Mr. Slater's Blog

Evaluating Limits Algebraically

Since the AP Calculus Exam is 65% non-calculator, and my retests are all done without a graphing calculator, there is a real need to be able to evaluate limits algebraically in this course.  Although we've looked at limits graphically (and had to deal with issues of zooming, resizing the window, pixel resolution, etc.) and numericqally (where we had to zoom, or look for a pattern with messy decimal results), algebraically evaluating a limit may be the surest way to know the result.  Unfortunately, when substituting in the value of x, it isnot uncommon to end up with an indeterminant form -- zero/zero, infinity/infinity, infinity - infinity, zero^zero, etc.  In that case, the function expression will need to be manipulated, and you know a lot of possibilities:  factoring and cancelling out common factors, rationalizing a denominator or numerator, getting a common denominator for fractions, using a trig identity, contemplating properties of exponents or logarithms, etc.  If your chosen technique does not seen to work, then try another option.  Tonight's assignment gives you a lot of chances to manipulate indeterminant form limits -- Page 98: #7-51 odd.  (I especially liked problems # 29, 35, and 47.)

Continuity finished

Today's assignment finishes up continuity with the problems on Pages 92-93: #52, 55-58, 59-79 eoo, and 82-87.  In particular, pay attention to cmmonly seen AP questions like #52, 83, 84, and 85.

Day 2 of Continuity

We continued working with continuity today in class, going over homework questions and allowing students to finish up last night's assignment (Pages 91-92: #2-4, 6, and 17-49 odd).  While students can look at a function graphically, the AP expectation is that students are examing the three parts of the formal continuity definition to decide where or where not a function is continuous.  (Remember as we discussed yesterday: in order for the T-SPLOST rail line to Cobb to be successful, there will have to be  a) a rail bridge built over the Chattahoochee River,  b) rail lines leaving from Cumberland Mall and from the Arts Center Station that go to the river, and  c) the rail lines must come to the river where the bridge was built. 

Section 2.4: Limits and Continuity

Today started our discussion of continuity, as we moved from the intuitive sense of continuous being can you draw a graph without picking your pencil up off the paper (with a domain defined everywhere) to the formal mathematical definition that has three conditions that must be met.  Definition:  A function f(x) is continuous at the point x = a iff:  i) f(a) exists,  ii) the limit of f(x) as x approaches a exists, and  iii) f(a) actual equals the limit value as x approached a.  Students began tonight's assignment on Pages 91-92: #2-4, 6, and 17-49 odd.

Section 2.3: Basic Limit Laws

Fortunately, much of what can be done to evaluate limits algebraically is common sense -- basic limits behave exacly the way we expect them to (the limit of a sum of terms is the sum of the individual limits, the product of a limit is the product of the individual limits, etc.).  Read through the example in this section to be sure you can work with all these limit properties (found summarized on Page 80).  Assignment is Pages82-83: #1-21 eoo (every other odd), 25-28, 31, and 43.  On tonight's assignment, #43 is the problem most conceptual like an actual AP question. 

Limits: Numerical and Graphical Approaches

Intuitively, a limit is a "prediction" - what do I expect the value of my function to be as x gets very close to a particular number.  To do this, I can examine the graph of my function and trace along; of course, I may need to adjust the viewing window or zoom in before I feel confident in making my prediction.  I could also look at a numerical table of values to see if there is a pattern.  In both cases, a key question may be how close to the x-value must one be?  Section 2.2 looks at determining a limit from either a graphical or numerical approach.  Tonight's assignment is Pages 76-78:  #3, 5, 8, 21-35 odd, 41-43 all, and 47-50 all.  (Note that problems 47-50 ask you to work backwards.  Instead of starting with a graph and determining a limit value, we are given information about several limit values and asked to sketch a possible graph of that function.)

Students also got their Chapter 1 Test back from Monday -- some students did extraordinarily well; then there were lots of A's and B's; and then some where we know we can do better!  There will be a ReTest opportunity held after school on Tuesday, September 6th, for anyone interested.  It will be the same format as the original test, will replace your original Chapter 1 Test grade (with the original grade now become a Quiz grade), and does not allow the use of graphing calculators (hence everyothing must be done algebraically!).  Reflect upon this first Test. and decide if you know the material better than this score indicated.

Rate of Change

Today we looked at one of the underlying concepts that is explored by calculus: rate of change.  We noted how there is a distinct difference between average rate of change (i.e., I drove exactly 100 miles to Macon and it took me exactly 2 hours = 100/2 = 50 miles per hour) and instantaneous rate of change (at 5:17 p.m., what "speed" was I traveling at?).  Average rate of change can be thought of graphically as basically the slope of the line connecting the two endpoints (and, therefore, pretty easy to calculate), while instantanteous rate of change can be much trickier to come up with.  (HINT: This might be a good place to use limits, once we determine just what they are...  Tonight's assignment is Pages 66-69: #1, 7, 11, 15, 17, 19, 23, 24, 25, 30, and 32.

 

Test Today -- time to reflect

Now that you've taken your first test, it's time to evaluate the experience.  How prepared were you?  Was the test what you were expecting?  Did you know the material well enough to finish before the period was over?  Should you have prepared differently -- studied more (or less) the night before, done more problems above and beyond homework, etc.?  How much help did your graphing calculator provide -- were there problems on the test the calculator could not solve?  Which problems were easier -- multiple choice or free response?  You'll want to use this evaluation as you prepare for future tests and the AP Exam on May 9th.

Chapter 1 Test on Monday

Today was spent in answering questions getting ready for our first major Test on Monday, covering Chapter 1.  This chapter was all a review of prerequisite skills learned before entering Calculus, so nothing on this Test will be new material.  However, I will look for different ways to ask questions about these review topics, so that our standard of "thinking and problem solving" is addressed thoroughly.  The Test will consist of 16 questions, worth 7 points, so it totals a possible 112 points (and counts out of 100).  About 75% of the questions will be multiple-choice (where only the answer is checked), and the remainder will be free response (where the final answer is worth one point and the work justifying that answer counts for the other six points).  Some time should be spent this weekend in anticipating how might we see problems presented in unique ways.  Study Hard!

Day Four PreCalculus Review

The final section in this first chapter examines the power and/or shortcomings of technology as a tool.  If my graphing calculator works properly 99+% of the time, when does it fail, and why?  When are window settings an issue, or pixel resolution a problem?  Why does the calculator produce such different looking graphs for y = sin x, just because I change the window slightly?  Just because we're looking at more interesting equations now, must I still think?  (YES! -- your brain is about ten times the size of your calculator; don't look like a fool next to the machine!)  As examples, do Page 57: #5, 7, 11, and 16 -- and look for other instances where the calculator might lead you astray!