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INFORMATION FOR TEST II This will happen on Tuesday Nov 14 during class period. Kiran will proctor it. The inclass part counts as 50% of your grade. The take home part is the other 50% You have 75 minutes to take Turn in the take home part before you start the inclass part. In test II you will find questions assessing the following items:
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| 11.09 | 23 | Discuss question on quiz on integrals. We also discussed problemes related with Simpson's rule, leading towar test preparation. We then continued the discussion on sequences. Now you have all the background to start discussing series.
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| 11.07 | 22 | Introduction to sequences. Definition and geometric interpretation. |
| 11.02 | 21 | Discussed in full detail Euler's method to solve differential equations. |
| 10.31 | 20 | Refresh the concept of what a differential equation and what a solution to a differential equation is. Gave an example of how to solve a differential equation using separation of variables. As application of solution of differential equations, we discussed what when two curves are orthogonal at a point and what orthogonal trajectories are. We showed that lines of the form y=mx are orthogonal to any circle centered at (0,0). In addition we found the orthogonal trajectories to parabolas of the from y = k x^2 . After we found the solution it was verfied graphically the orthogonal trajectories for y = x^2. At the end we discussed again vector fields and gave a preview of Euler's method. We will discuss in full length an example of Euler's method next class. Click here Euler's Method for a nice explanation about directional fields and Euler's method. Read it before next meeting. ABOUT THE TEST. In a way the test is comprehensive (Don't panic!!). I am interested in knowing that you master the main topics we have discussed. By Friday I will produce a list of topics to be assessed in the inclass part of the test. That part of the test will be administered on Nov. 13 (Tuesday). That day I am going to be visiting the NSF and somebody else will proctor the test. That part of the test will count as 50% of your grade. The other 50% will be a take home test. It will be available this Saturday by 8am and will be due on Friday Nov. 10 before noon. You have one full week to work on it. This part of the test can be done in collaboration with other classmate. However, you need to fully understand your work and be able to explain it to receive credit. I will be more specific on the rules when I make the test available. |
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Int this meeting we talked about the error estimating integrals. We discussed the midpoint rule, the trapezoid and Simpson's rule. Provided an example to determine the number of subdivisions to obtain the integral with any degree of accuracy with each of the three methods. | |
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We continued the explanation on improper integrals and showed that integrals of the form 1/x^p, p>1 converge on [1,infinity). We also solve some convergence problems by comparison. I spent some extra time justifying the meaning of convergence and divergence of improper integrals. |
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17 |
I did an example to illustrate how to solve a system of linear equations using matrices. Then we discuss improper integrals and what we mean by an improper integral to converge or diverge. I will explain how to estimate integrals next time. |
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16 |
I did more examples on integration of rational functions. For the last example, it is easier to solve the system of equations using matrices. I will attach the solution to the system using matrices . Next meeting we will discuss estimating integrals and improper integrals. |
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15 |
Continued working on trigonometric substitutions. At the end we introduced integration of rational functions. | |
| 10.10 | 14 | Give the test back. Results not very encouraging, but with hopes that performance can improve. I am thinking on ways to make it up to make sure you guys show what is expected. I appreciate the input about the test and possible causes for the low performance. I will work on it. The solutions to test I are available through the index page. We did discuss some problems where the use of trig substitutions is the technique to use. Most of these problems have the same format. Just practice. The quiz on Thursday has two problems. The second problem has three parts.
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| 10.5 | 13 | More Review for test I. More on integration of trigonometric functions |
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| 10.3 | 12 | We discussed problems related to the test. Then started talking about integrals of trigonometric functions. I did cover some simple cases and review the main trigonometrric identities to be used in this type of integrals. At this point you should be able to deal with most of the integrals involving sine and cosine. See homework |
| 9.28 | 11 | We spent most of the time just reviewing for the test. Enough I think. |
| LAB TEST I | ||
10 |
Continued the section on integration by parts. Examples whith the different instances were this technique is used were presented. In one of the problems we combined integration by parts with integration by substitution. I will review for the test tomorrow, Wednesday Sept. 27, from 11 am-1 pm in CI-345. This is the format for the test. You are allowed to bring a sheet of notes. Do not include examples.
