# Unit 5 – Analytical Applications of Differentiation

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.)

Graphing

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.

Extreme Values

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.

An Exploration of an Interesting Function = 2018 AB 6 Covers a number of differential equation topics. For an introduction see Summer Fun

Optimization – Reflections – The reflection properties of the conic sections.