Additional Mathematics Flashcards

What is the derivative of y = 3x⁴ - 2x² + 5?

Answer: 12x³ - 4x

Using the power rule dy/dx = nxⁿ⁻¹: d/dx(3x⁴) = 12x³, d/dx(-2x²) = -4x, d/dx(5) = 0. So dy/dx = 12x³ - 4x.

Solve the simultaneous equations: y = x + 1 and x² + y² = 25

Answer: x = 3, y = 4 and x = -4, y = -3

Substitute y = x + 1 into x² + y² = 25: x² + (x+1)² = 25, 2x² + 2x + 1 = 25, 2x² + 2x - 24 = 0, x² + x - 12 = 0, (x+4)(x-3) = 0. x = 3 → y = 4; x = -4 → y = -3.

What is lg (logarithm base 10) of 1000?

Answer: 3

lg 1000 = lg 10³ = 3. Since 10³ = 1000, the logarithm base 10 of 1000 is 3.

If f(x) = 2x + 3, what is f⁻¹(x), the inverse function?

Answer: (x - 3)/2

To find the inverse: let y = 2x + 3. Swap x and y: x = 2y + 3. Solve for y: y = (x - 3)/2. So f⁻¹(x) = (x - 3)/2.

What is the integral ∫(4x³ + 6x) dx?

Answer: x⁴ + 3x² + C

Using the power rule for integration: ∫4x³ dx = x⁴, ∫6x dx = 3x². So ∫(4x³ + 6x) dx = x⁴ + 3x² + C, where C is the constant of integration.

What is the remainder when f(x) = x³ - 2x² + 3x - 4 is divided by (x - 2)?

Answer: 2

By the Remainder Theorem, the remainder when f(x) is divided by (x - a) is f(a). f(2) = 8 - 8 + 6 - 4 = 2.