BQ#7- Unit V

                             Unit V Big Questions

Where does the formula for the difference quotient come from? Well know i now this question because i learned it from the wonderful Mrs. Kirch :D

       In this graph, we can see our understanding of where the difference quotient is derived from. The first point is (x, f(x)). There is delta x  and delta equals change in distance between the first and second point meaning the second point is (x + delta x, f(x + delta x)) that is what they used delta x but, for this we will reference to delta x as 'h' so our second point can also be written as (x+h, f(x+h) so that we can get a better understanding of a variable that we know so it wont be so confusing. The line that touches the graph at two points is called the secant line, very different from a  tangent line that only touches the graph once. To find the slope of the secant line we are going to use the slope formula. The slope formula is m = y2-y1 all divided by x2-x1. When we plug in our two points from the first graph we have f(x + delta x) - f(x) all over x + delta x - x. Then we substituted h for delta x and we get f(x+h) - f(x) all over x + h -x. Then we notice in the denominator we have a x and -x and what happens when you get a positive and negative number or variable is this case variable you CANCEL THEM, leaving only h. Then you get the Difference quotient f(x+h) - f(x) divided by the letter h, that's the difference quotient :D

There is a video that verbally shows you how to derive the difference quotient by clicking here

                                                     Works Cited 

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BQ#6- Unit U Concept 1-8

              Unit U Big Questions

1) What is continuity?
 Continuity is when something does not stop or is not interrupted by anything. A continuous function is predictable. It has no breaks in the graph, no holes, and no jumps. A function that is continuous can be drawn with a single, unbroken pencil stroke. In terms of limits, a function is continuous if the limit as x approaches a number of f(x) is equal to the height f(x) that reaches at that point.

This is a picture of a function that goes on forever in both directions and is not interrupted by anything, meaning there are no jumps, holes, or breaks in the graph.

What is discontinuity?

Discontinuity is when something doesn't go on forever in both directions and is not continuous. There are two families of discontinuous functions. They are removable and non-removable discontinuities. The removable discontinuity is just called point discontinuity, also known as a hole. The non-removable discontinuities are jump discontinuity, oscillating behavior, and infinite discontinuity, which is known as unbounded behavior and occurs where there is a vertical asymptote. The removable also means that there is going to be a limit that does exist while at non-removable discontinuities there all going to be the limit does not exist.

Removable: 

Non-removable:

2) What is a limit?  When does a limit exist? When does a limit not exist?  What is the difference between a limit and a value?
A limit is the intended height of a function. A limit exist when you reach the same height from both the left  and the right of the graph which only happens in a point discontinuity. A limit does not exist when its in the non-

removable family. Jump discontinuity DNE b/c of different L/R. Oscillating Behavior b/c of oscillating behavior, theres no name for it, and Infinite discontinuity b/c of unbounded behavior. The difference between a limit and a value is that, a limit is the intended height and the value is the actual height on a graph.

3) How do we evaluate limits numerically, graphically, and algebraically?
We evaluate limits numerically by using a table. With the table, we find the intended heights of the whole function. 

To evaluate limits graphically, we plug in our function into the y= screen on our graphing calculator, hit trace, and trace to the value we are looking for. If you hit trace and you don't get a value, the limit does not exist and the reasons for that situation are those three reasons.

Lastly, we evaluate limits algebraically by using three different methods: 

Direct substitution method
In which you just plug in the numbers that is approaching the limit of f(x) and solve it. You can get either a numerical answer, 0/# which equals to ZERO, #/0 which is UNDEFINED so the limit DNE,and if you get 0/0 it is a indeterminate form so then you have to use the next method which is divide and factor method.
Dividing out/Factoring method
We use this method when we get indeterminate form. We then factor both the numerator and denominator and cancel the common term to remove the zero in the denominator. Then we use direct substitution with the simplified expression. BUT always be sure to use direct substitution first!!
Rationalizing/Conjugate method
This method is also helpful with a fraction, especially if it has radicals. You multiply by the conjugate of either numerator or denominator depending where the radicals are at, then FOIL the conjugates and DO NOT multiply out the non-conjugate part, leave it factored because you then get something canceled out which is a good thing and then you simplify and once again use direct substitution with the simplified expression.

                           WORK CITED

Continuous Graph
Point Discontinuity
Oscillating Bahavior
Jump Discontinuity
INFINITE DISCONTINUITY
Evaluate Numarically