AP Calculus AB: 8.1.2 Three Big Theorems
This flashcard set covers the Intermediate Value Theorem, Rolle’s Theorem, and the Mean Value Theorem. It explains their conditions and implications, illustrating how these theorems guarantee values and slopes within intervals for continuous and differentiable functions.
Three Big Theorems
The intermediate value theorem: If f is continuous on a closed interval [a, b] and Q is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = Q.
Rolle’s theorem: If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there is at least one number c in (a, b) such that f (c) = 0.
The mean value theorem: If f is continuous on [a, b] and differentiable on (a, b), then there is at least one number c in (a, b) such that f(b) - f(a) / b - a = f’(c)
Key Terms
Three Big Theorems
The intermediate value theorem: If f is continuous on a closed interval [a, b] and Q is any number between f(a) and f(b), then there is at ...
note
The intermediate value theorem proves that continuous functions cannot jump around. If a continuous function goes through two different poi...
Given f (1) = 3 and f (5) = 11. If f is continuous on [1, 5], the intermediate value theorem guarantees which of the following statements about f for some c in the interval (1, 5).
f (c) = 9
Given f (1) = −3 and f (5) = 9. If f is continuous on [1, 5] and differentiable on (1, 5), the mean value theorem guarantees which of the following statements about f for some c in the interval (1, 5).
f ′(c) = 3
Which of the following theorems states that a function continuous on a closed interval takes on all intermediate range values?
The intermediate value theorem
Suppose you have a continuous and differentiable function on a closed interval from a to b. Which of the following statements must be true?
There is at least one point c such that the slope of the line tangent to the function at c is equal to the slope of the secant line through the fun...
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| Term | Definition |
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Three Big Theorems |
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Given f (1) = 3 and f (5) = 11. If f is continuous on [1, 5], the intermediate value theorem guarantees which of the following statements about f for some c in the interval (1, 5). | f (c) = 9 |
Given f (1) = −3 and f (5) = 9. If f is continuous on [1, 5] and differentiable on (1, 5), the mean value theorem guarantees which of the following statements about f for some c in the interval (1, 5). | f ′(c) = 3 |
Which of the following theorems states that a function continuous on a closed interval takes on all intermediate range values? | The intermediate value theorem |
Suppose you have a continuous and differentiable function on a closed interval from a to b. Which of the following statements must be true? | There is at least one point c such that the slope of the line tangent to the function at c is equal to the slope of the secant line through the function at a and b |
Given f (1) = 3 and f (5) = 11. If f is continuous on [1, 5] and differentiable on (1, 5), the mean value theorem guarantees which of the following statements about f for some c in the interval (1, 5) | f ′(c) = 2 |
Given f (3) = 7 and f (5) = 7. If f is continuous on [3, 5] and differentiable on (3, 5), Rolle’s theorem guarantees which of the following statements about f for some c in the interval (3, 5). | f ′(c) = 0 |
Given f (1) = −3 and f (5) = 9. If f is continuous on [1, 5], the intermediate value theorem guarantees which of the following statements about f for some c in the interval (1, 5). | f (c) = 0 |
Suppose you are given the parabola y = x^ 2. A point must exist on the graph of the function where the line tangent to the curve is horizontal. This observation is a consequence of which of the following theorems? | Rolle’s theorem |
Given f (−3) = −1 and f (3) = −1. If f is continuous on [−3, 3] and differentiable on (−3, 3), Rolle’s theorem guarantees which of the following statements about f for some c in the interval (−3, 3) | f ′(c) = 0 |
Suppose a function is continuous on the closed interval from a to b and is differentiable on the open interval from a to b everywhere except a single point. That there must be a point c where the slope of the line tangent to the function is equal to the slope of the secant line passing through a and b is a consequence of which theorem? | No theorem justifies this statement. |