As Crockett decided not to come to class today, I, Cole, have volunteered to take his place as scribe.
We began class today with an exploration worksheet of a piecewise function, seen below.
The first question examined f(x) when k=1. We were told to sketch the graph which should look like the image below.
The discontinuity seen at x=2 will be noticed as a jump discontinuity, when k=1. We then sought to determine the positive and negative limits of f(x) as x approaches 2, seen below.
As we have learned earlier, for a function to be continuous at a point, both of the single-sided limits must equal the value of f(x) at the x values which the limits approach. As the right limit is influenced by k, and the right limit was three times the value of the left limit, we reasoned that a k value of 1/3 would make the two limits approach the same value of 3 as x approaches two. When we implemented this hypothesis, we discovered that a k value of 1/3 did indeed make the function continuous at x=2. However, the function was still not locally linear at x=2, as there was a cusp at that value. A cusp, derived from the Latin cuspis - "point or apex," is the pointed meeting of two curves, seen below.
Once we had defined what a cusp was, we transitioned to looking over the Quiz. Two free points were given to account for small mistakes. The first four questions were simple limit questions which relied on factoring polynomials and finding clever ways to obtain the GOLDEN LIMIT (below)
Number two (below) was a little bit tricky, but if we remember the relationship between
sin(x)/x and
sin(2x)/x, then the expression can be simplified to a slope over slope relationship, 3/7. If limits are still a problem, the rules of limits can be found
here or on page 61 of our textbook.
We continued through, pausing at both five and six, which questioned our knowledge of the properties of limits and the classification of different discontinuities. These principles can be found right after the fire drill in
Scotty's scribe post. Seven and eight were not too eventful, and the Bonus was shown to be simple after a little bit of working (shown below).
After we finished the quiz, we moved on to IW #7, starting with four (below).
When we graphed this function, we observed that
x^2 was approaching 0 at twice the rate that
1-cos(x) was approaching zero. Therefore, the limit was equal to 1/2. An algebraic solution to this equation can be seen below.
Skipping over five, six, and seven, we moved on to eight and examined Instantaneous Rate of Change. The first part of eight asked us to find average rate of change, or slope, which is a rudimentary mathematical working which will not be gracing this post with its presence. However, we were then asked to find the rate of change at a point. This is done in a similar way to finding average rate of change, but its results are mich more useful. If we take the function g(x) and appraise it at two points, (0, g(0)) and (c,g(c)), and evaluate the limit as c approaches zero, we will find the Instantaneous Rate of Change.
How to find Derivatives
Limit Definition
Local Linearity (seen above with
g(c) and
g(0))
nDeriv(function, x, point) - on Graphic Calculator under the MATH menu
WolframAlpha
g'(2) on GeoGebra
The following is the "Ms. Damian Way" of finding limits.
Consider: v=t^2
Acceleration (Instantaneous Rate of Change): a=v'=2t
The following is the proof for this way of finding derivatives.
NOTE: The Ms. Damian Way does not play nicely with natural logarithms or trigonometric functions, it
only likes power (polynomial) functions. Here is a great
video that goes through this process step by step and gives a formula to find the derivative of a function (below).
NOTE: The notation (dy)/(dx) is what is used to show that you are taking the derivative of a function. You may notice that it is similar to (∆y)/(∆x), which we used to show
average rate of change. As the derivative shows
instantaneous rate of change, it requires its own notation.
If the concept of instantaneous rate of change is still a little hazy, this
interactive applet may be helpful. It evaluates the derivative and a secant of a parabolic, exponential, and hyperbolic equation as well as that of a sine curve. If you are confounded by the applet, which takes a little bit of examination to understand, then you can read this
post on M∆th Sc∞p which does a very good job of differentiating between secant and tangent lines and their application to derivatives.
UPDATE: Now, we no longer need any of these rules to differentiate equations. After IWs 10, and 11, we can tackle
any functions involving trig, inverse trig, exponential, or logarithmic functions. For example, even this monster (below) can now be solved using the rules we have learned.
Although we can solve this, after a few steps, it becomes rather insane, so it is much easier to use tech to solve this problem. nDeriv(
f(x),
x, any
x value) can evaluate this function at any point. Even though it does not yield a function equation, it can tell us what the derivative is of that monster at a point. For example, when
x=3 is ~35362601.44. Although it doesn't tell us anything about the graph, and shouldn't be relied on to find derivatives, nDeriv can be useful in evaluating complex derivatives at singular points.
END OF UPDATE
To finish off class, we received a practice version of a Finely Crafted Opportunity Day, which will serve as our eighth assignment. As many of our classmates will not be here friday, due to the very exciting Common Ground Fair, we are now on our own for studying for the
TEST ON MONDAY. Although we will be going over parts of the practice test on friday, as it is own independent work.
When we went over the practice test, a number of questions were focused on. The main skills that the problems focused on were the ability to simplify a limit by factoring and using conjugate FUFOO's, and the ability to graph a function based on a set of parameters involving single- and double-sided limits. Remember to look for the small negative signs on limits, as they are easy to miss.
As an aside, here is a
google doc that I have created which contains helpful LaTex formulas for weird functions. Anyone can edit and add formulas, so please contribute with your own formulas. Here is a
website that also has a number of helpful LaTex function formulas.
THE NEXT SCRIBE WILL STILL BE CROCKETT
(If he has the audacity to show up to class on Wednesday...)
One final thing, I found a cool animation of the instantaneous rate of change, or slope of a point, on a curve and thought I might add it in at the end. It plots slope over x values, which creates an interesting curve.
UPDATE: We now know that this animation is plotting the
derivative of a
function with a high power of at least
x^4. The line that moves on the function is tangent to the function, so its slope is the
instantaneous rate of change of the blue function.
. In order to make the slope field from this equation you simply start plugging in points! For this particular example we will start with the point (1,1). Plug this into the equation so 
=-1. This means that the slope at (1,1) is -1. How does this relate to our slope field?
. Mr. O'Brien told us that the calculator does not have mystical elves working inside to find the derivative for us, but instead it just uses this equation. In order to graph a derivative on your calculator use the syntax illustrated by the picture below:
:
/
(in red) and
, the derivative is 
. What a POWERful function!
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