Grade 12
#challengeoftheweek
Unit 2: Introduction to Calculus:

1. Which of the following is the definition of the derivative of a function at a point?
- A) The slope of the tangent line to the graph at that point.
- B) The average rate of change of the function over an interval.
- C) The instantaneous rate of change of the function at that point.
- D) The limit of the function as it approaches that point.
- Answer: A

2. What does the limit of a function at a point represent?
- A) The function's value at that point.
- B) The function's behavior as it approaches that point.
- C) The function's maximum value at that point.
- D) The function's minimum value at that point.
- Answer: B

3. Which of the following is a necessary condition for a function to be continuous at a point?
- A) The function must be differentiable at that point.
- B) The function must have a limit at that point.
- C) The function must be defined at that point.
- D) All of the above.
- Answer: D

4. The derivative of a function at a point gives information about:
- A) The function's concavity at that point.
- B) The function's slope at that point.
- C) The function's maximum value at that point.
- D) The function's minimum value at that point.
- Answer: B

5. Which of the following is the correct definition of the derivative of a function f at a point x = a?
- A) f'(a) = lim (h → 0) [f(a + h) - f(a)] / h
- B) f'(a) = lim (h → 0) [f(a) - f(a - h)] / h
- C) f'(a) = lim (h → 0) [f(a + h) - f(a - h)] / 2h
- D) f'(a) = lim (h → 0) [f(a) - f(a + h)] / h
- Answer: A

6. If a function f is differentiable at a point x = a, then it is also:
- A) Continuous at x = a.
- B) Continuous at x = a and has a limit at x = a.
- C) Continuous at x = a but does not necessarily have a limit at x = a.
- D) Not continuous at x = a.
- Answer: A

7. Which of the following is the power rule for differentiation?
- A) d/dx [x^n] = n * x^(n-1)
- B) d/dx [x^n] = n * x^(n+1)
- C) d/dx [x^n] = n * x^(n+1)
- D) d/dx [x^n] = n * x^(n-2)
- Answer: A

8. The chain rule is used to differentiate:
- A) Products of functions.
- B) Quotients of functions.
- C) Compositions of functions.
- D) Sums of functions.
-Answer: C

9. Which of the following is the product rule for differentiation?
- A) d/dx [u * v] = u' * v + u * v'
- B) d/dx [u * v] = u' * v'
- C) d/dx [u * v] = u' * v - u * v'
- D) d/dx [u * v] = u * v'
- Answer: A

10. Which of the following is the quotient rule for differentiation?
- A) d/dx [u / v] = (v * u' - u * v') / v^2
- B) d/dx [u / v] = (v * u' + u * v') / v^2
- C) d/dx [u / v] = (v * u' - u * v') / v
- D) d/dx [u / v] = (v * u' + u * v') / v
- Answer: A

11. The second derivative of a function provides information about:
- A) The function's concavity.
- B) The function's slope.
- C) The function's maximum value.
- D) The function's minimum value.
- Answer: A

12. Which of the following is the correct definition of the second derivative of a function f at a point x = a?
- A) f''(a) = lim (h → 0) [f(a + h) - 2f(a) + f(a - h)] / h^2
- B) f''(a) = lim (h → 0) [f(a + h) - f(a)] / h
- C) f''(a) = lim (h → 0) [f(a + h) - 2f(a) + f(a - h)] / h
- D) f''(a) = lim (h → 0) [f(a + h) - f(a - h)] / h
- Answer: A


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