📚 Topic Summary
The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that allows us to determine if a continuous function takes on a specific value within a given interval. Specifically, if a function $f(x)$ is continuous on the closed interval $[a, b]$, then for any value $k$ between $f(a)$ and $f(b)$, there exists at least one value $c$ in the interval $(a, b)$ such that $f(c) = k$. In simpler terms, if you can draw the graph of a function without lifting your pen, the function must take on every value between any two points you pick on the graph.
This theorem is useful for proving the existence of roots (zeros) of a function. If $f(a)$ and $f(b)$ have opposite signs, then there must be at least one value $c$ between $a$ and $b$ where $f(c) = 0$.
🧠 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Continuous Function |
A. A function whose limit exists at a point. |
| 2. Intermediate Value Theorem |
B. A function that can be drawn without lifting your pen. |
| 3. Closed Interval |
C. An interval that includes its endpoints. |
| 4. Root |
D. A theorem guaranteeing a function takes on every value between any two points. |
| 5. Limit |
E. A value $x$ such that $f(x) = 0$. |
✍️ Part B: Fill in the Blanks
The Intermediate Value Theorem states that if a function $f(x)$ is ________ on the closed interval $[a, b]$, then for any value $k$ between $f(a)$ and $f(b)$, there exists at least one value $c$ in the interval $(a, b)$ such that $f(c) = $ ________. This theorem is often used to prove the ________ of roots of a function.
🤔 Part C: Critical Thinking
Explain, in your own words, how the Intermediate Value Theorem can be used to show that a function has a root within a given interval. Provide an example.