Part a) Trapezoidal Rule

The trapezoidal rule for approximating a definite integral can be expressed as:

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

where:

• $a = 1$ and $b = 6$,
• $n = 6$ (the number of subintervals),
• $h = \frac{b - a}{n} = \frac{6 - 1}{6} = \frac{5}{6}$.

So the subintervals are:

$x_0 = 1, \, x_1 = 1 + \frac{5}{6}, \, x_2 = 1 + 2\frac{5}{6}, \, \dots, \, x_6 = 6$

Now we can calculate $f(x) = \frac{\cos(x)}{x}$ at these points. Let's evaluate it.

$x_0 = 1, \quad x_1 = 1.8333, \quad x_2 = 2.6667, \quad x_3 = 3.5, \quad x_4 = 4.3333, \quad x_5 = 5.1667, \quad x_6 = 6$

We will use the trapezoidal rule formula to estimate the integral. I'll calculate the sum for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can calculate the values of $f(x) = \frac{\cos(x)}{x}$ at the subintervals $x_0, x_1, \dots, x_6$ and apply the trapezoidal rule formula. Here's a quick breakdown of the steps:

1. Evaluate $f(x) = \frac{\cos(x)}{x}$ at the points $x_0 = 1, x_1 = 1.8333, \dots, x_6 = 6$.
2. Plug these values into the trapezoidal rule formula: $\frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$ where $h = \frac{5}{6}$.

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Part b) Error Bound for Trapezoidal Rule

The error bound $E_T$ for the trapezoidal rule is given by:

$|E_T| \leq \frac{(b - a)^3}{12n^2} \cdot \max |f''(x)|$

To find this error:

1. Compute $f''(x)$ for $f(x) = \frac{\cos(x)}{x}$.
2. Find the maximum value of $|f''(x)|$ on the interval $[1, 6]$.
3. Plug these values into the formula above.

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Would you like further help with the specific steps or details? Here are some follow-up questions that can help:

1. How do we evaluate the function $f(x)$ at specific points?
2. What is the formula for the second derivative $f''(x)$?
3. How is the error bound of the trapezoidal rule derived?
4. How do we calculate the maximum of $f''(x)$ on an interval?
5. Why does the error depend on the second derivative of the function?

Tip: When applying numerical methods, knowing the behavior of the function's derivatives is key to understanding accuracy.