To estimate the integral of $g(x)$ from $0$ to $1.6$ using the trapezoidal rule, I will need a table of values for $g(x)$ at various points. If you provide that table, I can help apply the trapezoidal rule.

Trapezoidal Rule Formula

The trapezoidal rule for approximating an integral is given by:

$\int_a^b g(x) \, dx \approx \frac{h}{2} \left[ g(x_0) + 2 \sum_{i=1}^{n-1} g(x_i) + g(x_n) \right]$

where:

• $[a, b]$ is the interval over which we are integrating,
• $h = \frac{b - a}{n}$ is the step size,
• $g(x_0), g(x_1), \dots, g(x_n)$ are the values of $g(x)$ at points $x_0, x_1, \dots, x_n$.

Absolute Error Estimation

The error bound for the trapezoidal rule is given by:

$\text{Error} = -\frac{(b - a)^3}{12n^2} \cdot \max |g''(x)|$

However, in this case, you have provided information about the fourth derivative $g^{(4)}(x)$ being bounded:

$-5 \leq g^{(4)}(x) \leq 2$

The error in the approximation of the integral for the trapezoidal rule based on the fourth derivative can be estimated as:

$\left| \text{Error} \right| \leq \frac{(b - a)^5}{180 n^4} \cdot \max |g^{(4)}(x)|$

In this case, $\max |g^{(4)}(x)|$ is at most $5$.

Steps to Complete:

1. Provide the table of values for $g(x)$ between $0$ and $1.6$.
2. Apply the trapezoidal rule with the values from the table.
3. Use the given bounds for $g^{(4)}(x)$ to estimate the absolute error.

Feel free to upload the data or provide the values of $g(x)$, and I can compute both the estimate and the error for you.

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Relative Questions:

1. What is the function $g(x)$, or what values are given for $g(x)$?
2. How many intervals do you want to use for the trapezoidal rule?
3. Would you like to compute a midpoint or Simpson's rule estimate as well for comparison?
4. Do you want more details on how the trapezoidal rule handles approximation?
5. Would you like a graphical representation of the trapezoidal rule applied to your data?

Tip: The trapezoidal rule becomes more accurate as the number of subintervals increases. Using more intervals provides a better estimate for the integral.