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Working toward Rigor versus Bare-bones justification in Calculus

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

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Working toward Rigor versus Bare-bones justification in Calculus

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  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 Both AB and BC topics are listed below. • 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 / 1st and 2nd 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

  4. References for problems See attached handout for justification outlines • 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/ • As we work through problems, I will address certain points and thoughts given in document 2. • Email for questions: todd.ericson@fortbendisd.com

  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 morerigorous 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)

  16. Extrema Problem 3 Solution • 3)

  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) • Find where the acceleration must be equal to 4 mile per hour2 and justify. • 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 P2(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

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