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- Find the solution of the limit using polar coordinate system. Lim (x,y)->(0,0)√x^2+y^2/x^2+y^2vestigate whether the function u(x, y, z) = e^(3x+4y)cos(5z) is a harmonic function, where harmonic functions are functions that satisfy the Laplace equation: ∂^2u/∂x^2 + ∂^2u/∂y^2 + ∂^2u/∂z^2 = 0 Provide the result of your investigation.How do we find the linearization of g(x, y) = tan−1(xy2) at the point (1, 1). and use linearization to estimate g(1.08, 1.037). thank you.
- Prove that arctan x + arctan y = arctan(x + y) / (1 − xy), xy ≠ 1. Use the formula derived to show that arctan (1/2) + arctan (1/3) = /4Suppose a particle begins to move from the origin(t = 0) following the curve in space. r(t) = {5sint, 5 cos t, 12t} Find the location of the particle on the curve at a distance26 π units along the curve from the origin in the direction of increasing t.lim y→0 sin3y/4y
- Dg 118. Calculus on Manifolds by Michael Spivak, Section 4. Integration of Chains, page 75Use polar coordinates to find the limit. [Hint: Let x = r cos and y = r sin , and note that (x, y) (0, 0) implies r 0.]Use polar coordinates to find the limit. [Hint: Let x = r cos(θ) and y = r sin(θ), and note that (x, y) → (0, 0) implies r → 0.]
- Determine the concavity of y = 3 + sin x on [0, 2π]w = sqrt(25 - 4x2- 4y2), x=r cos(theta), y=r sin(theta)(a) Find del w / del r and del w / del ? using the appropriate Chain Rule.A particle is traveling along a circualr path defined by x^2 + y^2 = 4, where x and y are measured in centimeters. If the particle starts at the point (2,0) and moves in a counterclockwise direction at a speed of 6cm/sec, what will be the coodeinate fo the particle's position after 10 seconds? (show detailed diagrams)