Reference no: EM131077500
Math 121A: Midterm 1-
1. The function f(x, y, z) = x2 + 3y2 + 5z2 + 2xy + 4yz - 4x - 2z, has one minimum point. Find its location.
2. (a) Calculate the derivative of f(x) = log(log(log(x))).
(b) By using an appropriate series test, determine whether
n=3Σ∞(1/n log(n)log(log(n)))
converges or diverges.
(c) By using an appropriate series test, determine whether
n=3Σ∞((-1)n/n log(n)log(log(n)))
converges or diverges.
3. By using Lagrange multipliers, find the smallest possible surface area (including both ends) of a cylinder with volume V.
4. (a) By considering appropriate powers of eiθ = cosθ + i sinθ or otherwise, determine an expression for sin3θ as a linear combination of terms with the form sin kθ.
(b) Consider the annulus A defined as a ≤ r ≤ b in polar coordinates, where 0 < a < b. Show that for any integer k, the function r±ksin kθ is a solution to the Laplace equation ∇2φ = 0 in A.
(c) Find a solution to ∇2φ = 0 in A that satisfies the boundary conditions
φ(a, θ) = 4 sin3θ, φ(b, θ) = 0.
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Find the smallest possible surface area
: Math 121A: Midterm 1. By using Lagrange multipliers, find the smallest possible surface area (including both ends) of a cylinder with volume V
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