The function is a polynomial and is differentiable everywhere, hence yes.

Using the limit definition, f'(0) = lim (h -> 0) [(h^2 sin(1/h) - 0)/h] = 0. Thus, f(x) is differentiable at x = 0.

The function is a polynomial and hence differentiable everywhere, including at x = 1.

Both limits as x approaches 0 from the left and right are equal to 1, hence f(x) is continuous at x = 0.

Both limits as x approaches 0 from the left and right are equal to 0, hence f(x) is continuous at x = 0.

Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.

The left limit is -1 and the right limit is 1, which are not equal. Therefore, f(x) is not continuous at x = 0.

Lenz's law states that the direction of induced current will oppose the change in magnetic flux that produced it.

The pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.

The ordered pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, there are 6 ordered pairs.

R is reflexive, antisymmetric, and transitive, thus it is a partial order.

Using the quadratic formula x = [8 ± √(64 - 48)] / 4 = [8 ± 4] / 4, giving x = 3 or x = 1.

Expanding gives 3x - 3 = 2x + 8. Rearranging gives x = 11.

Dividing both sides by 3 gives x - 2 = 4, thus x = 6.

Expanding both sides gives 3x - 6 = 2x + 2. Rearranging gives x = 8.

Subtracting 3x from both sides gives 2x + 2 = 10, then subtracting 2 gives 2x = 8, leading to x = 4.

Using properties of logarithms, log_3((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 3 => x + 1 = 3(x - 1) => x = 2.

log_3(x) = 2 implies x = 3^2 = 9.

Using properties of logarithms: log_5((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 5 => x + 1 = 5(x - 1) => 4x = 6 => x = 2.

log_5(x) = 2 implies x = 5^2 = 25.

Subtracting 8 from both sides gives 4y = 16, then dividing by 4 gives y = 4.

This is a linear first-order differential equation. The integrating factor is e^(2x). Solving gives y = 2 - Ce^(-2x).

Integrating both sides gives y = x^3 + C.

This is a non-linear differential equation. The solution can be found using substitution methods.

Using the integrating factor method, we find y = Ce^(3x) + 2.

The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).

The characteristic equation is r^2 - 5r + 6 = 0, which factors to (r - 2)(r - 3) = 0, giving the solution y = C1 e^(2x) + C2 e^(3x).

Rearranging gives sin(x) = -√3/2, so x = 4π/3 and x = 5π/3.

The solution is x = π/2.

Rearranging gives cos^2(x) = 1/3, so x = π/3 and 2π/3.