MAT 136 – 02 Calculus I
MATH 137 CALCULUS 2
Spring 2008
Project 1 Integral Approximations and Error Bounds
Due Friday, February 1
This project will be an individual effort. Be thorough and explicit in your work. Most of the work you do for this project will be hand written, but everything must be neat and compiled neatly into a single document.
1. Consider the integral [pic]. Produce a computer generated graph of the integrand in an appropriate window (a useful (and free) program for this is GraphCalc. It appears that their website () isn't working at the moment, but if you Google Graphcalc (one word), there are several places where you can download it). Actually, you'll need four copies of the graph. Then, with n=8, sketch the approximating rectangles generated by the Left Endpoint, Right Endpoint, Midpoint, and Trapezoidal approximation methods. Based on your drawings (that is, without actually calculating anything), list the estimates L8, R8, M8, and T8 in descending order, that is, if it looks like R8 would give the largest number, list it first, then whichever is next largest, all the way through the list. Finally, decide which method would produce the most accurate estimate for this integral.
2. Based on the following data, estimate [pic]by calculating L16, R16, T16, M8.
x |0.03 |0.04 |0.05 |0.06 |0.07 |0.08 |0.09 |0.1 |0.11 |0.12 |0.13 |0.14 |0.15 |0.16 |0.17 |0.18 |0.19 | |f(x) |0.33 |-0.17 |0.43 |0.12 |0.15 |0.17 |0.46 |0.35 |-0.15 |0.02 |0.19 |0.21 |0.43 |0.23 |0.53 |0.51 |0.32 | |
3. Estimate the integral [pic] with n=8 using the Trapezoid, Midpoint, and Simpson's methods. You don't have to graph/draw these estimates, just calculate them. Next, calculate the actual value of the integral (be sure to show the integration) and find ET, EM, and ES.
4. Using the formulas for error bounds for the Midpoint and Trapezoid methods, find the value of n that will guarantee the approximation of [pic]will be within 0.0001 of the actual integral value. (It might be helpful to utilize a graphing program or calculator to find the appropriate K).
[pic]
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.