# 4.1: Xtreme Calculus

**At Grade**Created by: CK-12

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

## Max & Min

In this activity, you will explore relative maximums and minimums by drawing a tangent line to a curve and making observations about the slope of the tangent line.

Press **APPS**, select the *Text Editor* application, and open *xtreme1*. Press \begin{align*}F4\end{align*}*xtreme1.89t* and *tanimat2.89p* must be installed on your TI-89.)

When the *tanimate2* program begins, choose **Interactive**. Press **ALPHA** 10 to find the slope 10 times. (You can choose a smaller number next time.) Select the **Tangent Only** option. Arrow to a point where you want to find the slope and press **ENTER**, or type a value and press **ENTER** twice. Observe the slope of the tangent line and determine the critical number(s) of the function.

1. What is (are) the critical number(s) of \begin{align*}y1(x)\end{align*}

2. What occurs at each of the critical numbers of \begin{align*}y1(x)\end{align*}

From the Main Menu arrow to the right, select **QUIT** and press **ENTER**. You will return to the script. Continue pressing \begin{align*}F4\end{align*}

For the graph that has a cusp:

3. What is (are) the critical number(s) of \begin{align*}y1(x)\end{align*}

4. What occurs at each of the critical numbers of \begin{align*}y1(x)\end{align*}

For the cubic function:

5. What is (are) the critical number(s) of \begin{align*}y1(x)\end{align*}

6. What occurs at the critical numbers of \begin{align*}y1(x)\end{align*}

7. Does a relative extreme value occur at every critical number? Predict a way to help determine whether or not there is a relative extreme value at a critical number.

Think about how you can tell if a critical number will be at a relative maximum, a relative minimum, or neither. When using *tanimat2* for the quadratic opening down, move the point of tangency along the curve. Find the slope of the tangent at many locations to observe what happens.

8. When the point of tangency is to the left side of the relative maximum, will the slope of the tangent line be positive, negative, or zero?

9. What about when the point of tangency is to the right of the relative maximum?

Quit *tanimat2* and return to the *xtreme1* script. Repeat the process for the quadratic function opening up.

10. For this function, when the point of tangency is to the left side of the relative minimum, will the slope of the tangent line be positive, negative, or zero?

11. What about when the point of tangency is to the right of the relative minimum?

Fill in the blanks of the following sentences.

12. Let \begin{align*}f\end{align*}

13. Let \begin{align*}f\end{align*} have a critical number at \begin{align*}x = c\end{align*}. If \begin{align*}f'(x) < 0 \end{align*} on an open interval extending left from \begin{align*}c\end{align*}, and \begin{align*}f'(x) > 0\end{align*} on an open interval extending from right of \begin{align*}c\end{align*}, then \begin{align*}f\end{align*} has a _______________ at \begin{align*}x = c\end{align*}.

14. Let \begin{align*}f\end{align*} have a critical number at \begin{align*}x = c\end{align*}. If \begin{align*}f'(x)\end{align*} has the same sign on an open interval extending left from \begin{align*}c\end{align*} and on an open interval extending right from \begin{align*}c\end{align*}, then \begin{align*}f\end{align*} has a _______________ at \begin{align*}x = c\end{align*}.

## Extension

15. How many relative extrema can an *n*th degree polynomial have? Explain.