Gaokao Exam Gaokao Mathematics: Functions and Calculus 1

Question 1

Let f(x) = x³ - 3x² + 2. The function f(x) is monotonically decreasing on which interval?

Accepted Answer

(0, 2)

Answer

(-∞, 0)

Answer

(2, +∞)

Answer

(-∞, +∞)

Question 2

If f(x) = ln(x) + (1/2)x², find the equation of the tangent line to f(x) at x = 1.

Accepted Answer

y = 2x - 1

Answer

y = 2x + 1

Answer

y = x + 1

Answer

y = x - 1

Question 3

Given f(x) = e^x - x - 1, which of the following is true?

Accepted Answer

f(x) ≥ 0 for all x, with equality only at x = 0

Answer

f(x) < 0 for all x ≠ 0

Answer

f(x) has two zeros

Answer

f(x) is decreasing on (0, +∞)

Question 4

Evaluate the definite integral ∫₀² (3x² - 2x + 1) dx.

Accepted Answer

6

Answer

8

Answer

10

Answer

4

Question 5

The function f(x) = ax² + bx + c (a ≠ 0) has a local maximum at x = 1 with f(1) = 3. If f(0) = 0, what is the value of a + b?

Accepted Answer

0

Answer

-3

Answer

3

Answer

6

Question 6

Let f(x) = x·ln(x) for x > 0. Which interval contains the only zero of f'(x)?

Accepted Answer

f'(x) has no zeros

Answer

(0, 1/e)

Answer

(1/e, 1)

Answer

(1, e)

Question 7

The area enclosed by f(x) = sin(x) and the x-axis from x = 0 to x = π is:

Accepted Answer

2

Answer

0

Answer

1

Answer

π