The derivative y' = 3x^2. At x = 1, y' = 3(1^2) = 3.

This is separable. Integrating gives y = Cx.

Integrating both sides gives y = x^3 + C, where C is the constant of integration.

Integrating gives y = (6/2)x^2 + C = 3x^2 + C.

This is separable. Separating and integrating gives y = Ce^(x^2/2).

Expanding gives 2x - 6 = 8. Adding 6 gives 2x = 14, thus x = 7.

Adding 4 to both sides gives 2x = 14. Dividing by 2 gives x = 7.

Subtracting 8 from both sides gives 4x = 16. Dividing by 4 gives x = 4.

Adding 8 to both sides gives 4x = 8. Dividing by 4 gives x = 2.

The general solution is y = Ce^(4x). Using y(1) = 2, we find C = 2e^(-4).

Specific heat capacity (c) = Q / (m * ΔT) = 500 J / (2 kg * 10°C) = 25 J/kg°C.

To find the sum of the coefficients, substitute x = 1: (2*1 - 3)^4 = (-1)^4 = 1. The sum of coefficients is 81.

The sum of the coefficients in the expansion of (x + 1)^4 is (1 + 1)^4 = 2^4 = 16.

The sum of the coefficients in the expansion of (x + 1)^n is given by (1 + 1)^n = 2^n. Here, n=8, so the sum is 2^8 = 256.

To find the sum, we add the real parts and the imaginary parts: (2 + 3) + (2 - 4)i = 5 - 2i.

The sum of two complex numbers z1 + z2 = (2 + 3i) + (4 - 2i) = (2 + 4) + (3 - 2)i = 6 + i.

The first five prime numbers are 2, 3, 5, 7, and 11. Their sum is 2 + 3 + 5 + 7 + 11 = 28.

Sum of interior angles = (n-2) * 180° = (6-2) * 180° = 720 degrees

The sum of the interior angles of a polygon is given by (n-2) * 180°, where n is the number of sides. For a pentagon, n = 5, so (5-2) * 180° = 540°.

The sum of the interior angles of any triangle is always 180 degrees.

The sum of the interior angles of any triangle is always 180 degrees.

Using Vieta's formulas, the sum of the roots is -(-8)/2 = 4.

The sum of the roots is given by -b/a = 5/1 = 5.

The sum of the squares of the roots is given by (sum of roots)^2 - 2(product of roots). Here, sum = 5, product = 6. So, 5^2 - 2*6 = 25 - 12 = 13.

'Abundant' means plentiful or existing in large quantities. Therefore, the synonym is 'plentiful'.

'Candid' means truthful and straightforward; frank.

'Elated' means in high spirits; extremely happy and excited.

'Frugal' means being economical or avoiding waste, making 'thrifty' the synonym.

'Gregarious' means fond of company; sociable.

'Meticulous' means showing great attention to detail; very careful and precise.