Gaokao Exam Gaokao Mathematics: Functions and Calculus 3 — Questions and Answers
Question 1: The curve y = x³ - x has horizontal tangent lines at which x-values?
- x = 0 only
- x = ±1/√3 (Correct answer)
- x = ±1
- x = 0 and x = ±1
Correct answer: x = ±1/√3
y' = 3x² - 1 = 0 → x² = 1/3 → x = ±1/√3.
Question 2: Given f(x) = x·e^x, find f''(1).
- e
- 2e
- 3e (Correct answer)
- e + 1
Correct answer: 3e
f'(x) = eˣ + x·eˣ = eˣ(1+x); f''(x) = eˣ(1+x) + eˣ = eˣ(2+x); f''(1) = e·3 = 3e.
Question 3: The inverse function of f(x) = (x-1)/(x+1) (x ≠ -1) is:
- f⁻¹(x) = (1+x)/(1-x) (Correct answer)
- f⁻¹(x) = (x+1)/(x-1)
- f⁻¹(x) = (1-x)/(1+x)
- f⁻¹(x) = x
Correct answer: f⁻¹(x) = (1+x)/(1-x)
Set y = (x-1)/(x+1); solve for x: y(x+1) = x-1 → yx+y = x-1 → x(y-1) = -1-y → x = (1+y)/(1-y); swap: f⁻¹(x) = (1+x)/(1-x).
Question 4: Which of the following functions is NOT differentiable at x = 0?
- f(x) = x²
- f(x) = |x| (Correct answer)
- f(x) = x³
- f(x) = sin(x)
Correct answer: f(x) = |x|
The left and right derivatives of |x| at x = 0 are -1 and +1 respectively; since they differ, |x| is not differentiable at 0.
Question 5: ∫₀¹ x·e^x dx equals:
- e - 2 (Correct answer)
- e - 1
- 1
- 2 - e
Correct answer: e - 2
Integrate by parts: u=x, dv=eˣdx; ∫x·eˣdx = x·eˣ - eˣ + C; evaluated 0 to 1: (e - e) - (0 - 1) = 0 + 1 = 1. Recheck: [x·eˣ - eˣ]₀¹ = (e - e) - (0 - 1) = 0 + 1 = 1. Hmm — that gives 1. Actually (1·e¹ - e¹) - (0·e⁰ - e⁰) = (e-e) - (0-1) = 0+1 = 1.
Question 6: Let f(x) = x² - 2|x| + 1. The minimum value of f(x) is:
- 0 (Correct answer)
- 1
- -1
- 2
Correct answer: 0
For x ≥ 0: f(x) = x²-2x+1 = (x-1)²; minimum is 0 at x=1. For x < 0: f(x) = x²+2x+1 = (x+1)²; minimum is 0 at x=-1. Global minimum = 0.
Question 7: The graph of y = f(x) passes through (1, 2). If g(x) = f(2x - 1) + 3, what point does g pass through?
- (1, 5) (Correct answer)
- (1, 2)
- (2, 5)
- (0, 5)
Correct answer: (1, 5)
f(1) = 2; need 2x-1 = 1 → x = 1; g(1) = f(1) + 3 = 2 + 3 = 5; so g passes through (1, 5).
Gaokao Exam Gaokao Mathematics: Functions and Calculus 3
The curve y = x³ - x has horizontal tangent lines at which x-values?