This is a day 2 lesson-activity for AP Calculus.  In just 5 days you can introduce your calculus students to the five many concepts they will be learning about.  

What is in the lesson-activity?

This lesson is made up of an 40-50 minute activity and a 2-page homework assignment with 2 questions similar to those in the activity. Answers to all activities are included. 

How have I used it?   

Behavior of Functions was designed to have students think about how the behavior of a function is changing.  This lessons has the students study the behavior of four functions they have seen since Algebra 2.  I encourage students to see if they can make a sketch without a graphing calculator, but then I also encourage they them to use their graphing calculator to build a window that matches the given graph and then make a sketch from their calculator.  (Notice these windows are not standard windows - a skill students must use on the AP test.)  After creating the four graphs, students are asked to draw a tangent line on each graph at x = 1 and use their graphs to approximate the slope of the tangent line.  Students are asked to describe how the behavior of the function at x = 1 on each graph.  Depending on the slope students could say the graph is increasing, decreasing or neither and relate that to the slope of their tangent line. 

This is a one day lesson-activity that is part of a larger package known as a Jet Tour of Calculus.  

The Jet Tour of Calculus is a set of five one-day lessons, where students are introduced to the five main topics of Calculus,  AB Calculus, and BC Calculus.   The titles of the five lessons are  

1. Instantaneous Rate of Change

2. Behavior of a Function

3. Definite Integrals

4. Using Formulas to find Definite Integrals

5. Limits

The five lessons can be purchased separately or in a bundle.  

Each lesson is designed to engage students in using their graphing calculator to building understanding for the five topics graphically, numerically, analytically, and verbally.  The concept of the five topics is emphasized but the notation does not need to be emphasized since you will be returning to each of these topics to study them further. Upon completion of the five lessons students gain a strong understanding of what they will be studying in the year of calculus that is ahead for them.

The product is for sale at https://www.teacherspayteachers.com/Product/Behavior-of-Functions-Day-2-of-a-Jet-Tour-of-Calculus-1406541 

This is a one day lesson that is part of a larger package known as a Jet Tour of Calculus.

The Jet Tour of Calculus is a set of 5 one day lessons, where students are introduced to the five main topics of Calculus,  AB Calculus, and BC Calculus.   The five lessons are:

1. Instantaneous Rate of Change
2. Behavior of a Function
3. Definite Integrals
4. Using Formulas to find Definite Integrals
5. Limits

The five lessons can be purchased separately or in a bundle.

Using Instantaneous Rate of Change:

The lesson can be used several different ways.  You can first separate the students into groups or pairs and have them work through a question about the height of a rocket.  Then bring the class back together for a discussion about what the tables of values are describing about the various average rates of change.  Or you can guide the students through the questions as a class, allowing them to use their graphing calculators to complete the tables.

This first lesson, Instantaneous Rate of Change, introduces the concept that average rates of change can lead to finding the instantaneous rate of change on any function by making the change in x smaller and smaller.  The first real world problem is focused around studying the height of a homemade rocket after it fired from a platform 400 feet above the ground at an initial speed of 300 ft/sec.    Students are asked to make a sketch of a possible graph for the height of the rocket as a function of time.  Then students are presented with a possible equation for the height of the rocket as a function of time, which they are asked to use their graphing calculators to find the graph.  Students are then challenged to think if they can find the instantaneous rate of change of the rocket at exactly 1 second after it was fired from the platform.

Using four different sets of table values, students notice they can approximate the instantaneous rate of change of the rocket at 1 second after the rocket is fired, by making decrease from the change from 0.1 to 0.01 to 0.001 to 0.0001 to 0.000001 to 0.0000001.  Then students are challenged to use similar tables to approximate the instantaneous rate of change at the rate at time when the rocket is at the same height it was at 1 second?  Students notice that the two approximations are opposite of each other because at the first time the rocket is rising and at the second time the rocket is falling.

This lesson is made up of an 40-50 minute activity and a 2-page homework assignment with 2 questions similar to those in the activity. Answers to all activities are included.

Each lesson is designed to engage students in using their graphing calculator to building understanding for the five topics graphically, numerically, analytically, and verbally.  The concepts of the five topics are emphasized but the notation does not need to be emphasized since you will be returning to each of these topics to study them further. Upon completion of the five lessons students gain a strong understanding of what they will be studying in the year of calculus that is ahead for them.

Comment from a buyer:  I liked this lesson,so I returned to buy the set of all 5-lessons.

Another buyer stated that they enjoyed using the five day jet tour because it is a great foundation for their students.