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) = -2, which is always negative, indicating concave down everywhere.

f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = -1, 1. f'(x) > 0 for x > 1.

f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). f'(x) > 0 for x in (0, 3).

f'(x) = 4x^3 - 12x^2. Setting f'(x) = 0 gives x = 0, 3. Testing intervals shows local minima at (0, 2).

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, ∞).

The length of the latus rectum for the parabola y^2 = 4px is given by 4p. Here, p = 4, so the length is 16.

Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 5. Thus, lim (x -> 0) (tan(5x)/x) = 5.

Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^2 + x + 1. Evaluating at x = 1 gives 3.

Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Evaluating at x = 1 gives 4.

f'(x) = 2x - 4. Setting f'(x) = 0 gives x = 2. f''(x) = 2 > 0 indicates a local minimum at x = 2.

f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = ±1. f''(1) = 6 > 0 (min), f''(-1) = 6 > 0 (min). Local maxima at (0, 0).

f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f''(x) = 12x^2 - 16. Minima at x = 0.

f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1, 3. f''(1) > 0 (min), f''(3) < 0 (max).

f'(x) = 4x^3 - 12x^2 + 4. Setting f'(x) = 0 gives x = 0 and x = 2. f''(0) = 4 > 0 (min), f''(2) = -8 < 0 (max).

f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. f(2) = -2^2 + 4(2) = 4.

f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x = 0 or x = 2. f(2) = 5 is a local maximum.

Find f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 12 is a local maximum.

f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. Since f''(x) = -2 < 0, it is a maxima.