1. Working toward Rigor versus Bare-bones justification in Calculus Todd Ericson

2. Background Info ► Fort Bend Clements HS ► 25 years at CHS after leaving University of Michigan ► 4 years BC Calculus / Multivariable Calculus ► 2014 School Statistics: 2650 Total Students 45 Multivariable Calculus Students 110 BC Calculus students 200 AB Calculus students ► 2013: 28 National Merit Finalists ► BC Calculus AP Scores from 2011 – 2014 5’s : 316 4’s : 44 3’s : 11 2’s : 2 1’s : 0 Coached the 5A Texas State Championship for Men’s Soccer 2014.

3. Common Topics involving Justification ► Topics and Outline of Justifications: ► Continuity at a point ► Differentiability at a point ► IVT and MVT (Applied to data sets) ► Extrema (Both Relative and Absolute) and Critical values / 1 st and 2 nd Der. Tests ► Concavity/Increasing decreasing Graph behavior including Points of Inflection ► Justification of over or under estimates (First for Linear Approx, then Riemann Sums) ► Behavior of particle motion (At rest, motion: up,down, left, right) ► Error of an alternating Series ► Lagrange Error for a Series ► Convergence of a series ► Justification of L’Hopital’s Rule Both AB and BC topics are listed below.

4. References for problems ► Justification WS is 3 page document handed out as you entered. ► All documents will be uploaded to my wikispaces account. Feel free to use or edit as necessary. ► http://rangercalculus.wikispaces.com/ http://rangercalculus.wikispaces.com/ ► As we work through problems, I will address certain points and thoughts given in document 2. ► Email for questions: todd.ericson@fortbendisd.com See attached handout for justification outlines

5. Sample Problem 1

6. Continuity Problem 1 1) Given this piecewise function, justify that the function is continuous at x = 2

7. Continuity Problem 1 Initial Solution (How can we create a more rigorous solution? ► 1)

8. Continuity Problem 1 Solution ► 1)

9. Sample Problem 2

10. Differentiability Problem 2 ► 2) Given this piecewise function, justify that the function is not differentiable at x = 2

11. Differentiability Problem 2 Solution(How can we create a more rigorous solution)? ► 2) ► Or ► The function is not continuous at x = 2 therefore it cannot be differentiable at x = 2.

12. Differentiability Problem 2 Solution ► 2) ► Or

13. Sample Problem 3

14. Extrema Problem 3 ► 3) Find the absolute maximum and minimum value of the function in the interval from

15. Extrema Problem 3 Solution(How can we create a more rigorous solution)? ► 3) xy 01 e 1

16. Extrema Problem 3 Solution ► 3) xy 01 e 1

17. Sample Problem 4

18. IVT/MVT - Overestimate Problem 4 4) Given this set of data is taken from a function v(t) and assuming it is continuous over the interval [0,10] and is twice differentiable over the interval (0,10) T=0 hoursT=1 hourT=2 hoursT=4 hoursT=6 hoursT=10 hours Vel=50mphVel=60mphVel=30mphVel=38mphVel=50mphVel=70mph a)Find where the acceleration must be equal to 4 mile per hour 2 and justify. b)Find the minimum number of times the velocity was equal to 35mph and justify. c)Approximate the total distance travelled over the 6 hour time frame starting at t = 4 using a trapezoidal Riemann sum with 2 subintervals. d) Assuming that the acceleration from 4 to 10 hours is strictly increasing. State whether the approximation is an over or under estimate and why.

19. IVT/MVT - Overestimate Problem 4 Solution ► a) Given that the function v(t) is continuous over the interval [0,10] and differentiable over the interval (0,10) and since and there must exist at least one c value between hours 2 and 4 such that by the Mean value theorem. ► b) Given the function v(t) is continuous over the interval [0,10] and since v(1)=60 and v(2) = 30 and since v(2)=30 and v(4) = 38 there must exist at least one value of c between hour 1 and hour 2 and at least one value between hour 2 and hour 4 so that v(c)=35 at least twice by the Intermediate Value theorem. ► c) ► d) Since the function v(t) is concave up and above the x-axis (because the derivative of velocity is increasing). The top side of the trapezoid will lie above the curve and therefore the approximation will be an over estimate.

20. Sample Problem 5

21. Taylor Series Problem 5 ► 5) Given the functions ► a)Find the second degree Taylor Polynomial P 2 (x) centered at zero for ► b) Approximate the value of using a second degree Taylor Polynomial centered at 0. ► c) Find the maximum error of the approximation for if we used 2 terms of the Taylor series to approximate the value.

22. Taylor Series Problem 5 Solution

23. Additional Time - Additional Problem

24. Additional Problem 2014 Problem 3

25. Additional Problem 2014 Problem 3