Working toward rigor versus bare bones justification in calculus
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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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Background info
Background Info Calculus

  • 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.


Common topics involving justification
Common Topics involving Justification Calculus

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


References for problems
References for problems Calculus

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: [email protected]



Continuity
Continuity Calculus

Problem 1

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


Continuity1
Continuity Calculus

Problem 1 Solution

  • 1)



Differentiability
Differentiability Calculus

Problem 2

  • 2) Given this piecewise function, justify that the function is not differentiable at x = 2


Differentiability1
Differentiability Calculus

Problem 2 Solution

  • 2)

  • Or

  • f(x) is not continuous at x = 2 since , therefore f(x) cannot be differentiable at x = 2.



  • Extrema
    Extrema Calculus

    Problem 3

    • 3) Find the absolute maximum and minimum value of

      the function in the interval from


    Extrema1
    Extrema Calculus

    Problem 3 Solution

    • 3)



    Ivt mvt overestimate
    IVT/MVT - Overestimate Calculus

    Problem 4

    4) Given the set of data and assuming it is continousover 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.


    Ivt mvt overestimate1
    IVT/MVT - Overestimate Calculus

    Problem 4 Solution

    • a) Given that the function 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) This must be an overestimate since the function is concave up (because the derivative of velocity is increasing) evaluted under a trapezoidal Riemann sum.



    Taylor series
    Taylor Series Calculus

    Problem 5

    • 5) Given the function

    • a)Find the second degree Taylor Polynomial P2(x) centered at zero for f(x)

    • b) Approximate the value of using a second degree Taylor Polynomial centered at 0.

    • c) Find and justify your solution


    Taylor series1
    Taylor Series Calculus

    Problem 5 Solution



    Additional problem
    Additional Problem Calculus

    2014 Problem 3


    Additional problem1
    Additional Problem Calculus

    2014 Problem 3


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