- This is what I wrote during the final exam.
- We skipped 16.6. But we did 16.7. The assigned problems for 16.7 are 1-6, 7, 11, 15, 17, 19-26.
- Please notice that I went through the details of Project 1 in Projects for Chapter 16 on page 876.
- Here is a tentative list of topics for the final exam. I marked three topics that will definitely appear on the final.
- Here is the Mathematica notebook which I used in class today.
- This is what I wrote during the exam.
- The assigned problems for 16.5 are 1-20 (all, but in particular 9, 10, 11, 19, 20), 21-35, 36, 37, 38, 40, 42-45, 47, 49-52, 55-58.
- The assigned problems for 16.4 are 1-7, 9, 12-15, 16-19, 20, 22, 24, 25, 27, 28, 29, 31, 32, 33, 35.
- Some relevant Chapter 16 Review exercises and problems are 14, 19, 20, 48, 49.
- The assigned problems for 16.3 are 1-4, 5, 7, 8, 11, 12, 16, 18, 20-23, 25, 27, 32, 35, 36, 37, 38-55, 57, 58.
- Here is the Mathematica notebook which I used in class today.
- Here is an Excel file in which I calculated an approximate value of an integral that we did in class on Friday.
- The assigned problems for 16.1 are 1, 3, 5, 7, 9-24 (do most), 26, 30, 31.
- The assigned problems for 16.2 are 1-28 (do most), 31-36, 38, 40, 41, 50-54.
- The assigned problems from Chapter 15 Review are 27, 37 and 46. If you feel that you need more practice with finding critical points and classifying them do Exercises 3, 7, 8, 9. And for more practice with Lagrange multipliers do 11, 17, 18, 19, 20.
- I find Problem 46 in Review of Chapter 15 very interesting. Here is my discussion of this problem. It is amazing how these optimization problems often end up with a symmetric solution. In this problem the symmetry was not obvious at first.
- Problem 46 in Review of Chapter 15 reminded me of the following important problem: Prove that among all triangles with a fixed area the equilateral triangle has the smallest perimeter. Here is a solution. I solved this problem using the method of Lagrange multipliers.
- We did 15.3 today. The assigned problems are 1-18 (do most, in particular 4, 8, 14, 15, 17), 19, 20, 22, 24, 25, 26, 28, 33, 38, 40.
- I did Exercise 4 in class. It turns out to be a very interesting exercise. We used the Lagrange multiplier method and found that the maximum and the minimum are among the following six points: (The points are listed clockwise starting from the top.)
\[ \bigl(0,2\bigr), \quad \left( \frac{1}{\sqrt{3}}, \sqrt{3} \right), \quad \bigl(1, 1 \bigr), \quad \bigl( 0, -2 \bigr), \quad \left( - \frac{1}{\sqrt{3}}, - \sqrt{3} \right), \quad \bigl( -1, -1 \bigr). \]
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- This is what I wrote during the exam.
- The assigned problems for Section 15.1 are 1-18, 20, 24 - 29, 34.
- The assigned problems for Section 15.2 are 1-15, 18-25, 27-30.
- The second exam is on Monday. It covers 14.1 through 14.7, 15.1 and 15.2.
- Here is the Mathematica file that I used today. I added a 3D plot with tangent lines. It is called Walking.nb. (Teach a point to walk in a coordinate system.) To start an animation double-click on the first picture.
- This is what I wrote during the exam.
- Here is the Mathematica file that I used today. It is called ParDerPlots.nb.
- Here is another Mathematica file with some instructions how to do plots in Mathematica. The file is called ClassPlots.nb.
- Notes on Chapter 13
- Here is the Excel spreadsheet that I will use in class tomorrow. An xls version is here. This is how I see vectors arising in the Excel setting. In the file that you can download, I have random numbers in the cell range A1:E5. In the cell K4 I put the formula =A1. In this way I have determined a "cell displacement vector": "look 10 cells left and 3 cells up". When I copy the cell K4 to the clipboard, what Excel remembers is this displacement: "look 10 cells left and 3 cells up". So when I paste at the cell L4, then "look 10 cells left and 3 cells up" means B2. Analogously, for all the cells in the range K4:O8. I repeat the same procedure for the cell range C13:G17. Here the displacement vector is "2 cells left and 12 cells up".
