# 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(π₯π₯) = οΏ½ sinc(π‘π‘)πππ‘π‘,

π₯π₯

0

where sinc(π‘π‘) = οΏ½sin(π‘π‘) /π‘π‘, π‘π‘ β 0

1, π‘π‘ = 0.

By using the Midpoint, Trapezoid, and Simpsonβs Rules with ππ = 20, you will approximate the value of Si(5). You will

also use a Taylor Series representation of Si(π₯π₯) 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 ππ = 20. Write the approximation here with at least 8 digits after

the decimal.

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

the decimal.

3. Use Simpsonβs Rule to approximate Si(5) with ππ = 20. Write the approximation here with at least 8 digits after the

decimal.

4. Starting with the Taylor Series representation of sin (π₯π₯) at ππ = 0, find the Taylor Series representation of Si(π₯π₯) at

ππ = 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(π₯π₯), and therefore all derivatives of

Si(π₯π₯), are bounded between β1 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)?