This unit includes the Mean Value Theorem, graphing, finding extreme values, and the Intermediate Value Theorem.
Mean Value Theorem (MVT):
Mean Numbers – Getting ready for the MVT.
Foreshadowing the MVT – An exploration to help students discover the MVT.
Discovering the MVT – A calculator exploration.
This series of 4 posts establishing the Mean Vale Theorem.
Fermat’s Penultimate Theorem – Extreme values occur at a function critical points.
Rolle’s Theorem – a special case of the MVT, used to prove the MVT.
The Mean Value Theorem – I – The analytic approach.
The Mean Value Theorem – II = Geometric demonstration.
Tables – Multiple-choice table questions using the MVT
The Intermediate Value Theorem (IVT)
Intermediate Weather – An unexpected result: weather obeys the IVT!
Darboux’s Theorem – The Intermediate Value Theorem applies to derivatives (Not an AP Calculus topic, but interesting.)
At Just the Right Time – Getting started with derivative applications
Concepts related to Graphs – The five shapes of a graph.
The Shapes of a Graph – Definitions of increasing, decreasing, concave up, and concave down.
Joining the Pieces of a Graph – Extreme values and points of inflection.
Extreme Values – The first and second derivative tests.
Did He, or Didn’t He? – Did Fermat know calculus? Finding the extreme value, the old-fashioned way.
Good Question 10 – The Cone Problem – A max/min problem.
Concavity – Not an easy thing to define; four ways to define concavity.
Teaching Concavity – Helping students with definitions or the lack thereof. Also, on including endpoints, or not.
Reading the Derivative’s Graph – What the graph of the derivative can tell you about the function. My most read post.
Using the Derivative to Graph a Function – Working from the derivative back to the function using a DEsmos app.
Tangents and Slopes – Using Desmos to explore the relationships between tangent lines, slopes, the derivative, and the graph of a function.
Good Question 1: 2008 AB 6 – Tangent line, critical points, POI, limits all in one question.
Asymptotes and the Derivative – How asymptotes may show up on the graph of the derivative.
Other Asymptotes – Horizontal Asymptotes the derivative’s graph.
Real “Real Life” Graph Reading – Derivative graphs in the news.
Open or Closed? – Do functions increase or decrease on open intervals or closed intervals? A common question with the answer and explanation.
Inequalities – Solving inequalities the easy way.
Extremes without Calculus – On the number of extreme values.
Soda Cans – Why are they not in the least-cost shape?
Far Out! – A scary limit problem explored using its extreme value and point of inflection.
The Marble and the Vase – A classic max/min problem.
Curves with Extrema? – Real life happens in the first quadrant. And very few functions describing real life situations have more than one or two extreme values. Here are two.
Optimization – Reflections – The reflection properties of the conic sections.