Multivariable Summary

Business Calculus · Axiom Academy

SUMMARY Multivariable Calculus Summary Review of key concepts and skills checklist Check off each skill as you master it: Visualize 3D surfaces Understand graphs of z = f(x,y), level curves, and cross-sections Compute partial derivatives Find f x and f y by treating other variables as constants Find second-order partials Calculate f xx , f yy , f xy and verify Clairaut's theorem Apply Cobb-Douglas production function Compute marginal products and analyze returns to scale Find critical points Solve systems where f x = 0 and f y = 0 simultaneously Apply the second derivative test Use discriminant D to classify critical points as max, min, or saddle Calculate MRTS Find the marginal rate of technical substitution and interpret it Set up Lagrangian problems Form L = f - λ(g - c) for constrained optimization Solve Lagrange multiplier problems Find optimal solutions subject to constraints Interpret the Lagrange multiplier Understand λ as the shadow price of the constraint z = f(x,y) creates a surface in 3D space. We analyze it using partial derivatives, level curves, and cross-sections. Measure rates of change in one direction while holding other variables constant. Essential for marginal analysis. Find critical points where all first partials are zero, then classify using the discriminant D. Lagrange multipliers let us optimize when resources are limited. The gradient condition ∇f = λ∇g is key.

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