Gaokao Mathematics: Functions and Calculus Flashcards

If f(x) = x² and the tangent at point P has slope 4, find the coordinates of P.

Answer: (2, 4)

f'(x) = 2x = 4 → x = 2; f(2) = 4; P = (2, 4).

Evaluate ∫₋₁¹ (x³ + x) dx.

Answer: 0

f(x) = x³ + x is an odd function; integral of any odd function over a symmetric interval [-a, a] is 0.

The function f(x) = e^(2x) - 2e^x + 1. The minimum value of f on ℝ is:

Answer: 0

Let u = eˣ > 0; f = u² - 2u + 1 = (u-1)²; minimum = 0 when u = 1, i.e., eˣ = 1 → x = 0.

If f''(a) = 0 and f'''(a) ≠ 0, then x = a is:

Answer: An inflection point

f''(a) = 0 with f'''(a) ≠ 0 means the concavity changes at x = a, making it an inflection point.

Let f(x) = x^(1/3). Find f'(8).

Answer: 1/12

f'(x) = (1/3)x^(-2/3); f'(8) = (1/3)·8^(-2/3) = (1/3)·(1/4) = 1/12.

If ∫₀^a x dx = 8, find a.

Answer: 4

∫₀^a x dx = a²/2 = 8 → a² = 16 → a = 4 (taking positive value).

For f(x) = ln(x) - x, identify the correct statement.

Answer: f has a global maximum at x = 1

f'(x) = 1/x - 1; f'(1) = 0; f''(x) = -1/x²; f''(1) = -1 < 0; so x=1 is a local (and global) maximum.