Gaokao Exam Gaokao Mathematics: Functions and Calculus 8

Question 1

The function f(x) = x/(1+|x|) has range:

Accepted Answer

(-1, 1)

Answer

For x ≥ 0: f = x/(1+x) ∈ [0,1); for x < 0: f = x/(1-x) ∈ (-1,0); overall range (-1,1), open endpoints.

Question 2

Given f(x) = x⁴ - 4x, find the value of x at which f attains its minimum on ℝ.

Accepted Answer

x = 1

Answer

f'(x) = 4x³ - 4 = 4(x³-1) = 0 → x = 1; f''(1) = 12 > 0, confirming minimum.

Question 3

Using the limit definition, which expression correctly represents f'(a)?

Accepted Answer

lim_{h→0} [f(a+h) - f(a)] / h

Answer

The derivative is defined as the limit of the difference quotient as h → 0.

Question 4

If f(x) = x²·sin(1/x) for x ≠ 0 and f(0) = 0, is f differentiable at x = 0?

Accepted Answer

Yes, f'(0) = 0

Answer

f'(0) = lim_{h→0} h²sin(1/h)/h = lim h·sin(1/h) = 0 (squeeze: |h·sin(1/h)| ≤ |h| → 0).

Question 5

The equation of the normal line to y = x² at (1, 1) is:

Accepted Answer

y = -x/2 + 3/2

Answer

Tangent slope at (1,1): y'=2x → 2; normal slope = -1/2; y - 1 = -1/2(x-1) → y = -x/2 + 3/2.

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