BQ#6 - Unit U

BQ #6

1. Q: What is a continuity? what is a discontinuity?

A continuity is a continuous function that is predictable and has no breaks, holes, or jumps. It also means that it can be drawn without your pen/pencil having to be lifted off the paper. A function is continuous if the intended height of a function and the actual height of a function are the same. 

Ex.

(This is a continuous function that has no breaks, holes, or jumps, it can be drawn w/o your pencil leaving the paper)

A discontinuity happens when the intended height of a function and the actual height of a function are different. This can result in one of three different discontinuities, either jump, infinite, or oscillating. A jump discontinuity is when there is a jump between two points that was caused between different left/rights. Secondly, is the oscillating discontinuity, which we mathematicians taught by Mrs. Kirch describe with our astounding vocabulary as, wiggly. Lastly, another discontinuity is an infinite discontinuity which is caused by a vertical asymptote which results in unbounded behavior. 

(This is an example of a jump discontinuity, as you can see there is a jump between 2 points and comes in from one left but it leaves in a different right)

(This is an example of an oscillating discontinuity which when looked at, is quite wiggly indeed)

(This is an example of an infinite discontinuity which as you can see has a vertical asymptote that results in it having unbounded behavior)

2. Q: What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and value? 

A limit is the intended height of a function. A limit exists as long as you reach the same height from both the left and the right. A limit does not exist if the left and right hand limits are not equal, this can usually be caused by breaks, jumps, or holes in the graph and function. A limit is the intended height of a function while a value is the actual height of a function.


(As you can see, even though we come in from different sides on the left and the right we end up at the same spot in the middle, this is an example of an existing limit.)


(This is an example of a limit NOT existing, as you can see we start from different sides but unlike the previous picture, we don't end up at the same place in the middle, instead there is a "jump discontinuity")

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

While multiple ways to evaluate limits, we will be focusing on three specific ways of evaluating them, those three being numerically, graphically, and algebraically. Numerically, we evaluate limits using tables which help us to determine the "x" and the "f(x)", which are the intended and actual height of a function. An example can be shown in the below picture.


(This picture shows a limit being evaluated numerically)

Another way of evaluating limits is by doing them is graphically. There are many ways to do it graphically but there are two main ways, those being having a picture of a graph and using your fingers, the other way is using a graphing calculator by plugging in the equation of the function. The following picture shows doing it by plugging in the equation of the function into your graphing calculator.

(This shows a picture of solving it graphically by plugging in the equation into your graphing calculator.)

The last way of evaluating the limit is by doing it algebraically. We do this by plugging the limit that is given into the function itself, substituting the x with the limit. All you would need is the limit and the function to evaluate the limit this way. The following picture shows an example of this being done in a step by step process.
(This picture shows a limit being evaluated algebraically)





Sources

Picture 1- http://www.mathsisfun.com/calculus/continuity.html

Picture 2- commons.wikimedia.org/wiki/File:Jump_discontinuity_cadlag.svg

Picture 3- http://web.cs.du.edu/~rjudd/calculus/calc1/notes/discontinuities/

Picture 4- http://www.wyzant.com/resources/lessons/math/calculus/limits/continuity

Picture 5- http://www.wyzant.com/resources/lessons/math/calculus/limits
Picture 6-http://curvebank.calstatela.edu/limit/limit.htm
Picture 7- https://www.youtube.com/watch?v=QQEeZB6m6e0
Picture 8- https://www.youtube.com/watch?v=eGEW67LdpRE
Picture 9- https://bccalculus.wikispaces.com/Algebraic+Methods+for+Evaluating+Limits

BQ#3 – Unit T Concepts 1-3

BQ #3

1). How do the graphs of sine and cosine relate to the others?

Tangent

Tangent equals sine/cos as a ratio identity so if one of the graphs is negative then tangent will be negative. If both of the graphs are either positive or negative then the tangent graph will be positive.

Cotangent

The relation is the same as tangent, except the ratio identity is the inverse of tangent, so the ratio identity would actually be cosine/sine.

Secant

It is related to a cosine graph because the reciprocal of cosine is secant.The reason asymptotes are there is once again because of a trig ratio, this time being that sec = 1/cos, and like before, since cosine is the denominator, any place where the cosine value is zero we will have an asymptote.

Cosecant

Just like secant is the reciprocal of cosine, so to are cosecant and sine related in that way. Cosecant is directly related to sine because the reciprical of cosecant is 1/sine.Whenever sine is equal to 0 on the graph, then you will find an asymptote for cosecant there.

BQ #4: Unit T Concepts 3: Tangent and Cotangent Graphs

BQ #4

1) Why is a "normal" tangent graph uphill but a "normal" cotangent graph is downhill?

They have different ratios, and that is why they go uphill and downhill. For example, the ratio for tangent is sine/cosine, this also means that whenever cosine is 0 on a graph that's where tangent will have an asymptote. Cotangent, has the inverse of that ratio, it's cosine/sine and that means that it will have an asymptote when cosine is equal to 0.The difference in the asymptote location causes for the differences in shape. Since cotangent and tangent follow the same ASTC pattern, they both are positive is quadrants I and III and negative in II and IV. In order for cotangent to follow this pattern, it must be downhill.

