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Calculus 151 Regression Project

This project examines the NJ Department of Education standardized test scores through various regression analyses, including cubic and quartic regressions, to identify trends in student proficiency in mathematics over the past nine years. It explores the average rate of change, instantaneous rates in 2003, and the implications of limits and derivatives in the context of these regressions. The study further incorporates area approximations under curves using left and right endpoints and utilizes the Fundamental Theorem of Calculus to ascertain the total area representing student proficiency percentages.

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Calculus 151 Regression Project

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  1. Calculus 151 Regression Project Data collected from the NJ Department of Education Website

  2. NJ Standardized Test Scores 76.8 – 75.2 1.6 Average Rate of Change = 02 - 11 = -9 = - .778

  3. Sine Regression Instantaneous Rate of Change at 2003 = -5.174

  4. Quartic RegressionR2 =.555 Instantaneous Rate of Change at 2003 = -1.826

  5. Split Regressions Limit x 6.5- 75.25657 Limit x 6.5+ 71.669602

  6. Continuous Split Regressions Limit x 6.5 73.463086 Limit x - ∞ ∞ Limit x ∞ DNE

  7. Derivative of Split Regressions

  8. Derivatives of exponential, logarithmic, and sine regressions Y’= 74.56303051 *.9993957215^x *ln(.9993957215) Y’= -.6345494264 x Y’=-6.784189065* cos(-2.304469566 x + 1.333706904)

  9. Newton’s Methodfinding zeros of the cubic regression X0=23.74251964 X0=23.74251964

  10. Mean Value Theorem f’(c) = 75.682- 76.565 11-2 f’(c) = - .883 9 f’(c) = -.098 c = X f(c) = Y4 f’(c) = Y5 Y= -.098(x – 3.4931) + 71.661 Y= -.098(x – 6.9124) + 75.325 Y= -.098(x – 9.67854) +72.782

  11. Error and Correlation

  12. Max and Min of Cubic Regression The Regression has a minimum at 5.4093854 and a maximum at 10.033224. It is increasing between [5.4093854, 10.033224] ,and is decreasing between (- ∞ , 5.4093854) U (10.033224, ∞).

  13. Second derivative of cubic regression Second Derivative Zero Inflection Point Concave up Concave down First Derivative Maximum

  14. Approximating area under a curve using left endpoints Estimate Area is 668.504 72.432 73.684 77.426 74.644 76.352 71.138 76.517 74.42 71.891

  15. Approximating area under a curve using right endpoints Estimate Area is 670.836 76.352 71.138 76.517 74.42 71.891 77.426 72.432 73.684 76.976

  16. Finding Area under the curve using the Fundamental Theorem of Calculus 11 Area=∫02 2.943926518sin(-2.304469566x+1.333706904) +74.26459702dx F(x)= 1.277485527cos(-2.304469566x +1.333706904)+74.26459702x F(11)- F(02)≈ 817.47-147.26≈ 670.21 Area ≈ 670.21

  17. Actual Area under the curve

  18. Average Value Area= the sum of the % of students proficient in Mathematics over the past 9 years Average % of students 670.69193 proficient in Mathematics = 9 ≈ 74.55% for each year

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