Gaokao Exam Gaokao Mathematics: Functions and Calculus 5 — Questions and Answers
Question 1: Let h(x) = f(x)·g(x) where f(x) = x² and g(x) = e^x. Find h''(0).
- 0
- 1
- 2 (Correct answer)
- 4
Correct answer: 2
h'(x) = 2x·eˣ + x²·eˣ = eˣ(2x+x²); h''(x) = eˣ(2x+x²) + eˣ(2+2x) = eˣ(x²+4x+2); h''(0) = 1·2 = 2.
Question 2: The function f(x) = x³ - 3x has how many real roots for the equation f(x) = 2?
- 0
- 1 (Correct answer)
- 2
- 3
Correct answer: 1
x³ - 3x - 2 = 0; factor: (x+1)²(x-2) = 0; roots x = -1 (double) and x = 2; distinct real roots: x = -1 and x = 2, giving 2 distinct values.
Question 3: Using the substitution u = x², evaluate ∫₀² x·e^(x²) dx.
- (e⁴-1)/2 (Correct answer)
- e⁴-1
- (e⁴+1)/2
- 2e⁴
Correct answer: (e⁴-1)/2
Let u = x², du = 2x dx; integral becomes (1/2)∫₀⁴ eᵘ du = (1/2)[eᵘ]₀⁴ = (e⁴-1)/2.
Question 4: For f(x) = arctan(x), which of the following is true?
- f is decreasing on ℝ
- f is an odd function with horizontal asymptotes at y = ±π/2 (Correct answer)
- f is an even function
- f has a vertical asymptote at x = 0
Correct answer: f is an odd function with horizontal asymptotes at y = ±π/2
arctan(-x) = -arctan(x) (odd); lim_{x→±∞} arctan(x) = ±π/2 (horizontal asymptotes); f'(x) = 1/(1+x²) > 0 so f is increasing.
Question 5: Let f(x) = {x² + 1, x ≤ 0; e^x, x > 0}. Find f(f(-1)).
- e² (Correct answer)
- e + 1
- e
- 5
Correct answer: e²
f(-1) = (-1)² + 1 = 2 (since -1 ≤ 0); f(2) = e² (since 2 > 0); so f(f(-1)) = e².
Question 6: If f(x) = x³ - 3x, the maximum value of f on [-2, 3] is:
- 2
- 9
- 18 (Correct answer)
- 20
Correct answer: 18
f'(x) = 3x²-3 = 3(x²-1) = 0 at x = ±1; f(-2) = -8+6 = -2; f(-1) = -1+3 = 2; f(1) = 1-3 = -2; f(3) = 27-9 = 18; maximum = 18.
Question 7: The area bounded by y = x² and y = x is:
- 1/3
- 1/6 (Correct answer)
- 1/2
- 1/4
Correct answer: 1/6
Intersection at x=0 and x=1; area = ∫₀¹ (x - x²)dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6.
Gaokao Exam Gaokao Mathematics: Functions and Calculus 5
Let h(x) = f(x)·g(x) where f(x) = x² and g(x) = e^x.
Find h''(0).