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 – ’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 ► 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 / 1 st and 2 nd 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 Both AB and BC topics are listed below.

References for problems ► 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. ► ► As we work through problems, I will address certain points and thoughts given in document 2. ► for questions: See attached handout for justification outlines

Sample Problem 1

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

Continuity Problem 1 Initial Solution (How can we create a more rigorous solution? ► 1)

Continuity Problem 1 Solution ► 1)

Sample Problem 2

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

Differentiability Problem 2 Solution(How can we create a more rigorous solution)? ► 2) ► Or ► The function is not continuous at x = 2 therefore it cannot be differentiable at x = 2.

Differentiability Problem 2 Solution ► 2) ► Or

Sample Problem 3

Extrema Problem 3 ► 3) Find the absolute maximum and minimum value of the function in the interval from

Extrema Problem 3 Solution(How can we create a more rigorous solution)? ► 3) xy 01 e 1

Extrema Problem 3 Solution ► 3) xy 01 e 1

Sample Problem 4

IVT/MVT - Overestimate Problem 4 4) Given this set of data is taken from a function v(t) and assuming it is continuous over the interval [0,10] and is twice differentiable over the interval (0,10) T=0 hoursT=1 hourT=2 hoursT=4 hoursT=6 hoursT=10 hours Vel=50mphVel=60mphVel=30mphVel=38mphVel=50mphVel=70mph a)Find where the acceleration must be equal to 4 mile per hour 2 and justify. b)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 v(t) 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) Since the function v(t) is concave up and above the x-axis (because the derivative of velocity is increasing). The top side of the trapezoid will lie above the curve and therefore the approximation will be an over estimate.

Sample Problem 5

Taylor Series Problem 5 ► 5) Given the functions ► a)Find the second degree Taylor Polynomial P 2 (x) centered at zero for ► b) Approximate the value of using a second degree Taylor Polynomial centered at 0. ► c) Find the maximum error of the approximation for if we used 2 terms of the Taylor series to approximate the value.

Taylor Series Problem 5 Solution

Additional Time - Additional Problem

Additional Problem 2014 Problem 3

Additional Problem 2014 Problem 3