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# 4.1: Xtreme Calculus

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Calculus, Chapter 3, Lesson 2.

ID: 11407

Time Required: 15 minutes

## Activity Overview

In this activity, students will explore relative maximums and minimums by drawing tangent lines to a curve and making observations about the slope of the tangent line. This activity uses both the script feature and a program that enables the drawing of tangent lines to be animated.

Topic: Relative Extrema

• Relative Minimum
• Relative Maximum
• Critical Numbers

Teacher Preparation and Notes

• Students will need two files, xtreme1.89t and tanimat2.89p. The xtreme1 script asks the program tanimat2 to run. The program tanimat2 has numerous functions that are not explored in this brief activity.
• Before beginning the activity, review with students the definitions of relative maximum, relative minimum, and critical numbers.

Associated Materials

## Introduction

After transferring the two files xtreme1 and tanimat2, students will run the script by pressing APPS, selecting the Text Editor application, and then opening xtreme1.

Students are instructed to press F4\begin{align*}F4\end{align*} to advance through the script. Reading the text will provide definitions and questions similar to the student handout.

## Graph 1 – Polynomial

The script sets up the window and defines a polynomial for y1(x)\begin{align*}y1(x)\end{align*}. The tanimat2 program uses whatever function is in y1(x)\begin{align*}y1(x)\end{align*} for its initial exploration. It is possible to change the function without exiting the program, but the xtreme1 script will set up the viewing window and next function for students.

Students are instructed to explore 10 slopes. Their goal should be to find the x\begin{align*}x-\end{align*}value for when the slope is zero.

Students are asked to find the critical numbers of y1(x)\begin{align*}y1(x)\end{align*}. Students are asked what occurs at each critical number of y1(x)\begin{align*}y1(x)\end{align*}.

Solutions

1. x=3,1,1\begin{align*}x = -3, -1, 1\end{align*}

2. a relative extreme value

## Graph 2 – Cusp

Students are asked to find the critical numbers of y1(x)\begin{align*}y1(x)\end{align*}. When the tanimat2 is used to explore the slope of the tangent at the cusp, a domain error message occurs.

The script leads students in using CAS to confirm that the derivative at a cusp is undefined.

Students are asked what occurs at each critical number of y1(x)\begin{align*}y1(x)\end{align*}.

Solutions

3. x=0,2\begin{align*}x = 0, 2\end{align*}

4. a relative extreme value

## Graph 3 – Cubic

Students are asked to find the critical numbers of y1(x)\begin{align*}y1(x)\end{align*} and what occurs at each critical number of y1(x)\begin{align*}y1(x)\end{align*}.

Solutions

5. x=1\begin{align*}x = 1\end{align*}

6. a plateau

7. A relative extreme value doesn’t occur at every critical number. A critical point is only an extrema if the function changes from increasing to decreasing or decreasing to increasing, i.e. when the derivative changes sign.

## Graph 4 – Negative Quadratic

Students are asked if relative extrema occur at every critical point.

Students are asked if the slope of the tangent line to the left of a relative maximum is positive, negative, or zero.

Solutions

8. positive

9. negative

Students are asked if the slope of the tangent line to the left of a relative minimum is positive, negative, or zero.

Students are asked if the slope of the tangent line to the right of a relative minimum is positive, negative, or zero.

This is the end of the script. Press \begin{align*}\forall\end{align*}.

Solutions

10. negative

11. positive

## Summing It All Up

Students are asked to summarize their findings about critical numbers and local extrema.

Student Solutions

12. relative maximum

13. relative minimum

14. plateau

## Extension

Students are asked how many extrema an n\begin{align*}n\end{align*}th degree polynomial can have. They are also asked to explain their answer.

Solution

15. n1\begin{align*}n - 1\end{align*}. An n\begin{align*}n\end{align*}th degree polynomial can only change from increasing to decreasing or vice versa n1\begin{align*}n - 1\end{align*} number of times.

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