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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 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.

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 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 Calculus

Problem 1

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

Continuity Calculus

Problem 1 Solution

• 1)

Differentiability Calculus

Problem 2

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

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 Calculus

Problem 3

• 3) Find the absolute maximum and minimum value of

the function in the interval from

Extrema Calculus

Problem 3 Solution

• 3)

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 - 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 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 Series Calculus

Problem 5 Solution