Daily Prep 2A -- MTH 201-04

--- tags: mth201, dailyprep --- # Daily Prep 2A -- MTH 201-04 ## Overview In this section we make first contact with the most central concept in the course: The notion of **the derivative of a function**. The derivative of a function at a point is essentially the same thing as the instantaneous velocity of an object at a single point in time, but the derivative is more general and encompasses a much wider range of applications than just motion. In the section, we define the derivative of a function at a point in terms of average rates of change and limits; introduce the "prime" notation for the derivative; discuss the units of measurement for the derivative; and connect the derivative to the geometric concept of a tangent line and especially its slope. ## 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. - State the formal definition of $f'(a)$, the derivative of a function $f$ at $x=a$. - Explain the meaning of the various parts of the definition of the derivative. - State the units of a derivative. - Interpret the meaning of a derivative in terms of instantaneous velocity and the slopes of tangent and secant lines. **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. - Compute the derivative of a function at a point by setting up and evaluating the limit in its definition. - Apply the derivative to questions about rates of change and tangent lines. ## Resources for learning **Reading:** Read through [Section 1.3](https://activecalculus.org/single/sec-1-3-derivative-pt.html) of the _Active Calculus_ text. **Video:** Watch the following videos and take notes. - Screencast 1.3.1: Quick review -- The derivative of a function at a point (2:22) https://www.youtube.com/watch?v=0zpQnwVaU28 - Screencast 1.3.2: The derivative of a function at a point (9:12) https://www.youtube.com/watch?v=fQ5yelPpFk0 - Screencast 1.3.3: Derivative of a function at a point using graphs (5:44) https://www.youtube.com/watch?v=0DJPSYeLFpc **Other:** Not required, but Desmos is a useful tool for exploring derivatives. + This interactive demo lets you enter a function, and you can move a point to see the derivative at that point: https://www.desmos.com/calculator/i1fxnah1ci + If you enter in a function as $f(x) =$ and then type the formula, in a later entry you can enter in $f'(a)$ to get the value of the derivative at $a$. Here's an example showing how to find derivative values for a particular function: https://www.desmos.com/calculator/gh62ucmxcp ## Exercises The exercises for this Daily Prep are packaged as a **Desmos activity**. To access it, go to student.desmos.com and log in with your Desmos account. If you have successfully joined the Desmos class roster as part of the *Tools Assignment*, you will see "Daily Prep 2A" available in a list. Click the link to begin, and continue through the end. Your work is saved as you enter it, so there is nothing to "submit" at the end. ## Submission and grading To submit your work, simply do the work in the Desmos activity. It will save 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.