Consider the integral approximation $T_{20}$ of ${\int }_0^{4}3e^{-\frac{x}{4}}dx$.
Does $T_{20}$ overestimate or underestimate the exact value?
A. underestimates
B. overestimates
Find the error bound for $T_{20}$ without calculating $T_N$ using the result that
$$\mathrm{Error}\left(T_N\right)\le \frac{M(b-a{)}^3}{12N^2}$$
where $M$ is the least upper bound for all absolute values of the second derivatives of the function $3e^{-\frac{x}{4}}$ on the interval $[a,b]$.
$\mathrm{Error}\left(T_{20}\right)\le$

Step 1 ：The first question asks whether the trapezoidal approximation $T_{20}$ overestimates or underestimates the exact value of the integral. The trapezoidal rule tends to overestimate the integral for concave down functions and underestimate for concave up functions. To determine the concavity of the function $3e^{-\frac{x}{4}}$, we need to find its second derivative and check its sign.

Step 2 ：The second derivative of the function $3e^{-\frac{x}{4}}$ is positive, which means the function is concave up. Therefore, the trapezoidal approximation $T_{20}$ underestimates the exact value of the integral.

Step 3 ：The second question asks for the error bound of $T_{20}$. To find this, we need to calculate the maximum value of the absolute value of the second derivative of the function on the interval $[0,4]$. This maximum value will be our $M$. We can then substitute $M$, $b-a=4-0=4$, and $N=20$ into the given error bound formula.

Step 4 ：The error bound for $T_{20}$ is $\frac{1}{400}$.

Step 5 ：Final Answer: The trapezoidal approximation $T_{20}$ $\boxed{{\displaystyle \text{underestimates}}}$ the exact value of the integral. The error bound for $T_{20}$ is $\boxed{{\displaystyle \frac{1}{400}}}$.