Focus on 5 Core Modules: (1) Multivariable Limits, Continuity tests at origin (e.g. (x^4 - y^2)/(x^4 + y^2)), Total derivative chain rule dw/dt; (2) Absolute extrema on bounded regions and Lagrange Multipliers constrained optimization; (3) Double and Triple Integrals (Cartesian, Polar, and Cylindrical coordinates); (4) Vector Calculus (Green’s Theorem, Conservative vector fields with scalar potential, Gauss Divergence Theorem); (5) Infinite Series convergence tests (D’Alembert’s Ratio Test, Cauchy’s Root Test, Alternating Series, Power Series Interval of Convergence, Maclaurin Series).
Key Points: Set grad(f) = lambda * grad(g). Grad f = (2, 1), grad g = (2x, 2y). So 2 = 2*lambda*x => x = 1/lambda; 1 = 2*lambda*y => y = 1/(2*lambda). Plug into constraint (1/lambda^2) + (1/4*lambda^2) = 5 => lambda = +-1/2. Points are (2, 1) [Max = 5] and (-2, -1) [Min = -5].
Key Points: Left side: Evaluate line integral along C1 (y=x^2 from 0 to 1) and C2 (y=x from 1 to 0). Right side: Double integral of (dQ/dx - dP/dy) = (2x - (x + 2y)) dx dy = (x - 2y) dx dy over region 0 <= x <= 1, x^2 <= y <= x. Both evaluate to -1/20.
Key Points: Compute div F = d(4xz)/dx + d(-y^2)/dy + d(yz)/dz = 4z - 2y + y = 4z - y. Use cylindrical coordinates: x = r cos theta, y = r sin theta, z = z. Integral from theta=0 to 2pi, r=0 to 2, z=0 to 3 of (4z - r sin theta) r dz dr d theta = 72 pi.
Key Points: Find limit |a_(n+1) / a_n| = limit (n+1)^2 / ((2n+2)(2n+1)) * x = (1/4) * x. By ratio test, series converges for x < 4, diverges for x > 4, test fails for x = 4 (apply Raabe’s test).
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