Consider the error in using the approximation $e^{-x} \approx 1-x$ on the interval $[-1, 1]$.

(a) Reasoning informally, on what interval is this approximation an overestimate?
An underestimate?
(For each, give your answer as an interval or list of intervals, e.g., to specify the intervals $-0.25 \le x < 0.5$ and $0.75 < x \le 1$, enter [-0.25, 0.75), (0.75,1] Enter none if there are no such intervals.)

(b) Use the Error Bound for Taylor Polynomials to find a good smallest bound for the error in approximating $e^{-x}$ with $1-x$ on this interval:
error bound =

Now, consider the error in using the approximation $e^{-x} \approx 1-x+\frac{x^{2}}{2!}$ on the same interval.

(c) Reasoning informally, on what interval is this approximation an overestimate?
An underestimate?
(For each, give your answer as an interval or list of intervals, e.g., to specify the intervals $-0.25 \le x < 0.5$ and $0.75 < x \le 1$, enter [-0.25, 0.75), (0.75,1] Enter none if there are no such intervals.)

(d) Use the Error Bound for Taylor Polynomials to find a good smallest bound for the error in approximating $e^{-x}$ with $1-x+\frac{x^{2}}{2!}$ on this interval:
error bound =