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
  • 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

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

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

Problem 1

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

continuity1
Continuity

Problem 1 Solution

  • 1)
differentiability
Differentiability

Problem 2

  • 2) Given this piecewise function, justify that the function is not differentiable at x = 2
differentiability1
Differentiability

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

Problem 3

  • 3) Find the absolute maximum and minimum value of

the function in the interval from

extrema1
Extrema

Problem 3 Solution

  • 3)
ivt mvt overestimate
IVT/MVT - Overestimate

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

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

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

Problem 5 Solution

additional problem
Additional Problem

2014 Problem 3

additional problem1
Additional Problem

2014 Problem 3

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