at a critical point — a peak, valley, or saddle.

How do you compute a directional derivative?

The directional derivative of f in the direction of a unit vector û = (u₁, u₂) is Dᵤf = ∇f · û = fₓu₁ + fᵧu₂. It gives the rate of change of f as you move from the point in the direction û. Important steps: (1) if the direction vector is not a unit vector, normalize it first by dividing by its magnitude. (2) Compute the gradient ∇f. (3) Take the dot product. The maximum directional derivative equals |∇f| and occurs in the direction of ∇f itself. The minimum equals −|∇f| and occurs in the direction −∇f.

How do you find critical points of a function of two variables?

Critical points of f(x, y) occur where both partial derivatives are zero simultaneously, or where one or both partial derivatives do not exist. To find them: (1) compute fₓ and fᵧ, (2) set both equal to zero: fₓ = 0 and fᵧ = 0, (3) solve the system of equations for (x, y). Example: for f(x, y) = x³ − 3x + y² − 4y, set fₓ = 3x² − 3 = 0 → x = ±1, and fᵧ = 2y − 4 = 0 → y = 2. Critical points are (1, 2) and (−1, 2). Then use the second derivative test to classify each one.

What is the second derivative test for functions of two variables?

At a critical point (a, b), compute the discriminant D = fₓₓ(a,b)·fᵧᵧ(a,b) − [fₓᵧ(a,b)]². Then: if D > 0 and fₓₓ > 0, the point is a local minimum. If D > 0 and fₓₓ < 0, it is a local maximum. If D < 0, the point is a saddle point (neither max nor min). If D = 0, the test is inconclusive and other methods are needed. The discriminant measures whether the surface curves up in every direction (D > 0, like a bowl) or curves up in some directions and down in others (D < 0, like a saddle).

What are Lagrange multipliers and when do you use them?

Lagrange multipliers solve constrained optimization problems: find the extrema of f(x, y) subject to a constraint g(x, y) = k. The key insight is that at a constrained extremum, the gradients of f and g must be parallel: ∇f = λ∇g for some scalar λ (the Lagrange multiplier). This gives the system: fₓ = λgₓ, fᵧ = λgᵧ, and g(x, y) = k — three equations for three unknowns (x, y, λ). Solve the system, then evaluate f at each solution to find the maximum or minimum. Use Lagrange multipliers when direct substitution from the constraint is algebraically difficult.

What are mixed partial derivatives and does order matter?

Mixed partial derivatives differentiate with respect to two different variables in sequence. For example, fₓᵧ means differentiate first with respect to x, then with respect to y: fₓᵧ = ∂/∂y(∂f/∂x). By Clairaut's theorem (also called Schwarz's theorem), if f and its partial derivatives are continuous, then fₓᵧ = fᵧₓ — the order of differentiation does not matter. This equality holds for virtually all functions encountered in calculus courses. The mixed partials appear in the discriminant D of the second derivative test, reflecting how the surface twists.