Structured Active Recall for Complex Coursework.
Turn your lecture PDFs into flashcards, adaptive quizzes, and spaced repetition reviews in seconds.
Start Free Sprint →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).
Upload your Single and Multivariate Calculus lecture slides to generate spaced repetition study cards in seconds.
Turn your lecture PDFs into flashcards, adaptive quizzes, and spaced repetition reviews in seconds.
Start Free Sprint →