We've spent almost three weeks now looking at series - the different types, how to test for convergence, etc. Today we went back to functions, and asked the question "Can we rewrite a function as a power series?" Since the expression 1/(1-r) is the summation formula for an infinite geometric series, then any function that can be somehow written in the form of a1/(1-x) can be expressed as a power series. The task is how does one rewrite the function into that form, so that the first term and the common ratio can be identified. This skill is addressed in tonight's assignment on Pages 603-604: #15-35 odd.
Starting Systems of Equations, today was a review from the 8th grade curriculum of the three methods students have already seen to solve: 1) Graphing. 2) Substitution, and 3) Elimination. We worked through a quick set of problems (Practice Worksheet 2-1) to review these three procedures - answers are posted on the wall outside my room. For homework, students are working on Practice Worksheet 3-1 (Solve graphically) and Practice Worksheet 3-2 (Solve algebraically - substitution or elimination). Answers for these homework worksheets are also posted on the wall outside my room - come by to check tomorrow morning! We'll check this homework before we progress to solving systems of three equations in three unknowns tomorrow.
The first half of Chapter 10 has been spent looking at various tests of convergence. We've seen 7-8 different ways to check a series of numbers to see whether it converges or diverges. Now we're looking at series containing variable terms, starting with power series which come in the form of: Sum (an(x-c)n). Since these series are vaiable, then we can actually solve for x to determine which values of x will make the series converge or diverge - prumarily using the ratio or root test. (Note that since the ratio or root tests are inconclusive if the limit is equal to one, you will need to test the endpoints of your domain separately when checking.) Tonight's assignment is found on Page 603: #3, 5, 7-14 all.
Students finished up the Conics unit today with the calculator part of their test. Next week will start a new chapter with Systems of Equations and Linear Programming. Some of this will be a review (especially solving systems of linear equations), and some will be new as we look at intersections of conic graphs. This will be a quick chapter (about a week-and-a-half), and is a transition into the next major unit on Matrices.
Today was spent summarizing the 6-8 different convergence tests we've studied at over the past week and a half - looking at their requirements for use, how to apply them, what the results mean, etc. Students could use class time to continue working on problems from Sections 10.1-10.5, check their solutions, and clear up any remaining questions as they work on their proficiency with these convergence tests. Next week will be devoted to studying special types of series - power, Taylor, Mclaurin, etc. Enjoy the Prom this weekend - I'll see you at the Fox Saturday evening! Have fun, and stay safe...
Today, students did the non-calculator part of their Conics Test (25 questions) during today's shortened period. Tomorrow is the calculator part of the test - 17 higher thinking questions with "real-world" numbers. These problems may take an additional step or two of work to do, but the biggest difference is that these questions require more "honors level" thinking and problem solving. But there are few questions (17) to do than on Part One (25), students have more time (12 additional minutes) to work on them, and they can now use their graphing calculator and all its nuiltin features. Students should review the many application problems we've done, as well as the many instances in which we were given the graph or informationa about the graph (center, vertices, foci, etc.) and asked to work backwards to come up with the appropriate equation. Study Hard!
Continuing on in Section 10.5, we looked at the Root Test for convergence. Like the Ratio Test from yesterday, there are three possibilities when you examine the limit. If the limit is any number less than one, the series converges absolutely. If the limit is greater than one, then the series diverges. If the limit actually equals one (which happens a fair number of times), then theis test is inconclusive and we have to employ a different test for convergence - Integral Test, p-series, Comparison Test, Limit Comparison Test, Partial Sums, Alternating Series Test, etc. Tonight's problems might use any of these tests to look at convergence. The assignment is on Page 593: #21-51 odd.
