Tuesday, May 27, 2014

BQ # 7: Unit V: The Difference Quotient and The Derivative

1). Where the difference quotient actually comes from

If we take a normal graph, such as the one above, we can easily derive the difference quotient formula. We make a point with an x value of  'x' and the distance from that point to the origin is therefore 'x'. The y-coordinate of the point would be f(x), which just means whatever y would be when 'x' is plugged into the equation of the graph. If a distance of 'h' is added to the distance of 'x', we get an overall distance of x + h where another point with an x- value of 'x+h' is located. The y-value of that second point would be f(x+h).

Next, all we do is apply the slope formula, which is (ysub2 - ysub1 / xsub2 - xsub1). As shown in red in the picture, when we apply this formula to our two points on the graph, we get the difference quotient formula after simplification.

2). How this is relevant to our lessons of derivatives

Since the difference quotient  basically represents the slope of a line that touches two points on a graph (shown above), and a secant line is a line that meets a graph in two points at given slopes, the difference quotient could be used to find the slope of secant lines wherever those two points may be. In other words, in Calculus the difference quotient represents the slope of secant lines.


The difference quotient may give us the slope of secant lines, however, they fail to give us the slope of lines that touch ONLY 1 POINT, which are known as tangent lines. In order to do this, we must find what is known as the derivative, which is the equation that represents the slope of tangent lines. In order to get this we must evaluate the limit as h approaches zero of the equation we got from the difference quotient. We do this because 'h' (as noted above) is the distance between the two points on the graph. In a secant line there is two points, and in a tangent there is only one, so how do we get the derivative from a secant line? The answer is clear: we make h, the distance between the two points as close to 0 as possible, because when h is zero, there is only one point, but h can never actually reach zero in a secant line so thus we evaluate the limit as h approaches zero in order to get the derivative which represents the tangent line.

A tangent line of a function with a point at (0, 0)
A secant line of the same function where h = .2
A secant line of the same function where h = 2.5

As displayed above, the closer h is to zero, the closer the secant line resembles the tangent line. This is the reason why we evaluate the limit as h approaches zero of the equation from the difference quotient, which represents the secant line approaching zero, or becoming the tangent line.


Last three pictures: desmos.com

Monday, May 19, 2014

BQ #6: Unit U: Continuity and Limits

1). What is continuity? What is discontinuity?

  • Continuity in Calculus basically means that a graph is continuous which means that the graph has 4 characteristics: 1). the graph can be predicted, 2). the graph contains no breaks, no holes, no jumps, and no oscillating behavior, 3). the value and the limit must be the same, and 4). the graph must be able to be drawn without the lifting of the pencil from the paper.
An example of a continuous graph with all 4 characteristics of continuity
  • Discontinuity in Calculus is, consequently, something that that causes a graph to be discontinuous. There are 4 types of discontinuities, which are divided into 2 families, these being Removable and Non-Removable.Removable discontinuities include point discontinuities, which are essentially just holes. Non-Removable include jump discontinuities, which cause breaks in the graph, infinite discontinuities, which result in unbounding behavior, and oscillating behavior, which creates unpredictable, random "wiggly lines". 
A point discontinuity, creating a hole
A jump discontinuity resulting in a break in the graph

A vertical asymptote causes an infinite discontinuity
Oscillating behavior causing discontinuity

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 EXISTS when the specific interval of the graph where the limit is located is CONTINUOUS. In other words, if a limit is located at a part of the graph that has all 4 characteristics of a continuous function, then the limit will exist. The reason behind this is that only when a graph is continuous does the graph have a consistent intended height; when a graph is discontinuous it has certain abnormalities that cause the intended height to be inconsistent and undefinable.
  • A limit, however, does not exist when the intended height of a function is unclear or inconsistent due to the presence of any of the three Non-Removable discontinuities.There are 3 specific cases of this happening. In the first case, the graph as it approaches the limit shows different left and right behavior, meaning that as the graph is getting to the limit from the left and from the right two different locations are reached and thus two different intended heights are apparent and thus a single limit does not exist. This is because of a jump discontinuity.In case 2, the graph may exhibit unbounded behavior due to an infinite discontinuity due to a vertical asymptote. This results in the graph to go in unbounded directions, and as a result the limit will never exist where the asymptote is because a limit can never intend to reach a height of infinity. Thirdly, if there is oscillating behavior, the graph is so unpredictable that a graph has no real intended height.
  • It is important to discern between a value and a limit. A value is the actual height of a function whereas a limit is the intended height. Sometimes these are both the same thing, but a lot of other times they are not. For instance, in the case of a hole, the limit is at the hole, but there is no value at a hole. On the other hand, there can be a value but no limit, such as the case in the jump discontinuity below:
File:Jump discontinuity cadlag.svg
The value is at the closed circle, however at that same value of x, there is no limit because of different left and right behavior due to the jump.

3). How do we evaluate limits numerically, algebraically, and graphically?

  • A limit can be evaluated numerically through the means of a table very similar to a number line. What we do is we use what x is approaching and list out  x- values that are mere tenths, hundredths, and thousandths away from that number and then find the y values to see what the limit is. For instance, if we wanted to evaluate a limit of a function as x approaches 4, we would make a x-y table and put 4 in the middle of the x section and then on the far left put 3.9 and on the far right 4.1, then we simply get closer to 4 with the x values and we plug the equation into a graphing calculator and trace the values for our x's. We can then observe what the limit is.
  • A limit can be evaluated graphically evaluated very easily. Simply get a graph and move your fingers from the left and right of the where the limit should be. If they meet, then the limit exists. If not, the limit does not exist.
  • Evaluating algebraically can involve three different methods. These are direct substitution, factoring/dividing method, and the rationalizing/conjugate method
  • In direct substitution we just plug in what x is approaching into all the places an 'x' is in the function. We can get four answers from this: a number value, zero, undefined (meaning there is a vertical asymptote/infinite discontinuity or a hole), or indeterminate form (0 divided by 0) which means we got to use an additional method.
  • In the factoring/dividing out method we factor both denominator and numerator in order to see if a hole cancels out so we can use direct substitution with the result to get the limit.
  • In the rationalizing/conjugate method, we have to multiply by a conjugate to try to get something to cancel (something that we can't factor in the first place) and then we resume with direct substitution.