Gaokao Exam Gaokao Mathematics: Functions and Calculus 2

Question 1

If f(x) = x³ + x, then f(x) is:

Accepted Answer

Odd function only

Answer

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.

Accepted Answer

Local max at x = 0; local min at x = ±2

Answer

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:

Accepted Answer

(-2, -1) ∪ (-1, 1)

Answer

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.

Accepted Answer

(1, +∞)

Answer

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.

Accepted Answer

1 + (e²-1)/2

Answer

∫(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.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

Gaokao Exam Gaokao Mathematics: Functions and Calculus 2