Intermediate Value Theorem Worksheet

Intermediate Value Theorem Worksheet

Read each question carefully and answer thoroughly. Show all your work for full credit. Make sure to clearly state your reasoning, especially when applying the Intermediate Value Theorem.

1. What are the essential conditions for the Intermediate Value Theorem (IVT) to apply to a function f(x) on a closed interval [a, b]?

2. If a function f(x) is continuous on [0, 5] and f(0) = -3 and f(5) = 7, which of the following values MUST f(x) take on the interval (0, 5)?

3. The Intermediate Value Theorem guarantees 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   such that f(c) = k.

4. The Intermediate Value Theorem is often used to prove the existence of   for a continuous function within a given interval.

5. Consider the function f(x) = x³ - 2x² + x - 1. Show that there is a root of f(x) in the interval [1, 2].

6. If f(x) is a function such that f(1) = 5 and f(3) = 1, and f is continuous on [1, 3], then by the IVT, there must be a c in (1, 3) such that f(c) = 3.

7. The Intermediate Value Theorem can be used to find the exact value of c where f(c) = k.

8. Explain in your own words why the continuity condition is crucial for the Intermediate Value Theorem to hold. Provide an example of a discontinuous function where the IVT fails.

9. A climber starts at an elevation of 1000 feet at 8:00 AM and reaches the summit at 10,000 feet at 4:00 PM. On the next day, the climber starts at the summit at 8:00 AM and descends to 1000 feet at 4:00 PM. Use the Intermediate Value Theorem to explain why there must be a time of day when the climber was at the same elevation on both days.