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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

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

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

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

Problem 1

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

Differentiability

Problem 2

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

Differentiability

Problem 2 Solution

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

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 - 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

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 Series

Problem 5 Solution

Additional Problem

2014 Problem 3

Additional Problem

2014 Problem 3

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