EmSAT Mathematics Flashcards

What is the derivative of f(x) = x³ + 4x?

Answer: 3x² + 4

Using the power rule: d/dx(x³) = 3x² and d/dx(4x) = 4. Therefore f'(x) = 3x² + 4.

What is ∫2x dx?

Answer: x² + C

The integral of 2x is x² + C. Using the power rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, so ∫2x dx = 2 · x²/2 + C = x² + C.

Find the limit: lim(x→2) (x² − 4)/(x − 2)

Answer: 4

Factor the numerator: x² − 4 = (x+2)(x−2). Cancel (x−2): the limit becomes lim(x→2)(x+2) = 4.

What is the area under f(x) = x² from x = 0 to x = 3?

Answer: 9

∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9.

If g(x) = sin(x), what is g'(x)?

Answer: cos(x)

The derivative of sin(x) is cos(x). This is a standard derivative rule that must be memorized.

At what point does f(x) = x² − 4x + 4 have a minimum?

Answer: x = 2

Set f'(x) = 2x − 4 = 0, giving x = 2. Since f''(x) = 2 > 0, this is a minimum.