# Analytical Applications of Differentiation – Unit 5

Unit 5 covers the application of derivatives to the analysis of functions and graphs. Reasoning and justification of results are also important themes in this unit. (CED – 2019 p. 92 – 107). These topics account for about 15 – 18% of questions on the AB exam and 8 – 11% of the BC questions.…

# Graphing with Accumulation 1

Accumulation 3: Graphing Ideas in Accumulation – Increasing and decreasing Previously, we discussed how to determine features of the graph of a function from the graph of its derivative. This required knowing (memorizing) and understanding facts about the derivative (such as the derivative is negative) and how they related to the graph of the function…

# Open or Closed?

About this time of year you find someone, hopefully one of your students, asking, “If I’m finding where a function is increasing, is the interval open or closed?” Do you have an answer? This is a good time to teach some things about definitions and theorems. The place to start is to ask what it…

# Far Out!

A monster problem for Halloween. A while ago I suggested you look at  , which using the dominance idea is zero. Of course your students may try graphing or a table. Here’s the graph done by a TI-Nspire CAS. Note the scales. This is not the way to go. Since the function is increasing near the…

# Joining the Pieces of a Graph

In this post we will consider how the shapes discussed in the previous two posts can join together. Continuity and the derivative at the point where two shapes join tell us what’s going on. Graphs can change from one shape to another only at places where: The first derivative changes sign. For this to happen,…

# The Shapes of a Graph

In my last post we discussed the five shapes of a graph. Hopefully, that activity, which is posted under the Resources tab above, helped your students discover that A function is increasing and concave up, on any interval where its first derivative is positive and its second derivative is positive, like y = sin(x) on…

# For Any – For Every – For All

The universal quantifier  –  for any – for every – for all Many theorems and definition in mathematics use the phrases “for any”, “for every” or “for all.” The upside down A is the symbol. The three phrases all mean the same thing! For example, we have the definition “A function is increasing on an…