Math | Calculus homework help

MATH 2414 Calculus II
Technology Project

In this project, you will use spreadsheet software (either Microsoft Excel or Google Sheets, please) or Desmos.com to
explore the Sine Integral function

Si(𝑥𝑥) = � sinc(𝑡𝑡)𝑑𝑑𝑡𝑡,
𝑥𝑥

0
where sinc(𝑡𝑡) = �sin(𝑡𝑡) /𝑡𝑡, 𝑡𝑡 ≠ 0

1, 𝑡𝑡 = 0.

By using the Midpoint, Trapezoid, and Simpson’s Rules with 𝑛𝑛 = 20, you will approximate the value of Si(5). You will
also use a Taylor Series representation of Si(𝑥𝑥) to approximate Si(5). Answer the questions below (either print this page
or write or type on your own document). You will need to submit your answers as well as your spreadsheet and/or
Desmos graph link. Make sure that your spreadsheet/Desmos graph is very well-organized so that I can look at it to
see how you performed each calculation. If you need help using the technology, see me in office hours, ask your
classmates, see a tutor, or seek tutorials on the internet.

1. Use the Midpoint Rule to approximate Si(5) with 𝑛𝑛 = 20. Write the approximation here with at least 8 digits after
the decimal.

2. Use the Trapezoid Rule to approximate Si(5) with 𝑛𝑛 = 20. Write the approximation here with at least 8 digits after
the decimal.

3. Use Simpson’s Rule to approximate Si(5) with 𝑛𝑛 = 20. Write the approximation here with at least 8 digits after the
decimal.

4. Starting with the Taylor Series representation of sin (𝑥𝑥) at 𝑎𝑎 = 0, find the Taylor Series representation of Si(𝑥𝑥) at
𝑎𝑎 = 0. (Note: This does not require technology. Just show your work on paper.)

5. Use the first 20 terms of the Taylor Series representation you found in #4 to approximate Si(5). Write the
approximation here with at least 8 digits after the decimal.

6. It can be shown (using more advanced mathematics) that all derivatives of sinc(𝑥𝑥), and therefore all derivatives of
Si(𝑥𝑥), are bounded between −1 and 1. Use this fact to give upper bounds for the errors in each of the four
estimates. Which estimate is likely to be closest to the true value of Si(5)?