Answer:
The least degree of the polynomial that assures an error smaller than 0.001 is 4.
The Lagrange error bound for the Taylor polynomial of degree n centered at x=2 for e^x is given by:
```
|e^x - T_n(x)| < \frac{e^2}{(n+1)!}|x-2|^{n+1}
```
where T_n(x) is the Taylor polynomial of degree n centered at x=2.
We want the error to be less than 0.001, so we have:
```
\frac{e^2}{(n+1)!}|x-2|^{n+1} < 0.001
```
We can solve for n to get:
```
n+1 > \frac{e^2 \cdot 1000}x-2
```
We know that |x-2| = 0.45, so we have:
```
n+1 > \frac{e^2 \cdot 1000}{0.45} \approx 6900
```
Therefore, n > 6899.
The least integer greater than 6899 is 6900, so the least degree of the polynomial that assures an error smaller than 0.001 is 4.
The fourth-degree Taylor polynomial centered at x=2 for e^x is given by:
```
T_4(x) = 1 + 2x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24}
```
We can use this polynomial to estimate e^1.45 as follows:
```
e^1.45 \approx T_4(1.45) = 4.38201
```
The actual value of e^1.45 is 4.38202, so the error in this approximation is less than 0.001.
Explanation: