Intermediate Value Theorem Worksheet
Explore the Intermediate Value Theorem with this worksheet for Grade 11 Calculus students, covering its definition, application, and conditions.
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Intermediate Value Theorem
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Read each question carefully and provide clear, concise answers. Show all your work for full credit.
The Intermediate Value Theorem (IVT) states that if a function f is continuous on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.
1. Which of the following is a necessary condition for the Intermediate Value Theorem to apply to a function f on an interval [a, b]?
f is differentiable on (a, b)
f is continuous on [a, b]
f(a) = f(b)
f is increasing on [a, b]
2. Consider the function f(x) = x³ - x on the interval [0, 2]. If we want to find a value c such that f(c) = 6, which condition of the IVT must we check first?
f(0) and f(2)
If f is continuous on [0, 2]
If f is differentiable on (0, 2)
If f(x) is monotonic
3. The Intermediate Value Theorem guarantees the existence of a value c, but it does not provide a method for that value.
4. If a function f is not on a closed interval [a, b], the Intermediate Value Theorem may not apply.
5. Given the function f(x) = x² - 4x + 3 on the interval [0, 5]. Show that there exists a value c in (0, 5) such that f(c) = 0. Explain your reasoning using the Intermediate Value Theorem.
6. Why is the continuity of the function a crucial condition for the Intermediate Value Theorem to hold true?
7. The Intermediate Value Theorem can be used to prove that every polynomial of odd degree has at least one real root.
True
False
8. If f(a) < k < f(b), then there is exactly one value c between a and b such that f(c) = k.
True
False
9. A continuous function f has values as shown in the table below.
| x | 1 | 2 | 3 | 4 |
| f(x)| 5 | -2 | 8 | 1 |
Does the IVT guarantee a root (f(c) = 0) in the interval [1, 4]? If so, in which sub-interval(s) does it guarantee a root?