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

2. 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 –
Average Rate of Change = = =

4. Sine Regression
Instantaneous Rate of Change at 2003 =

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

6. Split Regressions
Limit x Limit x

7. Continuous Split Regressions
Limit x
Limit x ∞ ∞ Limit x ∞ DNE

8. 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

9. Derivatives of exponential, logarithmic, and sine regressions
Y’= * ^x *ln( )
Y’= x
Y’= * cos( x )

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

11. Mean Value Theorem c = X f(c) = Y4 f’(c) = Y5
11-2
f’(c) = 9
f’(c) = -.098
c = X
f(c) = Y4
f’(c) = Y5
Y= -.098(x – )
Y= -.098(x – )
Y= -.098(x – )

12. Error and Correlation Regression Correlation Error Linear .0058864321
+/-.0263
Quadratic +/
Cubic +/
Quartic +/
Logarithmic
+/-.0793
Exponential
+/-.0225
Power
+/-.0745
Sine
N/A +/

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

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

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

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

17. Finding Area under the curve using the Fundamental Theorem of Calculus11
Area=∫ sin( x ) dx
F(x)= cos( x ) x
F(11)- F(02)≈ ≈
Area ≈

18. Actual Area under the curve

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