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Final Exams

Final Exams. Final Semester grades have been posted. Check your grades, use the online form to report which two exams you would like dropped. Changes to the Course Requirements.

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Final Exams

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  1. Final Exams • Final Semester grades have been posted. • Check your grades, use the online form to report which two exams you would like dropped.

  2. Changes to the Course Requirements • Warm-ups will count for a total of 5% of your grade. You MUST be ready for the warm-up (with a booted computer) when the period begins. • Homework will count a total of 20% of the course grade.

  3. Approximations of Area Calculus Lecture Notes

  4. Left Rectangular Approx. LRAM

  5. Left Rectangular Approx. LRAM • The LEFT side of the partition is used to determine the height of the rectangle • With 4 partitions, the approximate area is what?

  6. Left Rectangular Approx. LRAM • The LEFT side of the partition is used to determine the height of the rectangle • With 4 partitions, the approximate area is 4.146 sq units • A better approximation can be obtained with more rectangles.

  7. Left Rectangular Approx. LRAM

  8. Left Rectangular Approx. LRAM

  9. Left Rectangular Approx. LRAM

  10. Left Rectangular Approx. LRAM

  11. Summary of Results 4 Rectangles give an area of 4.146 8 Rectangles give an area of 4.765 16 Rectangles give an area of 5.058 32 Rectangles give an area of 5.200 64 Rectangles give an area of 5.268 256 Rectangles give an area of 5.315 1024 Rectangles give an area of 5.328 2048 Rectangles give an area of 5.332

  12. Numerical Approximations • Your graphing utility can compute the approximate area. • TI 83/84 look in MATH menu • TI 86,86 look in CALC menu

  13. Actual Answer • Calculus will eventually lead us to the actual solution:

  14. Error in Approximations • The error of an approximation can be determined using the formula

  15. Example of Error • Using 8 rectangles, the LRAM method yields and area of 4.765

  16. Other Approximations: RRAM

  17. Summary of Results for RRAM 4 Rectangles give an area of 6.146 8 Rectangles give an area of 5.765 16 Rectangles give an area of 5.558 32 Rectangles give an area of 5.450 64 Rectangles give an area of 5.392 256 Rectangles give an area of 5.347 1024 Rectangles give an area of 5.336 2048 Rectangles give an area of 5.336

  18. A better way? • For LRAM with n=8, we get 4.765 • For RRAM with n=8, we get 5.765 • The average for the two methods is 5.27 • Only 1.3% error!

  19. Midpoint Approximation

  20. Midpoint Approximation • The height of the rectangle is determined by the value of the function at the midpoint of the partition • Essentially averages the left and right methods

  21. Midpoint Approximation n= 4, A=5.384 error=.95620% n= 8, A=5.350 error=.31880% n= 16, A=5.340 error=.13120% n= 32, A=5.336 error=.05625% n= 64, A=5.334 error=.01875%

  22. Trapezoid Approximation • Use the left and right side of partition to determine the area of the trapezoid.

  23. Trapezoid Approximation • Using 4 Reimann partitions, approximate the area under the graph from 0 to 4

  24. Trapezoid Approximation • Using 4 Riemann partitions, approximate area under the graph from 0 to 4 • A = 5.146

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