---
tags: mth201, dailyprep
---

# Daily Prep 7A -- MTH 201-04

## Overview

In Module 7, we start a deep dive into *applications of the derivative*. The first application is one that we've seen before --- finding peaks and troughs on the graph of a function. We can estimate these visually if we can see them; but since estimation is error-prone, we'd like a way to pinpoint the exact location of places where a function reaches a peak or a trough, and find *all* such locations, not just the ones that we see. We'll introduce the concept of the **critical number** of a function to help us, and then the **First** and **Second Derivative Tests** as tools for finding them.

## Learning objectives

**Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught.

+ Identify the global and local minimum and maximum values of a function, given the function's graph.
+ State the definition of _critical number_ and identify all the critical numbers of a function given the function's graph.
+ State the First Derivative Test.

**Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class.

+ Find all the critical numbers of a function algebraically using derivatives.
+ Use the First Derivative Test to determine whether a critical number of a function is a relative minimum, relative maximum, or neither.

## Resources for learning

**Reading:** In the _Active Calculus_ text, read in [Section 3.1](https://activecalculus.org/single/sec-3-1-tests.html) only up to, but _not_ including Example 3.1.10. (Module 7B will start with Example 3.1.10.)

**Video:** Watch the following from the GVSUMath Calculus playlist. This is about 30 minutes of video altogether.

- Screencast 3.1.1: Quick review -- Extreme values of functions (4:11) https://www.youtube.com/watch?v=Bzj_oXB6yIU&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=55
- Screencast 3.1.2: Finding local and global extrema (5:06) https://www.youtube.com/watch?v=tvF3Rq0urvI&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=56
- Screencast 3.1.3: Identifying critical values graphically (4:21) https://www.youtube.com/watch?v=7RcLQZdB0LM&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=57
- Screencast 3.1.4: Finding critical values (3:12) https://www.youtube.com/watch?v=x-efl3Br11E&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=58&t=24s
- Screencast 3.1.5: Using the First Derivative Test (11:09) https://www.youtube.com/watch?v=9rg5jpFVSFY&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=59

## Exercises

The exercises for this Daily Prep are found on student.desmos.com. Look for "Module 7A Daily Prep".

## Submission and grading

To submit your work, simply do the work in Desmos. It's saved as you go.

A **Pass** mark is given if the Daily Prep is turned in before its deadline and if each item on the Daily Prep has a response that represents a good faith effort to be right. **Mistakes are not penalized**. A **No Pass** is given if an item is left blank (even accidentally), has an answer but it shows insufficient effort (including responses like "I don't know"), or if the Daily Prep is late.