Calculus 151 Regression Project

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Presentation transcript:

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

NJ Standardized Test Scores Year % of students proficient in Mathematics 2002 76.8 2004 70.1 2005 75.5 2006 75.9 2007 73.4 2008 74.8 2009 72.7 2010 74.1 2011 75.2 76.8 – 75.2 1.6 Average Rate of Change = 02 - 11 = -9 = - .778

Sine Regression Instantaneous Rate of Change at 2003 = -5.174

Quartic Regression R2 =.555 Instantaneous Rate of Change at 2003 = -1.826

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

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

Derivative of Split Regressions dy/dx of data points 2002 7.58 2004 1.634 2005 -.1321 2006 -1.093 2007 12.347 2008 -15.39 2009 8.2643 2010 4.3229 2011 -14.05

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)

Newton’s Method finding zeros of the cubic regression X0 =23.74251964 X0 =23.74251964

Mean Value Theorem c = X f(c) = Y4 f’(c) = Y5 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

Error and Correlation Regression Correlation Error Linear .0058864321 +/-.0263 Quadratic .0872759121 +/-.21465 Cubic .1213999181 +/-.24505 Quartic .5550642502 +/-.51315 Logarithmic .0285639439 +/-.0793 Exponential .0041896058 +/-.0225 Power .0244578151 +/-.0745 Sine N/A +/-.10239

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, ∞).

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

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

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

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

Actual Area under the curve

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