Gaokao Mathematics: Functions and Calculus Flashcards

If f(x) = x³ + x, then f(x) is:

Answer: Odd function only

f(-x) = (-x)³ + (-x) = -x³ - x = -(x³ + x) = -f(x), so f is an odd function.

Find all local extrema of g(x) = x⁴ - 8x² + 3.

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 min).

The domain of f(x) = √(1 - x²) / ln(x + 2) is:

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].

Given f'(x) = 3x² - 6x, identify all intervals on which f is concave up.

Answer: (1, +∞)

f''(x) = 6x - 6; f''(x) > 0 when x > 1, so f is concave up on (1, +∞).

Evaluate ∫₁^e (1/x + x) dx.

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.

If f(x) = 2^x and g(x) = log₂(x), find f(g(8)).

Answer: 8

g(8) = log₂(8) = 3; f(3) = 2³ = 8.

The function f(x) = kx + b is the derivative of F(x) = x² + 2x - 5. Find k + b.

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.