AP Calculus AB Derivatives And Limits Quiz

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5. Evaluate lim(x→0) (1 - cos(x))/x².

0

1/2

1

∞

6. Find the derivative of f(x) = (3x² + 1)/(x - 1) using the quotient rule.

(6x(x - 1) - (3x² + 1))/(x - 1)²

(6x)/(x - 1)

(3x² - 6x - 1)/(x - 1)²

(3x² + 6x)/(x - 1)²

7. A function has a vertical asymptote at x = a if lim(x→a) f(x) approaches ±∞.

True

False

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9. If f(x) = x³ - 2x, find f'(2).

10

8

6

4

10. What is the limit of sin(x)/x as x approaches 0?

0

1

∞

Undefined

11. What is the limit of (x² - 4)/(x - 2) as x approaches 2?

0

2

4

Does not exist

12. Find the derivative of f(x) = x · cos(x).

Cos(x) - x·sin(x)

Sin(x)

-sin(x)

X·sin(x)

13. Which statement is true: If f is differentiable at x = c, then f is continuous at x = c.

True

False

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15. If f(x) = ln(x), what is f'(x)?

1/x

1

X

E^x

16. Evaluate lim(x→∞) (5x² + 3x)/(2x² - x + 1).

0

5/2

∞

3

17. What is the derivative of f(x) = e^x · sin(x)?

E^x · cos(x)

E^x(sin(x) + cos(x))

E^x · sin(x)

E^x · tan(x)

18. A function is continuous at x = a if which condition holds?

F(a) is defined and lim(x→a) f(x) = f(a)

F(a) exists

Lim(x→a) f(x) exists

The function is differentiable at x = a

19. Find the derivative of f(x) = (2x + 1)³ using the chain rule.

6(2x + 1)²

3(2x + 1)²

6(2x + 1)³

12(2x + 1)²

20. Which of the following is the derivative of f(x) = 3x⁴ - 2x² + 5?

12x³ - 4x

12x⁴ - 4x²

3x³ - 2x

12x - 4

AP Calculus AB Derivatives and Limits Quiz

1. Using the definition of the derivative, lim(h→0) [f(x+h) - f(x)]/h represents____.

2.

What first name or nickname would you like us to use?

2. If f(x) = a^x where a > 0, then f'(x) equals____.

3. The second derivative f''(x) represents the rate of change of____.

4. If f is continuous and differentiable on [a, b], the Mean Value Theorem guarantees a point c where f'(c) = ____ .

5. Evaluate lim(x→0) (1 - cos(x))/x².

6. Find the derivative of f(x) = (3x² + 1)/(x - 1) using the quotient rule.

7. A function has a vertical asymptote at x = a if lim(x→a) f(x) approaches ±∞.

8. The derivative of f(x) = sec(x) is____.

9. If f(x) = x³ - 2x, find f'(2).

10. What is the limit of sin(x)/x as x approaches 0?

11. What is the limit of (x² - 4)/(x - 2) as x approaches 2?

12. Find the derivative of f(x) = x · cos(x).

13. Which statement is true: If f is differentiable at x = c, then f is continuous at x = c.

14. The derivative of f(x) = 1/x² is____.

15. If f(x) = ln(x), what is f'(x)?

16. Evaluate lim(x→∞) (5x² + 3x)/(2x² - x + 1).

17. What is the derivative of f(x) = e^x · sin(x)?

18. A function is continuous at x = a if which condition holds?

19. Find the derivative of f(x) = (2x + 1)³ using the chain rule.

20. Which of the following is the derivative of f(x) = 3x⁴ - 2x² + 5?