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Answers by Roman C.

If F = ∇U then we find that:   U = xey + f(y,z)   U = xey + yez + g(x,z)   U = yez + h(x,y)   We can take:   f(y,z) = yez   g(x,z) = 0   h(x,y) = xey   This means that F(x,y,z)...

Let's neaten things up:   f(x) = 2x3 - 6x2 + 3x + 1   f'(x) = 6x2 - 12x + 3   We are iterating:   xn+1 = xn - (2x3 - 6x2 + 3x + 1)/(6x2 - 12x + 3)   Note that convergence is quadratic. That is, there is...

First let's get the local maximums over the real number line.   The the critical points of y = ax3 + bx2 + cx + d are at   x = [-b ±√(b2 - 3ac)] / (3a).   Note that it looks almost identical to the quadratic formula. There is a reason for this...

(5 - 3i) ÷ (4 + 3i)   = [(5 - 3i)(4 - 3i)] ÷ [(4 + 3i)(4 - 3i)]   = (11 - 27i) ÷ 25   = 11/25 - (27/25) i

Add the equations together to eliminate y:   4x - 2y = -18 3x + 2y = 11 ---------------- 7x          = -7   x = -1   Back-substitute:   -3 + 2y = 11   2y = 14   y = 7   If...

Note: √3 / 2 = sin 60° = sin 120° sin 4t = √3 / 2   4t = 360k + 60 or 4t = 360k + 120   t = 90k + 15 or 90k + 30   In the [0°,360°], you get:   15° , 30° , 105° , 120° , 195° , 210° , 285° , 300°

Use Snell's Law.   We have:   n1 = 1.25   n2 = 1.44   θ1 = 72°   We need θ2.   n1 sin θ1 = n2 sin θ2.   1.25 sin 72° = 1.44 sin θ2.   θ2 = sin-1 ((1.25/1.44) sin 72°) = 55.6°

Vector Fields (answer)

a. ∇f(x,y,z,w) = 〈sin y + w cos x + 1, sin z + x cos y + 1, sin w + y cos z, sin x + z cos w〉   This means the following must all be f(x,y,z,w):   x sin y + w sin x + x + h1(y,z,w)   y sin z + x sin y + y +...

Original sequence {an} for n=1,2,3,...:   1,9,19,31,45,61,...   First differences: Δan = an+1 - an.   8,10,12,14,16,...   Second differences: Δ2an = Δan+1 - Δan.   2,2,2,2,...   So the formula has the form an =...