EmSAT EmSAT Mathematics 2 — Questions and Answers
Question 1: What is the derivative of f(x) = x³ + 4x?
- 3x² + 4 (Correct answer)
- 3x + 4
- x² + 4
- 3x² − 4
Correct answer: 3x² + 4
Using the power rule: d/dx(x³) = 3x² and d/dx(4x) = 4. Therefore f'(x) = 3x² + 4.
Question 2: What is ∫2x dx?
- 2
- x² + C (Correct answer)
- 2x² + C
- x + C
Correct 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.
Question 3: Find the limit: lim(x→2) (x² − 4)/(x − 2)
- 0
- 2
- 4 (Correct answer)
- Undefined
Correct answer: 4
Factor the numerator: x² − 4 = (x+2)(x−2). Cancel (x−2): the limit becomes lim(x→2)(x+2) = 4.
Question 4: What is the area under f(x) = x² from x = 0 to x = 3?
- 9 (Correct answer)
- 27
- 3
- 6
Correct answer: 9
∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9.
Question 5: If g(x) = sin(x), what is g'(x)?
- −sin(x)
- cos(x) (Correct answer)
- −cos(x)
- tan(x)
Correct answer: cos(x)
The derivative of sin(x) is cos(x). This is a standard derivative rule that must be memorized.
Question 6: At what point does f(x) = x² − 4x + 4 have a minimum?
- x = 0
- x = 2 (Correct answer)
- x = 4
- x = −2
Correct answer: x = 2
Set f'(x) = 2x − 4 = 0, giving x = 2. Since f''(x) = 2 > 0, this is a minimum.
EmSAT EmSAT Mathematics 2
What is the derivative of f(x) = x³ + 4x?