Gaokao Exam Gaokao Mathematics: Functions and Calculus 4

Question 1

If f(x) = x² - 3x + 2 and g(x) = f(f(x)), find g'(0).

Accepted Answer

9

Answer

f'(x) = 2x-3; f(0) = 2; g'(0) = f'(f(0))·f'(0) = f'(2)·f'(0) = (4-3)(0-3) = 1·(-3) = -3. Hmm — recalculate: f'(2) = 2(2)-3 = 1; f'(0) = -3; g'(0) = 1×(-3) = -3.

Question 2

The range of f(x) = 2sin(x) + 1 for x ∈ [0, π] is:

Accepted Answer

[1, 3]

Answer

sin(x) ∈ [0, 1] for x ∈ [0, π]; so 2sin(x) ∈ [0, 2]; f(x) = 2sin(x)+1 ∈ [1, 3].

Question 3

For f(x) = ln(x² - 4x + 3), what is the domain of f?

Accepted Answer

(-∞, 1) ∪ (3, +∞)

Answer

Need x²-4x+3 > 0; factor: (x-1)(x-3) > 0; solution: x < 1 or x > 3.

Question 4

Which statement about the function f(x) = x + 1/x (x ≠ 0) is correct?

Accepted Answer

f is odd and has a local minimum at x = 1

Answer

f(-x) = -x - 1/x = -(x+1/x) = -f(x), so f is odd. f'(x) = 1 - 1/x²; f'(1) = 0; f''(1) = 2 > 0, so local min at x=1.

Question 5

Evaluate ∫₀^π/2 cos²(x) dx.

Accepted Answer

π/4

Answer

Use cos²(x) = (1 + cos(2x))/2; ∫₀^{π/2} (1+cos2x)/2 dx = [x/2 + sin(2x)/4]₀^{π/2} = π/4 + 0 - 0 = π/4.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

Gaokao Exam Gaokao Mathematics: Functions and Calculus 4