BQ# 7: Unit V

     The origins of the the difference quotient comes from a graph and using an old equation from early this year.

Here is the graph f(x) and the line that's barely touching it at x is the tangent line. So the coordinates for the graph is ( x, f(x) ). 

This is another graph that has a secant line going through it. It still has the original point as the first one but it has another point in it. Since we moved a little bit from the original to the new point, then it's a change in the graph. That change can be written as delta x or as i put it h. So the new coordinates for this graph is ( x, f(x) ), ( x+h, f(x+h) ).

Highlighted in blue is the two coordinates from the graph. Highlighted in green is the slope formula that will help find the difference quotient. The one in pink has everything plugged in and in the denominator we see that the xs' cancel so there's only an  h left in the denominator. The last one in purple is the difference quotient and that is how we get the equation.

BQ #6 : Unit U Concepts 1-8

1) A continuity graph is a graph that's predictable, it has not breaks, holes, and jumps. Also the limit and the value are the same, the limit is the intended height while the value is the actual height. There are two groups, the removable and non-removable, a continuous graph is in the removable group.

A continuous graph is in the removable group because it has a limit. This can be considered a continuous graph because it has no breaks, hole, or jumps.

 Discontinuity graphs are in the non-removable group because these graphs have no limits. These graphs have jumps, breaks, and not predictable at one point. There are three types of discontinuity graphs, jump discontinuity, oscillating behavior, and infinite discontinuity.

Highlighted in green would be our jump discontinuity, orange is oscillating behavior, and in blue is infinite discontinuity. None of these graphs have a limit but only jump discontinuity can have a value. The other two have no limit or value. 

2) In this unit a limit is the intended height of a function. It only exits in a point discontinuity graph.

This graph is a point discontinuity and it has a limit. Now the first one is continuous but is still has a limit, the second one has a hole but it still has a limit because it's the intended height.

All of these don't have a limit so the limit does not exist for them.

Just like in the first question the difference between a limit and a value is that the limit is the intended height while the value is the actual height.

3)

To evaluate a limit numerically we do it by using a table that we can plug in to our graphing calculator. We can find the limit using the graphing calculator by tracing the graph. Then we can tell if the limit was reached.

 

When it's done graphically we can either use our fingers to see where the limit is if we come to the middle from the left and right. Then we just look at what type of discontinuity it is and if the limit exists or not. 

A

B

C

When it's done algebraically we can use three methods but the one we will always try first is the substitution method, know as picture A. Ifa problem can't be done using that method then we use the dividing/ factoring method. If those two don't work then we use the rationalizing/ conjugate method.