NSAA NSAA Mathematics - Calculus 2 — Questions and Answers
Question 1: If dy/dx > 0 for all x in an interval, the function is:
- Decreasing
- Constant
- Increasing (Correct answer)
- Concave
Correct answer: Increasing
A positive first derivative indicates the function is increasing over that interval.
Question 2: The chain rule states that if y = f(g(x)), then dy/dx =
- f'(x) · g'(x)
- f'(g(x)) · g'(x) (Correct answer)
- f(g'(x))
- f'(g(x)) + g'(x)
Correct answer: f'(g(x)) · g'(x)
The chain rule: dy/dx = f'(g(x)) · g'(x).
Question 3: Find the area under y = x² between x = 0 and x = 3.
- 3
- 6
- 9 (Correct answer)
- 27
Correct answer: 9
∫₀³ x² dx = [x³/3]₀³ = 27/3 - 0 = 9.
Question 4: What is the derivative of e^(3x)?
- e^(3x)
- 3e^(3x) (Correct answer)
- 3xe^(3x)
- e^(3x)/3
Correct answer: 3e^(3x)
Using chain rule: d/dx(e^(3x)) = 3e^(3x).
Question 5: At a maximum point, the second derivative f''(x) is:
- Positive
- Zero
- Negative (Correct answer)
- Undefined
Correct answer: Negative
At a local maximum, f''(x) < 0, indicating the curve is concave downward.
Question 6: What is the derivative of ln(x)?
- 1/x²
- 1/x (Correct answer)
- e^x
- x·ln(x)
Correct answer: 1/x
The derivative of ln(x) is 1/x.
NSAA NSAA Mathematics - Calculus 2
If dy/dx > 0 for all x in an interval, the function is: