Working toward Rigor versus Bare-bones justification in Calculus

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# Working toward Rigor versus Bare-bones justification in Calculus - PowerPoint PPT Presentation

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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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: todd.ericson@fortbendisd.com
Continuity

Problem 1

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

Continuity

Problem 1 Solution

• 1)
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.
Extrema

Problem 3

• 3) Find the absolute maximum and minimum value of

the function in the interval from

Extrema

Problem 3 Solution

• 3)
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