- Today in class I used two Excel files. The first one illustrated an example of a function which does not have a limit at (0,0). The second one illustrated an example of a function which does have a limit at (0,0).
- Here is an Excel file with the function studied in Example 1(b) in Section 12.6.
- Several floor related functions in a Mathematica file are here. As before, right-click on the underlined word "here"; in the pop-up menu that appears, your browser will offer you to save the file in your directory. Make sure that you save it with the name DiscontFun.nb. After saving the file follow the instructions below to open it and use it. Individual cells with plotting commands can be executed by placing the cursor in the cell and pressing Shift+Enter.
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Here is an animation which shows level surfaces of the function w = x^2 + y^2 - z^2 studied in Example 3 in Section 12.5.
Place the cursor over the image to start the animation.
Five of the above level surfaces.
- Today in class I demonstrated 3-d plots with level curves in Mathematica. Here is the Mathematica file that I used. It is called LevelCurves.nb. As before, right-click on the underlined word "Here"; in the pop-up menu that appears, your browser will offer you to save the file in your directory. Make sure that you save it with the exactly same name. After saving the file follow the instructions below to open it and use it. Individual cells with plotting commands can be executed by placing the cursor in the cell and pressing Shift+Enter.
- Here is an Excel file which you can use to explore linear functions.
- The homework for the weekend is 12.3 and 12.4, assuming that you have already done 12.1 and 12.2.
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Today in class I demonstrated few 3-d plots in Mathematica. Here is the Mathematica file that I used. It is called 3DGraphs.nb. As before, right-click on the underlined word "Here"; in the pop-up menu that appears, your browser will offer you to save the file in your directory. Make sure that you save it with the exactly same name. After saving the file you can open it with Mathematica. You will find Mathematica on all campus computers in
Start -> All Programs -> Math Applications -> Mathematica. Open Mathematica first, then open 3DGraphs.nb from Mathematica. You can execute the entire file by the following manu sequence (in Mathematica):Kernel -> Evaluation -> Evaluate Notebook. There are some more instructions in the file. - If you have problems running files that I posted please let me know. If you spend some time learning how to use these files you will enhance the understanding of math that we are studying and also learn the basics of these software packages.
- Here are some good Java applets for exploration of functions of two variables. At the end is a comprehensive list of web math tools from MIT.
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Here is an animation which shows level curves of the function z = y^3 + x*y.
Place the cursor over the image to start the animation.
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At the same time it is interesting to view the slices parallel to yz-plane and zx-plane. Here are the corresponding animations. The function is z = y^3 + x*y, as above.
Place the cursor over the image to start the animation.
- Here is the Excel file that I used in class today. In the file I give a step-by-step explanation how to create such files. You can explore other functions from Section 12.2 by creating similar Excel files.
- Here is an Excel file with a different function that we also studied in class today.
- To explore the above files right-click on the underlined word "Here"; in the pop-up menu that appears your browser will offer you to save the file in your directory. Make sure that the file is saved with exactly the extension "xlsx". After saving the files you can open them from Excel. To start Excel click on Start, then All Programs, then Microsoft Office and choose Excel.
- The information sheet
- Notes on Chapter 12
- Related Wikipedia links:
- Disk. Pay attention to the notation used here. That is standard mathematical notation that is unfortunately not used in our textbook.
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Sphere. There is too much stuff here. Some of it we will cover later on. Still, you can recognize few equations on this page. Pay attention to the sentence:
In higher mathematics, a careful distinction is made between the surface of a sphere (referred to as a “sphere”), and the inside of a sphere (referred to as a “ball”).
In mathematics words do have precise meanings. I find this aesthetically very pleasing. - Euclidean distance. This page can give you a teste how the distance that we talked about today fits into a broader context.
- The proof of the distance formula in space is based on the Pythagorean theorem. As you can see on the Wikipedia page, there are many different proofs of the Pythagorean theorem. Click here for a visual proof.