We're all familiar with the difference quotient, but where the heck does it come from and how does that affect derivatives? Origin of Difference Quotient: f(x+h) -f(x) all divided by h
In this graph, we have the visuals to aid us in our understanding of where the difference quotient is derived from. The first point is (x, f(x)). There is delta x (delta = change) distance between the first and second point meaning the second point is (x + delta x, f(x + delta x)). But for this class we will reference to delta x as 'h' so our second point can also be written as (x+h, f(x+h). The line connecting these two points is called the secant line, much different than the tangent line, which only touches the graph once and is depicted below. Now we want to find the slope of the secant line and for that we are going to use our favorite slope-finding formula: 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. Or as we substituted h for delta x, we have f(x+h) - f(x) all over x + h -x. We use the additive process in our denominator which cancels out the 2 x's, leaving only h. This brings us to the difference quotient we all know and love: f(x+h) - f(x) divided by the letter h, that's the difference quotient!
This video by Mathbyfives named Difference Quotient.mov (which can be found here) verbally goes through the process in detail and can provide further explanation.
Continuity in this unit means a continuous functions, no jumps, breaks, strange behavior. A continuous function means a continuous graph -- meaning you are able to draw the graph without lifting your pencil from the paper. Sometimes a graph will show up that seems to demonstrate a discontinuity but it is also a change in function -- a change in function does not make it any less continuous. Discontinuities is much more vast and the main concern of this unit. There are two families of discontinuities. There is removable discontinuities such as point discontinuity, also known as a hole. Then there is non-removable discontinuities such as jump discontinuity, oscillating behavior, and infinite discontinuity, also known as unbounded behavior and occurs at vertical asymptotes.
2. Limits (vs. Values)
A limit is the INTENDED height of a function while a value is the ACTUAL height of a function. When the limit and value of a function are the same then the graph is continuous meaning the absence of of one of discontinuities. A limit does exist at continuous graphs and removable discontinuities such as point discontinuity (also known as a hole) because although the value is undefined, the limit INTENDED on reaching that height. A limit does not exist at non-removable discontinuities such as jump discontinuity because of different left/right behavior, oscillating behavior because it does not approach any single value, and infinite discontinuity because of unbounded behavior due to a vertical asymptote.
In this photo we can note the difference between a limit and value. In the first graph, there is a point discontinuity -- the limit exists, but the value is undefined. In the second graph, the limit exists at the point discontinuity as well but the value exists elsewhere (the black dot.) And the last example is one of a jump discontinuity where the limit does not exist but the value exists at one of the one-sided limits.
3. Evaluating Limits through VANG (minus the Verbally -- just reading the limit notation out as "The limit as x approaches 'a number' of f(x) is eqaul to 'L' "
There are three methods to evaluating limits algebraically. -Direct substitution method In which you plug in the number that is approaching the limit of f(x) and solve to see what you get. Your possible answers are numerical, 0/# which is 0, #/0 which is undefined so the limit does not exist because of the presence of a vertical asymptote which means unbounded behavior, or 0/0 which is indeterminate form meaning we must use factoring or rationalizing method. -Dividing out/Factoring method We use this method when we get indeterminate form. In this case, we factor both the numerator and denominator and cancel the common term -- removing the 0 from the fraction. 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, FOIL the conjugates and DO NOT multiply out the non-conjugate part, leave it factored. Something ought to cancel and then you simplify and once again use direct substitution with the simplified expression.
Through this method we use a table that calculates how the limit approaches 'x' from the left and right by first subtracting 1/10 and then adding 1/10.
You may use your calculator by simply plugging in the limit's equation, hit TRACE and then trace to the value you are searching for. OR you can put your finger to the left and to the right of where you want to evaluate the limit --if your fingers meet, that is where the limit exists, if your fingers don't meet then the limit does not exist either due to different left/right behavior (jump discontinuity), it is interrupted by a vertical asymptote which leads to unbounded behavior (infinite discontinuity), or it does not approach any single value (oscillating behavior).