Daily Prep 3B -- MTH 201-04

--- tags: mth201, dailyprep --- # Daily Prep 3B -- MTH 201-04 ## Overview In Part B of Module 3 we introduce a powerful new tool for examining the rate of change in a function --- the **second derivative**. The second derivative is what you get when you find the derivative of a function, then find the derivative of *the result*. The second derivative gives information about a function's rate of change that is subtle but important --- namely, what we call the **concavity** of the function. Having information about a function, its derivative, and its second derivative gives us a complete "fingerprint" of how the function behaves. ## 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 definition of the *second derivative* of a function. + Determine whether a function is increasing, decreasing, or constant on an interval by examining the sign of its derivative. (Conversely, state the sign of a function's derivative using information about whether the function is increasing, decreasing, or constant.) + Determine whether a function is concave up or concave down on an interval by examining its graph. **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. + Explain the connection between concavity of a function and its first and second derivatives. + If a function is given with units for the input and output, state the units of the second derivative. + Given the graph of a function, make a reasonable sketch of the graph of its second derivative. ## Resources for learning **Reading:** Read through [Section 1.6](https://activecalculus.org/single/sec-1-6-second-d.html) of the *Active Calculus* textbook. Work through the examples and all interactive exercises found at the end of the section. **Video:** Watch the following videos and take notes. - Screencast 1.6.1: Quick review: The second derivative (3:07) https://www.youtube.com/watch?v=UtzyPEk2zxA - Screencast 1.6.3: Determining concavity from a graph (7:43) https://www.youtube.com/watch?v=o1_o4E-LGsA **Other:** These videos made outside GVSU are optional, but good: - Concavity, inflection points, and second derivative (Organic Chemistry Tutor) (12:49) https://www.youtube.com/watch?v=OhqNbQi9QPk - Concavity introduction (Khan Academy) (9:41) https://www.youtube.com/watch?v=LcEqOzNov4E - Concavity, inflection points, and second derivative (PatrickJMT) (10:23) https://www.youtube.com/watch?v=c1N8zyVhWxM - Not a screencast, but [a tutorial website at Khan Academy on concavity and the second derivative](https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6b/a/concavity-review). - [Another tutorial website from Lamar University on concavity](https://tutorial.math.lamar.edu/classes/calci/shapeofgraphptii.aspx). Nice pictures. **Also:** On Desmos, just as you learned earlier that you can enter a function and then type `f'(x)` to graph its derivative, you can type `f''(x)` to graph the second derivative. For example, here's a function with its graph and the graphs of its first and second derivatives: https://www.desmos.com/calculator/9vnh3wt0hf ## Exercises The exercises for this Daily Prep are found on student.desmos.com. ## Submission and grading To submit your work, simply do the work on Desmos, and it will be saved and submitted automatically as you work. 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.