The left limit as x approaches 1 is 1, and the right limit is also 1. Thus, f(1) = 1, making it continuous.

Centroid = ((0+6+3)/3, (0+0+6)/3) = (3, 2).

Setting f'(x) = 3x^2 - 3 = 0 gives x^2 = 1, so critical points are x = -1 and x = 1.

f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x = 0 and x = 2. Critical points are (0, 4) and (2, 1).

Setting f'(x) = 0 gives critical points at x = 0 and x = 3.

Setting f'(x) = 0 gives critical points at x = 1 and x = 2.

Setting f'(x) = 0 gives critical points at x = 0, ±2.

f'(x) = 4x^3 - 16x = 4x(x^2 - 4). Critical points are x = 0, ±2.

f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 1)(x - 3) = 0, so critical points are x = 1 and x = 3.

Using the power rule, f'(x) = -1/x^2.

Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).

Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x * (x^2 + 2x).

Equation of circle: (x - h)² + (y - k)² = r² => (x - 2)² + (y + 3)² = 5² = 25.

The slope m = (9 - 0) / (3 - 0) = 3. The equation is y = 3x.

f'(x) = 2x + 2. At x = 1, f'(1) = 4. The point is (1, 4). The tangent line is y - 4 = 4(x - 1) => y = 4x - 4 + 4 => y = 4x - 2.

The equation x^2 - y^2 = c represents a family of hyperbolas with varying values of c.

The equation x^2/a^2 + y^2/b^2 = 1 represents a family of ellipses with varying semi-major and semi-minor axes.

The equation y = ax^2 + bx + c represents a family of parabolas with varying coefficients a, b, and c.

The equation y = ax^3 + bx represents a family of cubic functions where a and b are constants.

The equation y = ax^3 + bx^2 + cx + d represents a family of cubic functions.

The equation y = e^(kx) represents a family of exponential curves with varying growth rates determined by k.

The equation y = k/x represents a family of hyperbolas with varying values of 'k'.

The equation y = kx^2 represents a family of parabolas that open upwards or downwards depending on the sign of k.

The equation x^2 = 4py gives 4p = 12, hence p = 3. The focus is at (0, p) = (0, 3).

The standard form of the parabola is x^2 = 4py. Here, 4p = 8, so p = 2. The focus is at (0, p) = (0, 2).

Boron in BF3 is sp2 hybridized, forming three equivalent sp2 hybrid orbitals.

The central atom in ozone (O3) is sp2 hybridized, forming a resonance structure.

Phosphorus in PCl5 is dsp3 hybridized, allowing it to form five bonds.

f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = ±1. f'(x) > 0 for x > 1, so f(x) is increasing on (1, ∞).

f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). The function is increasing where f'(x) > 0, which is in the intervals (-∞, 0) and (3, ∞).