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Questions and Answers
What are the conditions required for Rolli's Theorem to hold true for a function?
What are the conditions required for Rolli's Theorem to hold true for a function?
The function must be continuous on the closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b).
Using Rolli's Theorem, find the value of c for the function f(x) = 1/3 x^3 - 3x^2 + 2x + 2 on [0,1].
Using Rolli's Theorem, find the value of c for the function f(x) = 1/3 x^3 - 3x^2 + 2x + 2 on [0,1].
c = (1 - √3)/3, which is approximately 0.42265.
How does the Mean Value Theorem relate to the concept of the average rate of change of a function?
How does the Mean Value Theorem relate to the concept of the average rate of change of a function?
The Mean Value Theorem indicates that there exists at least one point c where the instantaneous rate of change (derivative) equals the average rate of change over the interval [a,b].
Explain the geometric interpretation of Rolli's Theorem.
Explain the geometric interpretation of Rolli's Theorem.
Demonstrate how to find the value of c in the context of the function f(x) = sin x on the interval [0, π] using Rolli's Theorem.
Demonstrate how to find the value of c in the context of the function f(x) = sin x on the interval [0, π] using Rolli's Theorem.
Flashcards
Rolle's Theorem
Rolle's Theorem
If a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and has the same value at both endpoints (f(a) = f(b)), then there exists a point c within the open interval (a, b) where the derivative is zero (f'(c) = 0).
Mean Value Theorem
Mean Value Theorem
If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c within the open interval (a, b) where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints (f'(c) = (f(b) - f(a)) / (b - a)).
Continuous function
Continuous function
A function that can be drawn without lifting a pen from the paper.
Differentiable function
Differentiable function
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Secant line
Secant line
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Study Notes
Rolle's Theorem
- If a function f is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there exists a number c ∈ (a, b) such that f'(c) = 0.
- Geometrically, this means there's a point c on the graph where the tangent line is parallel to the x-axis.
Example Applications
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Example 1: Find a value of c satisfying the conclusion of Rolle's Theorem for f(x) = x³ - 3x² + 2x + 2 on [0, 1].
- f(0) = 2
- f(1) = 2
- f'(x) = 3x² - 6x + 2
- f'(c) = 0 => 3c² - 6c + 2 = 0
- Solving for c: c = (6 ± √12) / 6 = 1 ± √(1/3) ≈ 0.423, 1.577
- Since c ∈ (0, 1), c ≈ 0.423.
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Example 2: Find a value of c satisfying the conclusion of Rolle's Theorem for f(x) = sin(x) on [0, π].
- f(0) = 0
- f(Ï€) = 0
- f'(x) = cos(x)
- f'(c) = 0 => cos(c) = 0
- This implies c = π/2.
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Example 3: Find a value of c satisfying the conclusion of Rolle's Theorem for f(x) = x³ - x + 1 on [0, 1].
- f(0) = 1
- f(1) = 1
- f'(x) = 3x² - 1
- f'(c) = 0 => 3c² - 1 = 0
- Solving for c, c = ±√(1/3) ≈ ±0.577
- Since c ∈ (0, 1), c ≈ 0.577.
Mean Value Theorem
- If a function f is continuous on [a, b] and differentiable on (a, b), then there exists a number c ∈ (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
- Geometrically, the theorem states that there's a point c on the graph where the tangent line is parallel to the secant line connecting f(a) and f(b).
Example Applications
- Example 1: Find a value of c satisfying the conclusion of the Mean Value Theorem for f(x) = x³ - x² + x - 1 on [0, 2].
- f(0) = -1
- f(2) = 1
- f'(x) = 3x² - 2x + 1
- f'(c) = (1 - (-1)) / (2 - 0) = 1 => 3c² - 2c + 1 = 1 Solving for c: 3c² - 2c = 0, c(3c - 2) = 0 Possible values are c = 0 (not in (0, 2)) and c = 2/3 (in (0, 2)).
- Example 2: Find a value of c satisfying the conclusion of the Mean Value Theorem for f(x) = x³ - 3x² + 1 on [-2, -1]
- f(-2) = -19
- f(-1) = 13 *
- f'(x) = 3x² - 6x
- f'(c) = (13 - (-19)) / ( -1 - (-2)) = 32
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