Month: December 2023 (Page 2 of 2)

Class Info

• Date: Wed Dec 6
• Meeting Info: 12p-1:40p, N700

Annoucements

WebWork:

• “Application – Monotonicity” – due Mon Dec 11
• “Application – Shapes of Polynomials” – due Mon Dec 11

Schedule for the rest of the semester:

• Exam #3 will consist of a take-home component which will be distributed in class on Monday (Dec 11) and will due the following Monday (Dec 18)
• We do not meet on Wed Dec 13 (Tues Dec 12 and Wed Dec 13 are reading days, so no classes will meet)
• There will also be an in-class exercises for Exam #3 which we will take that Monday Dec 18 (the take-home exercises will serve as preparation for the in-class exam); we will also review for the final exam that day
• The final exam is in-class on Wednesday Dec 20
• See the 1st board snapshot below for a day-by-day outline of the schedule

Topics

Here is the schedule, and also the list of topics which we have been discussing, and which will be covered on Exam #3:

We covered “Related Rates–this is an application of the Chain Rule, for “relating” two rates of change. A typical application involves some quantity (such as the volume of same shape) which is function of some other quantity (e.g., a dimension), both of which are changing over time.

We worked through three examples from the Final Exam Review sheet:

This topic is covered in Sec 4.1 of the textbook. You should read the explanation and examples there:

Class Info

• Date: Mon Dec 4
• Meeting Info: 12p-1:40p, N700

Topics

We reviewed how the signs of the 1st and 2nd derivatives give us information about the shape of the graph. This material is covered in Sec 4.3 and Sec 4.5. Please work on the WebWork sets “Application – Monotonicity” and “Application – Shape of Polynomials”, which cover these topics.

See also the previous Class Recap, which includes screenshots from the textbook and some videos on these topics. I also recommended working through #18 from the Final Exam Review sheet:

We outlined some of the exercises in the “Monotonicity” WebWork set, and used the last exercise to illustrate finding the critical point in order to identify the maximum of a function:

We then did two exercises from the Final Exam Review sheet on “applied optimization” (#9 and #10), which also involves finding the maximum or minimum of a given function by finding its critical point(s). The challenge in these problems is setting up the function from the given information:

You can also study the examples at the beginning of Sec 4.7: Applied Optimization.

Class Info

• Date: Wed Nov 29
• Meeting Info: 12p-1:40p, N700

Topics

We covered finding critical points (i.e., the points where f'(x) = 0), maxima/minima, and using the 1st and 2nd derivatives to figure out the shape of the graph. This material is covered in Sec 4.3 and Sec 4.5.

Please work on the WebWork sets “Application – Monotonicity” and “Application – Shape of Polynomials”, which cover these topics.

We revisited Exam #1, where you were asked to find the critical points of a cubic function. We now used f'(x) to figure out the intervals on which f(x) is increasing and decreasing. This allowed us to sketch the graph, and also classify the critical points as a maximum (at x = 1/3) and a minimum (at x = 3):

We then studied the additional information about the shape of the graph that the 2nd derivative of f gives us–this is called the “concavity”:

We then applied these ideas about the 2nd derivative to the cubic function from Exam #1:

Here are the figures from Sec 4.5 of the textbook that summarize these ideas:

For this, recall that f’ increasing means f” > 0. Thus the two figures in the left column above, for concave up, correspond to f” > 0; and the two figures on the right, for concave down, correspond to f” < 0.

These four “basic shapes” are “glued together” in the following example graph:

Here are a few videos which go through these ideas, with examples: