Lab 3: Integral approximations - part 2


Lab Preparation:
Find the first four derivatives of \(\displaystyle f(x) = e^{-x^2}\).
Hint: It will be helpful to factor out \(\displaystyle e^{-x^2}\) in each derivative before taking the next. Do you see why?

 

Instructions: We encourage you to collaborate both in and out of class, but you must write up your responses individually. Your work must be neat and include sufficient exposition to make the solution clear to another student who has not seen the assignment (for example, a sequence of equations without explanation will most likely receive zero credit). Pay particular attention to places where explanations using multiple representations are requested, and explicitly discuss the connections between your explanations using different representations. Type or write all of your work legibly on 8\(\frac{1}{2}\)''×11" paper with no spiral fringe, at least one-inch margins on all sides free of writing except your name, date, and assignment number, and staple all pages together.

 

The error for each method of approximating a definite integral \(\displaystyle \int_a^b f(x) dx\) with \(n\) subintervals can be bound using the following formulas:

 

Method

Error Bound

Left-hand or Right-hand Sum

error \(\displaystyle < \frac{(b-a)^2}{2n}\cdot \max|f'(x)|\)

Trapeziod Rule

error \(\displaystyle < \frac{(b-a)^3}{12n^2}\cdot \max|f''(x)|\)

Midpoint Rule

error \(\displaystyle < \frac{(b-a)^3}{24n^2}\cdot \max|f''(x)|\)

Simpson's Rule*

error \(\displaystyle < \frac{(b-a)^5}{2880n^4}\cdot \max|f^{(4)}(x)|\)

*Note: Using standard nomenclature, the error bound for Simpson's method is \(\displaystyle \frac{(b-a)^5}{180n^4}\cdot \max|f^{(4)}(x)| \)

 

 Lab:

  1. Compute the error bounds for your approximations of \(\displaystyle\int_{-1}^2 2^{-x} dx\) in Lab 2 using \(n=4\) with the formulas above. Show all work needed to obtain the error bounds. Compare the sizes of your actual errors to the error bounds to make sure your computations are reasonable.

  2. Find the approximations for with 20 subintervals using

    1. A left-hand sum

    2. A right-hand sum

    3. The midpoint method

    4. The trapezoid method

    5. Simpson’s rule

    Note: You cannot find an antiderivative using elementary functions. Thus you won’t be able to compute the integral exactly or determine exact values for the errors.

    Hint: Use the sum(seq(...)) command on your calculator.

  3. Compute the error bounds for your approximations of \(\displaystyle\int_{-1}^4 e^{-x^2} dx\).

    Hint: You should be able to quickly determine where \(\max|f'(x)|\) occurs. To save you time on the others, \(\max|f''(x)|\) and \(\max|f^{(4)}(x)|\) both occur at \(x = 0\).

  4. For each method, determine \(n\), the minimum number of subintervals required to guarantee accuracy to 5 decimal places (i.e., error \(\leq 5 \times 10^{-6}\)) according to the error bounds.

  5. Use Simpson's rule to determine \(\displaystyle\int_{-1}^4 e^{-x^2} dx\) to 5 decimal places.