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)...

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...

For the question on the mystery number, let x be the number. x + 1/x = 13/6 6x2 - 13x + 6 = 0 (2x - 3)(3x - 2) = 0 x = 3/2 or 2/3 For the question with the secretary: lcm(5,6) = 30 In...

rs = 2GM/c2 ≈ 2(6.67408 × 10-11 Nm2/kg2)(7.95 × 1028 kg) / (299792458 m/s)2 ≈ 118 m

The discriminant is: Δ = b2 - 4ac = (2m+1)2 - 4(2m-1) = 4m2 - 4m + 5 = (2m - 1)2 + 4 We need this to be a perfect square. Since 32 - 22 > 4, we know that Δ < 32 = 9 It is easy to then check that it must be Δ = 4. However, this...

Find the absolute max and min values (answer)

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...

Find dy/dx in terms of x and y (answer)

x3y - x - 3y - 10 = 0 3x2y + x3dy/dx - 1 - 3dy/dx = 0 (x3 - 3)dy/dx = 1 - 3x2y dy/dx = (1 - 3x2y) / (x3 - 3)

Complex number division (answer)

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

Using the convention that the potential has limit 0 infinitely far away, we have the following: The potential on the outside of the sphere is V(r) = -GM/r, where r is the distance from the center. Thus if the sphere has radius R, the surface potential is V(R) = -GM/R Next,...

It's linear equations by elimination (answer)

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°

Note that the first given plane x - y - 4z = 2 already contains (2,4,-1) and so it is the answer to this problem. I will outline the general procedure here for how to find such a plane in case you are given a version of the problem where neither plane given has the...

The minimum (or maximum) value achieved for y = ax2 + bx + c is y = -Δ/(4a) where Δ = b2 - 4ac is the discriminant. In your case: Δ = b2 - 4ac = (-5)2 - 4(3)(-k) = 25 + 12k ymin = -(25 + 12k)/(4·3) = -25/12 - k -25/12 -...

Yes. For every integer N≠0, we have that each of N and -N is both a multiple and factor of N simultaneously.

The GPA is calculated as follows: Each grade possible corresponds to a Grade Point amount, which is typically as follows: 4.00 | A 3.75 | A- 3.25 | B+ 3.00 | B 2.75 | B- 2.25 | C+ 2.00 | C 1.75 | C- 1.25 | D+ 1.00 | D 0...

Surface: Break it into the parts above the xy-plane and below it. Above: Domain is the washer 9 ≤ x2 + y2 ≤ 25. Use two integrals here. Vector: S=〈 √(25 - x2 - y2) / x , √(25 - x2 - y2) / y, 1 〉 Curl F = | i ...

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°

f'(x)=1/(2√(x-1)) Since the endpoints of the part of the function in this interval are (1,0) and (3,√2), the average rate of change in[1,3] is √2 / 2 f'(c) = √2 / 2 1/√(c-1) = √2 1/(c-1) = 2 c - 1 = 1/2 c =...

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 =...

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