given 1/sqrt x dx [1,8] suppose we plan to use simpsons rule with n subintervals to estimate the definite integral. how large do we need to choose n so that the Sn approximation is accurate to within 1.0E-5

To determine how large \( n \) must be for Simpson's Rule to approximate the integral

\[
\int_1^8 \frac{1}{\sqrt{x}} \, dx
\]

with an accuracy of at least \( 1.0 \times 10^{-5} \), we need to use the error bound formula for Simpson’s Rule. The error bound \( E_S \) for Simpson's Rule is given by:

\[
E_S \leq \frac{M(b - a)^5}{180n^4}
\]

where:

- \( a \) and \( b \) are the limits of integration (\( a = 1 \) and \( b = 8 \)),
- \( n \) is the number of subintervals (which must be even),
- \( M \) is the maximum value of the absolute value of the fourth derivative of the integrand on the interval \([a, b]\).

### Step 1: Find the fourth derivative of \( f(x) = \frac{1}{\sqrt{x}} \).

Let’s compute the derivatives:

1. First derivative:
   \[
   f'(x) = -\frac{1}{2} x^{-3/2}
   \]

2. Second derivative:
   \[
   f''(x) = \frac{3}{4} x^{-5/2}
   \]

3. Third derivative:
   \[
   f'''(x) = -\frac{15}{8} x^{-7/2}
   \]

4. Fourth derivative:
   \[
   f^{(4)}(x) = \frac{105}{16} x^{-9/2}
   \]

### Step 2: Estimate \( M \)

To find \( M \), we need to determine the maximum of \( |f^{(4)}(x)| \) over the interval \([1, 8]\). Since the fourth derivative \( f^{(4)}(x) = \frac{105}{16} x^{-9/2} \) is a decreasing function (as \( x^{-9/2} \) decreases as \( x \) increases), its maximum occurs at \( x = 1 \).

At \( x = 1 \):
\[
f^{(4)}(1) = \frac{105}{16}
\]

So, we take \( M = \frac{105}{16} \).

### Step 3: Apply the error formula

We want the error to be less than \( 1.0 \times 10^{-5} \). Therefore, we set up the inequality:

\[
\frac{\frac{105}{16}(8 - 1)^5}{180n^4} \leq 1.0 \times 10^{-5}
\]

Simplifying:

\[
\frac{105 \times 7^5}{16 \times 180n^4} \leq 1.0 \times 10^{-5}
\]

First, calculate \( 7^5 = 16807 \), so:

\[
\frac{105 \times 16807}{16 \times 180n^4} \leq 1.0 \times 10^{-5}
\]

Simplifying the numerator:

\[
\frac{1764735}{2880n^4} \leq 1.0 \times 10^{-5}
\]

Multiply both sides by \( n^4 \):

\[
1764735 \leq 2880 \times 10^{-5} n^4
\]

\[
1764735 \leq 0.0288n^4
\]

Divide both sides by \( 0.0288 \):

\[
n^4 \geq \frac{1764735}{0.0288} \approx 61280729.17
\]

Taking the fourth root:

\[
n \geq \sqrt[4]{61280729.17} \approx 54.56
\]

Thus, \( n \) must be at least 55 (since \( n \) must be even, the smallest possible value for \( n \) is 56).

### Conclusion:
To ensure the error is within \( 1.0 \times 10^{-5} \), you need to choose \( n = 56 \) subintervals.

---

Do you need further details or have any questions?

Here are 5 related questions:
1. What is Simpson's Rule and how does it work?
2. How do you calculate the derivatives of more complex functions?
3. How does Simpson's Rule compare to other numerical integration methods, like the trapezoidal rule?
4. What happens to the error if you choose an odd \( n \)?
5. Why is Simpson's Rule more accurate for smooth functions?

**Tip:** Simpson’s Rule is particularly effective when the function you're integrating is smooth and can be approximated well by quadratic polynomials.

Given 1/sqrt x dx [1,8], suppose we plan to use Simpson's rule with n subintervals to estimate the definite integral. How large do we need to choose n so that the Sn approximation is accurate to within 1.0E-5?

Learn how to determine the number of subintervals (n) needed for Simpson's Rule to estimate the integral of 1/sqrt(x) from 1 to 8 with an accuracy of 1.0E-5. This solution involves error bounds, the fourth derivative of the integrand, and key numerical integration concepts.

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