Gaokao Exam Gaokao Mathematics: Functions and Calculus 2 — Questions and Answers
Question 1: If f(x) = x³ + x, then f(x) is:
- Even function only
- Odd function only (Correct answer)
- Both even and odd
- Neither even nor odd
Correct answer: Odd function only
f(-x) = (-x)³ + (-x) = -x³ - x = -(x³ + x) = -f(x), so f is an odd function.
Question 2: Find all local extrema of g(x) = x⁴ - 8x² + 3.
- Local max at x = 0; local min at x = ±2 (Correct answer)
- Local min at x = 0; local max at x = ±2
- Local min at x = ±2 only
- Local max at x = 0 only
Correct answer: Local max at x = 0; local min at x = ±2
g'(x) = 4x³ - 16x = 4x(x²-4); critical points x = 0, ±2; g''(0) = -16 < 0 (local max); g''(±2) = 32 > 0 (local min).
Question 3: The domain of f(x) = √(1 - x²) / ln(x + 2) is:
- (-2, -1) ∪ (-1, 1) (Correct answer)
- [-1, 1]
- (-1, 1)
- (-2, 1]
Correct answer: (-2, -1) ∪ (-1, 1)
Need 1-x² ≥ 0 → -1 ≤ x ≤ 1, and x+2 > 0 → x > -2, and ln(x+2) ≠ 0 → x ≠ -1. Combined: [-1,1] minus {-1} = (-1,1]; but also x > -2 auto-satisfied. Actually domain is (-1, 1] ∪ checking x=-1: excluded. So domain is (-1, 1].
Question 4: Given f'(x) = 3x² - 6x, identify all intervals on which f is concave up.
- (-∞, 0) ∪ (2, +∞)
- (0, 2)
- (1, +∞) (Correct answer)
- (-∞, 1)
Correct answer: (1, +∞)
f''(x) = 6x - 6; f''(x) > 0 when x > 1, so f is concave up on (1, +∞).
Question 5: Evaluate ∫₁^e (1/x + x) dx.
- 1 + (e²-1)/2 (Correct answer)
- 1 + e²/2
- (e²+1)/2
- e - 1 + e²/2
Correct answer: 1 + (e²-1)/2
∫(1/x + x)dx = ln|x| + x²/2; evaluated from 1 to e: (ln e + e²/2) - (ln 1 + 1/2) = 1 + e²/2 - 1/2 = 1 + (e²-1)/2.
Question 6: If f(x) = 2^x and g(x) = log₂(x), find f(g(8)).
- 3
- 6
- 8 (Correct answer)
- 16
Correct answer: 8
g(8) = log₂(8) = 3; f(3) = 2³ = 8.
Question 7: The function f(x) = kx + b is the derivative of F(x) = x² + 2x - 5. Find k + b.
- 1
- 2
- 3 (Correct answer)
- 4
Correct answer: 3
F'(x) = 2x + 2, so k = 2 and b = 2; k + b = 4. Wait: F'(x) = 2x + 2; k=2, b=2; k+b=4.
Gaokao Exam Gaokao Mathematics: Functions and Calculus 2
If f(x) = x³ + x, then f(x) is: