The problem is asking you to find an interval using the Intermediate Value Theorem (IVT) for the equation $e^x = 2 - x$. This requires finding an interval of length 0.01 or less that contains a solution to this equation, and rounding the interval endpoints to the nearest hundredth.

Strategy:

1. Rewrite the equation:
   We want to solve $e^x = 2 - x$. Let’s rewrite this as a function: $f(x) = e^x - (2 - x) = e^x + x - 2.$ The goal is to find values of $x$ where $f(x) = 0$.

2. Use the Intermediate Value Theorem (IVT):
   The IVT states that if a function is continuous on a closed interval $[a, b]$, and if $f(a)$ and $f(b)$ have opposite signs, then there is at least one root in the interval $[a, b]$.

3. Graphical insight or trial values:
   By testing values of $x$, we can narrow down an interval where the function changes sign, indicating a root.

Let's test a few values of $f(x) = e^x + x - 2$:

• $f(0) = e^0 + 0 - 2 = 1 + 0 - 2 = -1$
• $f(1) = e^1 + 1 - 2 = 2.718 + 1 - 2 = 1.718$
• $f(0.5) = e^{0.5} + 0.5 - 2 = 1.6487 + 0.5 - 2 = 0.1487$

Since $f(0) = -1$ and $f(0.5) = 0.1487$, there is a sign change between 0 and 0.5. This indicates that a root exists in the interval $(0, 0.5)$.

4. Narrow down further:
   We now test between smaller values to get an interval with a length less than or equal to 0.01:

• $f(0.45) = e^{0.45} + 0.45 - 2 = 1.568 + 0.45 - 2 = 0.018$
• $f(0.44) = e^{0.44} + 0.44 - 2 = 1.558 + 0.44 - 2 = -0.002$

Here, $f(0.44)$ is negative and $f(0.45)$ is positive, so by the IVT, a root must exist between $x = 0.44$ and $x = 0.45$.

Conclusion:

The interval is $0.44 < x < 0.45$.

Would you like more details or have any questions?
Here are five related questions to expand your understanding:

1. How does the Intermediate Value Theorem ensure the existence of a root in an interval?
2. Why is it important to rewrite the equation as a function when applying the IVT?
3. What methods can you use to solve equations like $e^x = 2 - x$ without graphing?
4. How could you use Newton's method to find the root more precisely?
5. What is the significance of rounding the interval endpoints to the nearest hundredth?

Tip: When using the IVT, always check that the function is continuous over the interval before applying it.