Elazar and Connor | Final Exam Presentation

WORK AND FORCE

Force:
A push or pull which acts on an object. This doesn't necessarily mean the object will move, but the force is existent.

Imagine a water bottle on a table, if you push the bottle lightly it might not move. This is an example of a force acting on a force and being counteracted by friction. This leads to the ideas of force diagrams.

Think of all the forces acting on you now. What keeps you on Earth? What keeps you from falling through the ground? Are these things acting on you?

Before we take a look at work, recall what we learned about displacement. It is important to know that displacement is the difference between the final position and the initial position. Where the particle goes in between is not important. So displacement=X(final)-X(initial)


Work:

A force acting on a body which results in a displacement.

Work is related to force through calculus. Just like position is the antiderivative of velocity, work is the antiderivative of force. The only difference is the variable. With velocity and position the change is the change in time, while with work the change is with respect to the axis of displacement.

Constant force: something which doesn't change over time, gravity is a great example of this.
Equation for work when the force is constant:

Variable force: a force which varies over time, given as an equation.
Equation for work when the force is variable:

QUIZ

1.25.13

1.25.13

We began by putting IW#4, arguably the easiest IW to date, on our desks to begin class. OB had the class look at the answers via computer, and then led the class in a rousing discussion about the IW.

We noted the use of the I.V.T when looking at when the particle is at rest in question 2, on example 1 of the IW. This is the kind of calculus reasoning we can use of the AP test.

Using fitPoly in Geogebra you can take a table as seen in example 1 and make a list of points and draw a function through all of them to show you what type of polynomial the function is. 

However, some of the time you wont have the entire function and you just have the values from the table. Then finding the slope can be done by finding acceleration, given a v(t) chart, since a(t) is v’(t).

For number 3 on the IW, Noah remembered the Symmetric Difference Quotient, which is just nDeriv. Taking a point on either side and calculating the slope off the line from point to point. OB pointed out than units for acceleration come from the fact that  which gives you units of , Which is . So in terms of motion, if the acceleration -1/2, the particle is slowing down at that point.

Using a graph we can see 4 places where the graph changes from increasing to decreasing, meaning there is 4 places where acceleration is zero, since acceleration is the derivative of velocity. 

OB asked if a function is differentiable, must its derivative be differentiable. He shockingly answered “no”. He said there was extra credit for everyone except Eliot if they found an example of this. 

The second page had little confusion and the answers can be seen on the Even Answers page of the blog. Its's important to note that a table only gets us so far OB says, you have to look at an instant in time to really tell.

SURPRISE! Quiz tuesday! YAAAAY! What a treat!

LEAP FROG TIME. SOLIDIFY THOSE BASICS. BLOW MINDS. LEAP FROGS. IW#5. CONFUSION. SPEED.

The goal for the leap frog game is to make us leave confused about speed. I suppose it was successful, since the game was quite easy in general, and then only questions people seemed to be confused on were when it asked about speed instead of velocity.

Scribe Post 9/10

We began class by taking a 40 minute quiz. After the quiz, we read through Francie's scribe post from last class. After that Mr. O'Brien put a warm up on the board in which we were instructed to find three limits.  The three equations are:

a.)

b.)

c.)

The class took a shot in the dark, and Mr. O'Brien stated that there were errors, only to realize he was in fact the culprit. Cooper scrawls a messy answer but Mr. O'Brien still insists the answers aren't perfect. While the class continued to solve the Warm-Up, we reviewed Francie's scribe post. Connor, with eyes like a hawk, spots a slight error in Francie's work. Together, the class watched a math related clip from one of the great classic films of our time, Mean Girls.

Turning back to the problems at hand, all the examples on the board are examples of indeterminate form. This is when you get 0/0 when using substitution.  When dealing with indeterminate form you cannot just substitute.  Note that this is different that the problem we earlier dealt with, , which can be rewritten as  . While the denominator would be 0 if we substituted 2 directly in for x, we avoided this by realizing we could factor first, which allows us to cancel out the (x-2)'s.  While this problem potentially gives a denominator of 0 when substituting, factoring avoids this.  We are left with (x+3), where 2 can be substituted in to give us a limit of 5.  Note that the three warm-up problems won't work this way since they are indeterminate form.

So really, what is the indeterminate form? 

The indeterminate form technically gives you an answer of anything since you are not just dividing by zero, but you have 0/0, which gives a possibility of anything.  Using an explanation on Khan Academy, we see that undefined means there is no possible value, while indeterminate means there could be any value and there is not enough information given.  This is explained in this video.

This is shown in warm-up b, , where you can factor out an x and then cancel to get , which, using substitution of 0 for x, since , gives us 1/0.  This is undefined as opposed to indeterminate.  1/0 is infinitely large, which is a non-existant limit.  The goal by the end of the day was to answer the first 3 warm-up problems without using a calculator.  To do this, we need to first know the sandwich theorem, which will help us solve problem a.  In an example, , we can let .  As x approaches 0, theta approaches 0 as well.  You can sub a 3 into the numerator and denominator and you get . We know that .  This is the same problem as a, which was but with x instead of theta.  Since the answer to a is 1 (which we know by using a calculator) , we can sub into to get 3 times 1, which is equal to 3.

Algebraically solving problem c, which is , if we substitute immediately we get we can use a FUFOO, or in this case, as Cooper expertly points out,.  Multiplying that FUFOO by c, we get , which simplifies to .  Then we can use sin rules to get .  This can be confirmed using a calculator.  Cooper triumphantly shouts "hell yeah" as he understands the concept. O'Brien points out that using a calculator on indeterminate form will always work.

Cooper asks how the limit of problem a = 1 algebraically. This excites O'Brien greatly, seeing as this is the entire key to what we have learned today.  This leads us to the Sandwich theorem, also known as the Squeeze theorem.

If you have three functions, f(x), h(x), and g(x), where f(x)≤g(x)≤h(x) on some open interval containing c (a range of x-values that contain endpoints) and in addition,  and  both equal L, then  MUST be L.  A picture speaks a thousand words O'Brien says, as he proceeds to draw a masterpiece that explains the Sandwich theorem.

You can use this theorem to find a function that is sandwiched between two other functions. An example is when you have , , , where .  Since the limits of the functions are all the same, we can graph these functions.  Note that f(x) is in red while h(x) is in blue.  This gives us the graph:

When we graph g(x) in green, we see that it is "sandwiched" between f(x) and h(x).

The key to this theorem is that you must know the limit of the two sandwiching functions, since their limit is the limit of the function that has been sandwiched.  If you want some more visuals on the sandwich theorem, check out this fun applet as well as this even more fun applet!

UPDATE: 

Tonight's IW (66/31, 33, 65, 72, 77) will focus on the newly learned Sandwich Theorem, where we will get to practice using it.

FUN FACT: Math team has it's first practice this wednesday, september 12th. The schedule is located here: http://math-ob.wikispaces.com/mathteam. The meeting is in O'Briens room. However, extra credit is iffy at the moment. However, some other benefits include bragging rights, great camaraderie, fun bus rides, and Ms. Orhon's and Mr. O'Brien's company.