Problem 1 (Calculus): Find the derivative of f(x) = 3x⁴ - 5x² + 2x - 7
Solution:
Using power rule: d/dx [xⁿ] = nxⁿ⁻¹
f'(x) = 3·4x³ - 5·2x¹ + 2·1x⁰ - 0
f'(x) = 12x³ - 10x + 2
Problem 2 (Algebra): Solve for x: 2x² + 7x - 15 = 0
Solution:
Using quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
Where a = 2, b = 7, c = -15
x = [-7 ± √(49 - 4·2·-15)] / 4
x = [-7 ± √(49 + 120)] / 4
x = [-7 ± √169] / 4
x = [-7 ± 13] / 4
x₁ = (6)/4 = 1.5 | x₂ = (-20)/4 = -5
Problem 3 (Statistics): Find the mean, median, and mode of: 4, 8, 6, 5, 3, 8, 9, 8
Solution:
Mean: (4+8+6+5+3+8+9+8) / 8 = 51 / 8 = 6.375
Sorted order: 3, 4, 5, 6, 8, 8, 8, 9
Median: (6 + 8) / 2 = 7
Mode: 8 (appears 3 times)
Problem 4 (Differential Equations): Solve dy/dx = 3y, y(0) = 5
Solution:
This is a separable ODE.
∫ (1/y) dy = ∫ 3 dx
ln|y| = 3x + C
y = e^(3x+C) = Ae^(3x), where A = e^C
Using initial condition y(0) = 5: 5 = A·e^0 = A
Therefore, y = 5e^(3x)
