BQ #7: Unit V: Derivatives and the Area Problem

How is the Difference Quotient derived ?

Throughout this entire year, we have practiced and memorized the difference quotient through song, and rhythm. In order to understand it, one must know that it is derived from the slope of a secant line, that touches two points on the function.  

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0413.JPG

A secant line is when it touches two points of the functions graphed. touching two points we know that the slope is not steep, but is negative or positive, depending on where it points. 

http://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Secant_line_and_curve_basic.svg/450px-Secant_line_and_curve_basic.svg.png

A tangent line is when it touches the graph only ONCE! This shows that the graph is steep, and or the slope is equal to 0, that means the line is horizontal. 

http://mathinsight.org/media/image/image/neuron_firing_rate_kink_tangent_line.png

BQ #6: Unit U

1. What is Continuity ?

A continuous function is predictable. It has no holes, no breaks, and no jumps. It can be drawn with a single, unbroken pencil stroke. A continuous graph has to have the SAME limit and value; meaning the limit can only be reached with a continuous graph.

What is Discontinuity? 

A discontinuity is any type of break, hole, or jump from a graph. The limit and the value of a discontinuous graph is always different from the value. There are two families of discontinuity: Removable and Non- Removable. Removable discontinuities (also known as holes), the limit does exist, but the value id undefined. Non- Removable discontinuities, the limit Does Not Exist, but the value can.

http://www.zweigmedia.com/RealWorld/calctopic1/pics3/pic1.gif

2. What is a limit? 

 A limit is the intended height of a graph. the limit can be a numerical, zero, undefined, and indeterminate. The limit DNE in Non-Removable discontinuities: jumps (comparing left and right behavior), oscillating, and infinite (unbounded behavior).

When does a limit exist?

 A limit exist when a height is being shown with the exceptions of : two opened circles, and any open circle in general.

http://i45.tinypic.com/152hn36.jpg

When does a limit not exist? 

A limit does not exist when we have opened circles, and wherever there may be vertical asymptotes. If the answer to a limit is 0/a number, the limit DNE. having different points on an X-point (jump) the limit DNE because of different Left and Right. Oscillating behavior, the limit DNE.

What is the difference between a limit and a value?

A limit is the INTENDED height of a graph, does not mean the value is the same.
A value is the ACTUAL height of a graph, it can be determined as undefined at times.

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

Numerically is evaluated with a table where we get VERY, VERY, VERY, VERY, VERY, close to the point, but never reach it.

https://finitemathematics.wikispaces.hcpss.org/file/view/limit_table.PNG/239144381/575x194/limit_table.PNG

We evaluate Graphically by using two fingers and following our graph to see if they meet. The identification of the graph and its discontinuities, and continuities.

http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/images/21-graphing-04.gif

We evaluate Algebraically by using the direct substitution, dividing out/ factoring method, and rationalizing/ conjugate method.

http://www.analyzemath.com/calculus/limits/find_limit_3_2.gif

BQ #4: Unit T Concept 1-3: Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill?

Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill?

For a Tangent graph we know that it is positive only in quadrant I and III based off the Unit Circle. A tangent graph has the ratio of sine/cosine, and for an asymptote to appear the denominator must be zero; meaning that the value of cosine must equal 0.To complete a full period, an asymptote must go from a negative to a positive, creating an uphill graph.

A cotangent graph is bit similar to that of a tangent graph, but a few things differ. For instance the ratio is cosine/sine; meaning that the sine must equal 0 to give it an asymptote. so in order for the cotangent to complete a full period with an asymptote, the graph has to start from a positive and move to a negative, creating a downhill graph.

BQ #3: Unit T Concepts 1-3: How do the graphs of sine and cosine relate to each of the others?

Tangent's ratio is, as we all know, Cosine/Sine, which relates to the reason to why it has asymptotes. Sine has the possibility of having the value of 0, make the ratio undefined, creating an asymptote in the graph. According to the Unit Circle, Sine is positive in the I and II quadrants, Cosine is positive in the I and IV quadrants and Tangent is positive in the I and III quadrants. Cotangent is but the reciprocal of Tangent, meaning that the ratio is Sine/Cosine. Cotangent is also another that includes an asymptote in it's graph because of the undefined ratio solution. 

Secant is the reciprocal of Cosine, meaning the ratio is R/X giving it an asymptote if the x (cosine) is equal to 0. 

Cosecant is the reciprocal of Sine, meaning that the ratio is R/Y giving it an asymptote if the y (sine) is equal to 0 

According to this graph, there are no asymptotes present, but it does give a small interpretation of how the Unit Circle unfolds onto a graph. the loop going up is divided in half and the first half is the first quadrant and the second half is the second quadrant, so on and so on. these quadrants represent the value of the sign that the trig functions have in the graph. 

BQ #5: Unit T Concept 1-3: Why do cosine and sine not have asympototes, but the other four trig functions do?

According to the Unit Circle, Sine and Cosine were the only trig functions that were not able to equal zero with their ratio. Sine being y/r and Cosine being x/r, the r representing 1, NEVER can the ratios equal zero. An asympototes only appear when the ratio of a trig function is equal to zero, which is why every other trig function ( CSC, SEC, TAN, COT) do have asymptotes.