BQ #5: Unit T Concepts 1-3: Asymptotes

Big Question Blog Post

1. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain

A: An asymptote isA line whose distance to a given curve tends to zero. If we look back at the unit circle we'll see that the trig ratios for sine and cosine are over r, yet r will always be constant, that constant being r=1. Therefore we'll never be able to get asymptotes with sine and cosine. Sine and cosine are never undefined and so they never approach infinity....thus vertical asymptotes for them do not exist. The other trig functions do have asymptotes because they don't have a constant value for the denominator.

BQ #2: Unit T Concept 1 Intro: Trig Graphs and the Unit Circle

Big Question Blog Post: Unit T

1. How do the trig graphs relate to the Unit Circle?

1) The period of a trig function is the horizontal length of one complete cycle. The trig functions of sine and cosine have a period of 2π because their waves repeats every 2π units.  The same can be said of periods of tangeant and cotangeant because their waves repeat at every pi and so that is their period. The following picture shows this. 

Picture illustrates period

2) The amplitude of a sinusoidal function is one-half of the positive difference between the maximum and minimum values of a function. It is the peak from the center of  the graph. At first, trig functions were just related to triangles but now we can relate them to amplitudes and unit circles.  The curves go one unit above and below their midlines (here, the x-axis). This value of "1" is called the "amplitude". It is also illustrated in the above graph

References: http://www.purplemath.com/modules/grphtrig.htm 

Reflection: Unit Q

Reflection

1. To verify a trig function is to make sure that both sides are equal to each other, the same thing, and that it is true, for example 1=1 is true while 2=1 is not true, but instead of simple numbers it's trig identities.

2. One extremely helpful hint was finding out that there isn't just one way to find something. So you and your partner on the test don't have to use the same method to find a problem, but can still get the same answer. It is a good way to check your work.

3.First I would try to convert everything or see if any trig identities can be substituted then I'll try to make the statements equal.

SP7

I/D #3: Unit Q Concept 1: Pythagorean Identities

I. Inquiry Activity Summary

1. Finding the Origin:

We can infer that the Pythagorean Theorem and the Pythagorean Identity have something in common seeing as they both share Pythagorean. We would be correct in assuming that because it is the Pythagorean Identity that originates from the Pythagorean Theorem. In the following steps,we will show how this is true.

II. Inquiry Activity Reflection

1. One connection that I see between units N, O, P, and Q are that they were all sequential, in that they all built upon one another. By that I mean, in Unit N we found out about degrees and radians, also learning how to convert the two into the other. In Unit Q, we use identities but the answer is in radians and degrees. So at first, we learned the answer and its format, while later we learnt how to get to that answer. Another connection that I see between the units is that they come back to the unit circle which we learned about in Unit N, where we first learned about it and the different values of it. Then in Unit Q it was used to find the answer.

2. Three words that I would use to describe them would be: convoluted, enigmatic, and intricate.

WPP #13-14: Unit P Concepts 6-7: Law of Sine and Law of Cosine

The Hundred Years War

BQ #1: Unit P Concepts 1-3 and 4-5: Law of Cosines (Derivation) and Area Formulas (Oblique Derivation)

Big Question Blog Post #1

Questions:

3. Law of Cosines - Why do we need it? How is it derived from what we already know?  The derivation must be shown either in a video or in multiple sequential pictures and it should include descriptions and information beyond what you can find in the SSS.

4. Area formulas - How is the “area of an oblique” triangle derived?  How does it relate to the area formula that you are familiar with?

Responses:

3. The Law of Cosines: First of all, before I explain why the Law of Cosines is needed, I must explain what it is; the Law of Cosines is a formula relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of Cosines can be used to compute the remaining sides of a triangle when two sides and an angle are known(SAS) or to calculate the remaining angles when you know all three sides of a triangle(SSS). According to this law the following is true,

The Law of Cosines will be explained in the example(where α is the interior angle at A, β is the interior angle at B,  is the interior angle at C and c is the line AB), however we will only use variables for now:

4. Area Formulas:

Normally, if we wanted to find the area of a triangle we would use, A= 1/2bh. However, with an oblique triangle we are not given the value of "h" and so must compute for the area of a triangle without it and substitute our regular area equation for h. The way of finding the area of an oblique triangle without being given "h" will be shown in the next few steps.

 Using the labels in the triangle above, the altitude is h = a sin . Substituting this in the formula Area = ½ × b × h derived above, the area of the triangle can be expressed as:

(where α is the interior angle at A, β is the interior angle at B,  is the interior angle at C and c is the line AB).

Furthermore, since sin α = sin (π − α) = sin (β + ), and similarly for the other two angles:

Works Cited

1st Picture: Law of Cosines-http://www.mathwarehouse.com/trigonometry/law-of-cosines-formula-examples.php

2nd Picture: Cosines Triangle Example- http://en.wikipedia.org/wiki/Law_of_cosines

3rd Picture: Area Formula Triangle Example-
http://en.wikipedia.org/wiki/Triangle#Computing_the_area_of_a_triangle

WPP #12: Unit O Concept 10

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