Intermediate Value Theorem Worksheet
Explore the Intermediate Value Theorem with this worksheet, featuring questions on continuity, existence of roots, and application to functions.
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Intermediate Value Theorem
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Read each question carefully and answer to the best of your ability. Show all your work for full credit.
1. Which of the following conditions is NOT required for the Intermediate Value Theorem to apply to a function f(x) on a closed interval [a, b]?
f(x) must be continuous on [a, b]
f(a) ≠ f(b)
k must be a value between f(a) and f(b)
The derivative of f(x) must exist on (a, b)
2. Given a continuous function f(x) on the interval [1, 5] where f(1) = -3 and f(5) = 7, which of the following values MUST f(x) take on within the interval [1, 5]?
-5
0
8
All of the above
3. The Intermediate Value Theorem guarantees that if a function f is 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.
4. If a continuous function f(x) has f(a) and f(b) with opposite signs, then the Intermediate Value Theorem guarantees the existence of a of f(x) between a and b.
5. Explain the Intermediate Value Theorem in your own words.
6. If a function is not continuous on a closed interval, the Intermediate Value Theorem cannot be applied.
True
False
7. The Intermediate Value Theorem can be used to find the exact value of c such that f(c) = k.
True
False
8. Consider the function f(x) = x³ - 4x + 1. Show that there is a root of f(x) in the interval [0, 1] using the Intermediate Value Theorem. Explain your steps clearly.
9. Examine the graph below. Does the Intermediate Value Theorem apply to the function shown on the interval [-2, 2] for k = 0?
Explain your reasoning, referencing the conditions of the theorem.