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More examples dealing with applications of integrals. Started integration by parts. | |
8 |
TEST HAS BEEN POSTPONED UNTIL NEXT WEEK. YOU WILL SPEND THE NEXT LAB WORKING ON PROBLEMS RELATED TO THE TEST. I WANT YOU DO TO WELL ON THIS TEST. ONLY PREPARATION WILL TAKE YOU THERE. Today we worked on one example to review how to find the volume using slices or cylindrical shells. I also did one example to calculate the volume of a solid (not a solid of revolution, where the cross sections were squares, and the length of the edges were determined by a parabola. We then move to explain how to calculate the arclenght of a curve on a given interval, and how to calculate the area of the surface of a solid of revolution. Usign this concept we move to set up integrals to calculate the area of the surface of a solid of revolution. To make sense out of these formulas understand how to calculate the length of a piece of a cord, or the area of a slice of the solid of revolution. |
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Continued the discussion about volume of solids. Concentrate more on how to calculate the volume of solids of revolution, revolving around any vertical or horizontal axis (line). We then move to explain the need for cylindrical shells. In the lab I did two examples: one where the only way to find the volume was by using cylindrical shells. The other where the volume could be calculated using slices or cylindrical shells. On Tuesday I will discuss how to set up integrals to find arc lenght and surface area of solids of revolution. After that we will review for Test I. It will cover up to Tuesday's discussion. |
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Started by calculating the area of a region. We did the integration with respect to x and then with respect to y. We then worked out another example of integration by substitution. We then moved to discuss how to calculate the volume of solids where we can find the area of a cross section. We derived the volume of a cylinder, and the volume of a cone. We will discuss in more detail how to calculate the volume of solids of revolution next time (by slices and by cylindrical shells). There will be a quiz during the lab: 1. Set up the integral to calculate the area of a given region. 2. Solve an integral by substitution. |
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We discussed the net change of a function when its derivative is given. In the case of distance and velocity, the integral gives gives you the net change in distance which corresponds to the displacement. On the other hand the integral of the absolute value of the velocity gives the distance traveled on the given time interval. the second item we discussed was integration by substitution, which is simply recognizing that the integrand is the derivative of the composition of two functions. There are a variety of problems in this section. You need to become an expert on this technique. Visit the integrator (go to links page) in case you want to double check your work. In the homework you can see some of these problems too. Finally, we worked on application of integrals to calculate the area of the region enclosed between two regions. The main point here is that in the region you need to identify the function above and the function below. If the functions intercept then you need to do the integrals by pieces, in each case determining the function above and the one below. I will not talk about section 5.6 except for the definition of Ln(x) on page 422. I will entertain questions about sections 5.5-6.1 on Tuesday and then we will start working on volumes. |
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Revisited the FTC version I and II. For version I we dealt with several examples where we had to use chain rule to calculate the derivative of the function. To use version II we revisited the concept of general antiderivative and its meaning. We use them to evaluate integrals. The table of general antiderivatives are found on page 409 We worked on problem #3 section 5.3. As part of the solution we used the FTC to calculate the integral more accurately. Also, we used the first derivative test to determine where the funciton is increasing/decreasing and find any local max/min In our next meeting we will deal with integration by substituion and how to use integrals to calculate area of regions. You can deal with most of the problems up to section 5.4. Try to understand the difference between displacement and total distance traveled by a car. This can be compared to the value of the integral of a function on an interval and the value of the absolute value of the funciton on the same interval. |
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3 |
Jillian showed us how to use the TI-83 to find summation. We formally defined the integral as the limit of Riemann Sums and interpreted the integral as area to deal with some of the properties. We calculated the integral of a linear function as area, the integral of sq(1-x^2) on [0,1], the integral of sin(x) on [0,2Pi], and estimated the integral of x^2 on [0,4] using the idea of area. For the last problem there were some good ideas as to how to estimate that integral. . Finally, we defined the FTC version I and II. To calculate definite integrals we will use version II. |
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2 |
I explained in detail why the sequence 1/n has limit 0 using the N-epsilon definition, and the geometric interpretation. We then move to summation notation and basic properties. We then discussed in detail the area under a curve using subdivisions, and choosing the left hand point of each subdivision to determine the height of the rectangles. The important point here is how to come up with the expression to estimate the area with n rectangles. We came up for the expression to estimate the area using n rectangles, or the function y=x^2 on the interval [1,5] and then found the limit of that expression to determine the area. I used the TI-89 heavily to find the summation. We could also use the calculator to find the limit but you should be able to do it by hand. Now you can finish all section 5.1. My goal for next class is to deal with sections 5.2-5.4. We may use some of the lab time to work on problems related to them. |
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Introduction. Map of the concepts in this course. Introduction to limit of Sequences through Riemann Sums and estimating of the area of a circle. Read section 5.1 and 5.2 from the book |
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