Gaokao Exam Gaokao Mathematics: Functions and Calculus 3

Question 1

The curve y = x³ - x has horizontal tangent lines at which x-values?

Accepted Answer

x = ±1/√3

Answer

y' = 3x² - 1 = 0 → x² = 1/3 → x = ±1/√3.

Question 2

Given f(x) = x·e^x, find f''(1).

Accepted Answer

3e

Answer

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:

Accepted Answer

f⁻¹(x) = (1+x)/(1-x)

Answer

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?

Accepted Answer

f(x) = |x|

Answer

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:

Accepted Answer

e - 2

Answer

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.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

Gaokao Exam Gaokao Mathematics: Functions and Calculus 3