Mr. Slater's Blog

Continuity - Day One

When is a function continuous at the point x = c?  Our intuitive idea of continuity from previous math courses is that one can draw the graph of the function without picking their pencil up off the paper.  Unfortunately, if the domain of the function is all Real numbers, you could never draw the entire graph from beginning to end - thus, we need a more formal definition.  Analogous to building a new highway from Marietta to Atlanta that pust include a new bridge over the Chattahoochee River, the definition of a function being continuous at the point x = c  has three conditions: 

1) Does the point f(c) exist?  (Is there actually a bridge over the river?)

2.) Does the limit as x approaches c exist?  (Is there actually a road from Marietta leading to the river that meets at the same spot as a road from Atlanta to the river?)

3.) Does the limit as x approaches c exist equal to f(c)?  (Does the road from Marietta to the river and from Atlanta to the river meet at where the bridge was built?)

Tonight's assignment is Pages 91-92: #2, 3, 4, 6, and 17-49 odd.

Basic Limit Laws

Having looked at evaluating limits numerically and graphically, we're now starting to evaluate limits algebraically.  This section developed half-a-dozen limit laws, which essentially allow us to substitute the value of c in for x into the function and "do the arithmetic" - assuming we don't get an indeterminant form of some sort.  (We'll address dealing with indeterminant forms algebraically in more detail in a couple of days.)  Tonight's assignment is found on Page2 82-83: #1-21 eoo (every other odd) - write these solutions out formally, 25-28 all, 31, and 43.

Solving Right Triangles

Last week we were given a right triangle and asked to find the basic trig ratios (sine, cosine, and tangent) for an acute angle.  The inverse trig buttons on our calculator (2nd sin, 2nd cos, or 2nd tan) allow us to work in reverse  - that is, given sides of a right triangle we can work to find the measure of the acute angles.  Thus, we can "solve any right triangle".  If we know any two facts about a right triangle (the lengths of two sides, or the measure of one angle and one side), we can find the lengths of all three sides and the measures of all three angles.  (You can also use prior knowledge to help out, such as applying the Pythagorean Theorem or knowing the angles of any triangle sum up to 180).  Tonight's assignment asks us to do exactly that - Pages 174-175: #1-15 and 23-27 all.  Also remember that you Quilt block final task is due tomorrow (if you have not already turned it in).

We will have a short Quiz over finding and using trigonmentric ratios on this Wednesday (August 18th).

Quilt Square Final Task

Students got their Quiz back from yesterday over Special Right Triangles.  Approximately 70% of the grades were A's, so many, many students are off to a great start!  These special right triangles (45-45-90 and 30-60-90) exhibit simple relationships between their sides which should be quick to identify and use, and are found in a multitude of applications.

We then reviewed our definitions of the three basic trigonometric ratios (sine, cosine, and tangent) we developed in yesterday's powerpoint presentation, and summarized these as "SOHCAHTOA".  Students then had all class period to work on either

1) Homework:  Pages 169-160: #1, 5, 9, 11, 13, 15, 17, 19, 21, 22 and Pages 168-169: #1, 9, 11-19 all (which will be checked in class on Monday), or

2) Quilt Square Task:  Students received their unique quilt square pattern, which they had to solve (i.e, calculate the lengths of all the segments).  Then they need to "make" the their quilt square using color paper, etc,  Students had access to use all the necessary equipment (colored paper, ruler, scissors, protractor, glue stick, etc.) today in class - which allowed several to finish and turn their square in.  The finalized Quilt Square is due by next Tuesday, August 17th, for a minor assessment grade.

Introduction to Rate of Change

After yesterday's Chapter 1 Test, today we began looking at the Essential Question for all of fall semester Calculus:  "How can one determine the rate at which things change?"  When determining how things change, sometimes we want an overall picture of the situation - What is the average rate of change over time?  Fortunately, we've known how to compute averages since middle school or earlier ((Y2-Y1)/(X2-X1)); graphically, this represents "slope".

On the other hand, how could we determine the rate of change at a specific moment - especially if it is not a constant rate of change?  ...and can we determine this rate of change numericaaly, graphically, or algebraically?  Tonight's assignment begins this exploration, on Pages 66-69:  #1, 7, 11, 15, 17, 19, 23, 24, 25, 30, 32.

Chapter 1 Test Today!

Today is our first major assessment - a Test over all that math one should know how to do from the past ten years leading up to Calculus.  Hopefully, I've created problems that tests these concepts creatively, without becoming too burdensome with excessive calculations, etc.  (Remember the assistance your graphing calculator could provide with its graphs, tables, etc.)  Although there was absolutely no new material in this chapter, this test should be a good indicator of how well one really knows what they've learned, and, hence, how much work should be necessary in the future to do well.  Tests should be returned early next week.