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Questions and Answers
What is the derivative of the function f(x) = x^2?
What is the derivative of the function f(x) = x^2?
- x^2
- x
- 2x (correct)
- 2
Which conditions must be satisfied to apply Rolle's theorem on the interval [-1, 1] for f(x) = x^2?
Which conditions must be satisfied to apply Rolle's theorem on the interval [-1, 1] for f(x) = x^2?
- f(x) must be decreasing on [-1, 1]
- f(x) must be concave up on [-1, 1]
- f(-1) must equal f(1) (correct)
- f(x) must have a maximum at the endpoints
What is the value of f(-1) and f(1) for the function f(x) = x^2?
What is the value of f(-1) and f(1) for the function f(x) = x^2?
- 1 and 0
- 0 and 1
- 0 and 0
- 1 and 1 (correct)
Where does the derivative of f(x) = x^2 equal zero in the interval [-1, 1]?
Where does the derivative of f(x) = x^2 equal zero in the interval [-1, 1]?
Which of the following best describes the nature of the function f(x) = x^2 in the interval [-1, 1]?
Which of the following best describes the nature of the function f(x) = x^2 in the interval [-1, 1]?
Which of the following correctly explains the continuity of f(x) = x^2 on the interval [-1, 1]?
Which of the following correctly explains the continuity of f(x) = x^2 on the interval [-1, 1]?
What does Rolle's theorem guarantee for the function f(x) = x^2 on the interval [-1, 1]?
What does Rolle's theorem guarantee for the function f(x) = x^2 on the interval [-1, 1]?
If Rolle's theorem applies to f(x) = x^2 in the interval [-1, 1], what can you say about the function's behavior in that interval?
If Rolle's theorem applies to f(x) = x^2 in the interval [-1, 1], what can you say about the function's behavior in that interval?
When evaluating f(x) = x^2 at the endpoints of [-1, 1], what value is obtained?
When evaluating f(x) = x^2 at the endpoints of [-1, 1], what value is obtained?
What is the derivative of the function f(x) = x^2 used to apply Rolle's theorem?
What is the derivative of the function f(x) = x^2 used to apply Rolle's theorem?
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Study Notes
Derivative and Evaluation
- The derivative of the function f(x) = x² is f'(x) = 2x.
- Evaluating the function at the endpoints, f(-1) = (-1)² = 1 and f(1) = 1² = 1.
Rolle's Theorem Conditions
- For Rolle's theorem to apply, the function must be continuous on the closed interval [-1, 1] and differentiable on the open interval (-1, 1).
- f(x) = x² is continuous everywhere and differentiable everywhere.
Application of Rolle's Theorem
- The conditions are met because f(-1) = f(1), both equal to 1.
- Rolle's theorem guarantees at least one c in (-1, 1) where f'(c) = 0.
Derivative and Critical Points
- Setting f'(x) = 0 gives 2x = 0, leading to x = 0 as the only critical point in the interval [-1, 1].
Nature of the Function
- The function f(x) = x² is a continuous and differentiable quadratic function, always non-negative, with a minimum at the vertex (0, 0) in the interval.
Theorem Implications
- Since the conditions of Rolle's theorem hold, f(x) = x² shows a smooth transition from 1 to 1 across [-1, 1] with a critical point at x = 0.
- The function is symmetric about the y-axis and opens upwards, indicating the behavior is parabolic and concave up throughout the interval.
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