Gaokao Exam Gaokao Mathematics: Functions and Calculus 8

Question 1

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

Accepted Answer

(-1, 1)

Answer

(-∞, +∞)

Answer

[-1, 1]

Answer

(0, 1)

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

x = 0

Answer

x = -1

Answer

x = 4

Question 3

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

Accepted Answer

lim_{h→0} [f(a+h) - f(a)] / h

Answer

lim_{x→a} f(x)/x

Answer

lim_{h→∞} [f(a+h) - f(a)] / h

Answer

f(a+1) - f(a)

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

Yes, f'(0) = 1

Answer

No, the limit does not exist

Answer

No, f is not continuous at 0

Question 5

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

Accepted Answer

y = -x/2 + 3/2

Answer

y = 2x - 1

Answer

y = x

Answer

y = -2x + 3

Question 6

Evaluate lim_{x→0} (e^x - 1 - x) / x².

Accepted Answer

1/2

Answer

0

Answer

1

Answer

∞

Question 7

The function f(x) = x³ - 6x² + 9x + 2 has local extrema at:

Accepted Answer

x = 1 (max) and x = 3 (min)

Answer

x = 3 (max) and x = 1 (min)

Answer

x = 0 (max) and x = 2 (min)

Answer

x = 1 only