Calculus 151 Regression Project

Data collected from the NJ Department of Education Website

NJ Standardized Test ScoresYear % 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

0 2 4 6 8 10 1266

68

70

72

74

76

78

76.8

70.1

75.575.9

73.4

74.8

72.7

74.1

75.2

NJ Standardized Math Scores

NJ Standardized Math Scores

Years

% o

f st

ud

en

ts p

rofi

cien

t in

Math

em

ati

cs

Sine Regression

Instantaneous Rate of Change at 2003 = -5.174

Quartic RegressionR2 =.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’= -.6345494264 x

Y’=-6.784189065* cos(-2.304469566 x + 1.333706904)

Y’= 74.56303051 *.9993957215^x *ln(.9993957215)

Newton’s Methodfinding zeros of the cubic regression

X0 =23.74251964

X0 =23.74251964

Mean Value Theoremf’(c) = 75.682- 76.565 11-2 f’(c) = - .883 9 f’(c) = -.098

c = Xf(c) = Y4

f’(c) = Y5

Y= -.098(x – 3.4931) + 71.661Y= -.098(x – 6.9124) + 75.325Y= -.098(x – 9.67854) +72.782

Error and CorrelationRegression 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

Concave up

Concave down

Inflection Point

Second Derivative Zero

First Derivative Maximum

Approximating area under a curve using left endpoints

74

.64

4

76

.35

2

71

.13

8

76

.51

7

74

.42

71

.89

1

77

.42

6

72

.43

2

73

.68

4

Estimate Area is 668.504

Approximating area under a curve using right endpoints7

6.3

52

71

.13

8

76

.51

7

74

.42

71

.89

1

77.4

26

72.4

32

73.6

84

76.9

76

Estimate Area is 670.836

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 670.69193proficient in Mathematics = 9 ≈

74.55%for each year