Intermediate Value Theorem (IVT) limits application examples

📚 Quick Study Guide

• 🌱 The Intermediate Value Theorem (IVT) states that if $f$ is a continuous function 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 $f(c) = k$.
• 📝 Conditions for IVT: The function $f(x)$ must be continuous on the closed interval $[a, b]$.
• 🎯 Application to Limits: IVT can be used to show the existence of a solution to an equation $f(x) = k$ within a certain interval, which can indirectly help evaluate limits.
• 💡 Key Idea: If you can find an interval where the function changes sign (i.e., goes from positive to negative or vice versa), then IVT guarantees there is a root (a point where $f(x) = 0$) within that interval.
• 📈 Example: To show a limit exists using IVT, demonstrate the function is continuous and find an interval where the function crosses the limit value.

Practice Quiz

1. Which of the following is a necessary condition for the Intermediate Value Theorem to apply to a function $f(x)$ on the interval $[a, b]$?
   1. A. $f(x)$ must be differentiable on $[a, b]$.
   2. B. $f(x)$ must be continuous on $[a, b]$.
   3. C. $f(x)$ must be increasing on $[a, b]$.
   4. D. $f(x)$ must be bounded on $[a, b]$.
2. Suppose $f(x)$ is continuous on $[1, 5]$, $f(1) = -2$, and $f(5) = 7$. According to the Intermediate Value Theorem, which of the following values must $f(x)$ take on at some point in the interval $(1, 5)$?
   1. A. -3
   2. B. 8
   3. C. -9
   4. D. 10
3. Let $f(x) = x^2 - 3$. On which of the following intervals does the Intermediate Value Theorem guarantee the existence of a value $c$ such that $f(c) = 0$?
   1. A. $[-2, -1]$
   2. B. $[-1, 0]$
   3. C. $[0, 1]$
   4. D. $[1, 2]$
4. Consider the function $f(x) = \frac{1}{x-2}$. Can the Intermediate Value Theorem be applied on the interval $[1, 3]$ to show that there exists a $c$ such that $f(c) = 0$?
   1. A. Yes, because $f(1) < 0$ and $f(3) > 0$.
   2. B. Yes, because $f(x)$ is defined on $[1, 3]$.
   3. C. No, because $f(x)$ is not continuous on $[1, 3]$.
   4. D. No, because $f(1) > 0$ and $f(3) < 0$.
5. If $g(x)$ is continuous on $[0, 4]$ with $g(0) = 1$ and $g(4) = 9$, what value $k$ could we *not* guarantee $g(c) = k$ for some $c$ in $(0, 4)$, according to the Intermediate Value Theorem?
   1. A. 3
   2. B. 5
   3. C. 1
   4. D. 7
6. Suppose $h(x)$ is a continuous function such that $h(2) = 5$ and $h(6) = -1$. Which of the following conclusions can be drawn using the Intermediate Value Theorem?
   1. A. $h(x)$ has a root in the interval $(2, 6)$.
   2. B. $h(x)$ is always decreasing on the interval $(2, 6)$.
   3. C. $h(x)$ has a maximum value of 5 on the interval $(2, 6)$.
   4. D. $h(x)$ is always positive on the interval $(2, 6)$.
7. Let $f(x) = x^3 + x - 5$. Use the IVT to determine which interval contains a root of $f(x)$.
   1. A. $[-2, -1]$
   2. B. $[-1, 0]$
   3. C. $[0, 1]$
   4. D. $[1, 2]$