Final answer:
To approximate the value of ln(2), we can use the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with n = 10, by dividing the interval from 1 to 2 into 10 parts and applying the respective formulas to the function 1/x.
Step-by-step explanation:
We are asked to calculate the approximate value of ln(2) using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with n = 10. The function we need to integrate is 1/x to find the natural log of 2 because the integral of 1/x dx from 1 to 2 is ln(2).
Using the Trapezoidal Rule with n = 10, we would divide the interval from 1 to 2 into 10 equal parts and use the formula:
Approximation = (h/2) * [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)], where h is the width of each subinterval and f(x_i) is our function evaluated at the ith point.
For the Midpoint Rule, we would use the midpoints of these intervals for our calculations:
Approximation = h * [f(x_1*) + f(x_2*) + ... + f(x_n*)], where x_i* is the midpoint of the ith interval.
Simpson's Rule Would combine the trapezoidal and midpoint approximations to give a better estimate:
Approximation = (h/3) * [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 4f(x_{n-1}) + f(x_n)], with the same definitions for h and f(x_i) as before.
Applying the given rules to the function and with n = 10, the approximations for ln(2) would be calculated by using the respective formulas for each rule, plugging in the values, and calculating the result.