Problem 1. Note that the exact root is: x = - 3√4 Machine precision: -1.5874010519681994 Using seed x0 = -2: f(x) = x3 + 4 so f'(x) = 3x2. Thus we iterate: x ← x - (x3 + 4) / (3x2) = 2x/3 - 4/(3x2) x0 = -2 x1...

Problem 1. Note that the exact root is: x = - 3√4 Machine precision: -1.5874010519681994 Using seed x0 = -2: f(x) = x3 + 4 so f'(x) = 3x2. Thus we iterate: x ← x - (x3 + 4) / (3x2) = 2x/3 - 4/(3x2) x0 = -2 x1...

Triple both sides and then regroup based on pairs which look like a product rule. 3(x2 - 4xy - 2y2)dx + 3(y2 - 4xy - 2x2) dy = 0 3y2 dy - (6y2 dx + 12xy dy) - (12xy dx + 6x2 dy) + 3x2 dx x3 - 6x2y - 6xy2 + y3 = C &n...

simplify the difference quotient (answer)

f(x) = 4 - 3x - x2 [f(x) - f(a)] / (x - a) = [(4 - 3x - x2) - (4 - 3a - a2)] / (x - a) = (-3x - x2 + 3a + a2) / (x - a) = [-3(x - a) - (x + a)(x - a)] / (x - a) = -3 - x - a

algebra 1 help please? (answer)

(z2 - 3z + 2)(z2 + 6z + 8) / (z2 + 3z - 4)(z2 - 4) = (z-1)(z-2)(z+2)(z+4) / (z+3)(z-1)(z+2)(z-2) = (z+4) / (z+3)

You can only give bounds for this number. Let's assume, the n hexagons must form a polyhex (like a polyomino but with regular hexagons). Maximum: The first hexagon contributes 6 vertices and each new hexagon after that contributes at most 4 vertices. MAX...

derivative w(x) = 2cotx sin^2x (answer)

w(x) = 2 cot x sin2 x = 2 cos x sin x = sin 2x dw/dx = 2 cos 2x

probability? (answer)

Coins are memoryless, so it is 1/2.

sin 3x - 1 = 0 sin 3x = 1 3x = 2kπ + π/2 where k ∈ Z x = 2kπ/3 + π/6 where k ∈ Z In the [0,2π), the solutions are π/6, 5π/6, and 3π/2.

x2 + y2 - x + y - 12 = 0 x2 + y2 - 4x - 2 = 0 The easiest way to do this is to subtract the second equation from the first, which yields a linear equation through the two intersection points. 3x + y - 10 = 0 So that means y...

More generally, for odd n, 1n + 2n + 3n + … + nn is a multiple of n. To show that, note that nn is a multiple of n and that we can split the sum into pairs of terms (n-k)n + kn summing to a multiple of n. We...

We can say a couple of things: 1. Since the directrix is vertical and the focus is to it's left, the parabola opens left. 2. We can find the parabola's equation in the form (y - k)2 = 4p(x - h). The vertex is halfway between the focus and directrix, so...

∂f/∂x = 1/(xy) ∂f/∂y = (-1/y2)ln x

Step 1. Use Mark's response to get 2x2 + 5x - 3 = 0 Then use Viete's formulas to get that the sum of the solutions is -5/2. Viete's formulas state that the set R={r1, … ,rn} of roots (real and complex) of anxn + an-1xn-1 + … + a1x + a0 = 0 satisfy...

calculus help (answer)

1. The total area of the squares is 13. The area of the disk is 4π < 4 × 3.25 = 13. Thus clearly, some squares must overlap 2. This is actually a Number Theory problem. If we choose 12 different two digit numbers...

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...