(http://www.sosmath.com/trig/Trig2/trig2/img6.gif)

BQ #2: Unit T Concept Introduction

How do the trig functions relate to the Unit Circle?

The trig functions relate to the unit circle based on the signs they hold in the Unit Circle that distinguish how the graph will look on the x and y graph. The x and y graph is the unfolded Unit Circle. Sin has a pattern of positive, positive, negative, negative. Cos has a pattern of positive, negative, negative, positive. Both sin and cos go through ONE cycle while covering 2pie units. The trig function of tan is different in the way that tan's pattern is positive, negative, positive, and negative and tan and cot go through ONE cycle while covering pie units.

How does the fact that sine and cosine have amplitude of one relate to what we know about the Unit Circle?

Sine and Cosine have amplitudes because of the ratio they share. Amplitudes are only found when the ratio is undefined, meaning that the denominator is 0. Sine and Cosine (CSC and SEC) have the ratio of R/ X or R/Y and with that the answer cannot equal zero, whereas Tan and Cot have the possibility of equaling and being left with a ratio of undefined, leaving it to have an Amplitude.

Reflection #1: Unit Q concepts 1-5: Identify Trig Functions

1. To verify a trig function means to break down an equation of trig functions to the smallest, most simplest trig function possible. that means to put all our knowledge of previous math to work. All the information we learned from the Unit Circle are being tested in this unit. All equations must break down to one trig function. That is your goal! 

2. The tips and tricks that i have found helpful to me is the Unit Circle and the identities that we have had to memorize. The Unit Circle's points and degrees are a part of Units 1-5. The trig identities are what come to finding our answers much faster than any other method. 

3. To begin breaking down a trig function, i first look to see if it is possible to split or substitute anything into the equation (You look for the fastest way to break down your equation). Next we see if any identities are found or can be made with what we have broken down. If not, we continue with what we have, either multiply or divide functions. The final thing i look for is the possibility if the function can be substituted, distributed, or identified (usually will be one of these). 

SP 7: Unit Q Concept 2: Finding trig functions using identities

Original Problem: 

tan(x) = -1, cos(x)= 1

In this step above we have the substitution of cot that substitutes with 1/tan. 

In this next step we have the identity that includes both tan and sec. This identity allows us to simplify to a one term answer. 

 Next we come to another identity to where both cot and csc are found together. This identity allows the answer to come out to be one term and the steps include: adding, and square rooting. 

This following step shows the substitution of csc to sin. the substitution is to solve with the answer known for the answer we are in search of. 

These are the answers we came to finding with all the steps above. 

I/D #3: Unit Q: Pythagorean Identities

 Pythagorean Identities

1. sin2x=cos2x=1 comes from the the unit circle ratios. The sin2x symbolizes sin= y/r and the cos2x symbolizes cos= x/r. Combined together in an equation y/r turns into y, x/r turns into x, and added together it is x+y=r, or x2+y2=r2 which equals to one.

2. sin2x+cos2x=1

If we look back at our Unit Circle we can relate sin equaling to the ratio of y/r; leaving the answer to be y, since the r=1. To continue with the reference of the Unit Circle, cos has the ratio of x/r, leaving us with x. Plugging that into our original equation, we get y^2 + x^2 = 1. Looks similar, it should because it is just like the Pythagorean Theorem of a^2 + b^2 = c^2.

                                   

                                     

WPP #13 & 14: Unit P Concept 6 & 7 Law of Sines and Cosines.

Please see my WPP13-14, made in collaboration with Anthony Lopez, by visiting their blog here.  Also be sure to check out the other awesome posts on their blog

BQ #1: Unit P Concept 1 & 4 - Law of Sines and Area Formula

1. Law of Sines 

The Law of Sines has great use when only given a few measurements of a not right triangle when needing all measurements. Figuring out the measurements of the angles and sides of a non right triangle is a bit more complex than with special right triangles because we cannot use SOH CAH TOA. Instead we use the law of sin to figure out one angle/side at a time. For example, in the picture below if we are only given Angle C, side b and Angle A, we can derive the law of sines to figure out the rest of the measurements. Given the measurements of two angles, we add them and divide by 180 to find the third angle. Once we find the third angle we match up sin (angle) B/ sin (side) b = sin (angle) of either A or C and continue that until all sides are found. 

(https://www.google.com/search?hl=en&site=imghp&tbm=isch&source=hp&biw=1680&bih=949&q=non+right+triangles&oq=non+right+triangles&gs_l=img.3..0.1425.6210.0.6666.23.21.2.0.0.0.150.1641.19j2.21.0....0...1ac.1.37.img..2.21.1450.ngevANF7olE#facrc=_&imgdii=_&imgrc=OMDCBtQcvuCROM%253A%3BibnpwnU4btR6lM%3Bhttp%253A%252F%252Fwww.drcruzan.com%252FImages%252FTrigNonRight%252FNonRightTriangleLabeling.png%3Bhttp%253A%252F%252Fwww.drcruzan.com%252FMathNonRightTrig.html%3B317%3B196) 

 4. Area formulas

Finding the area of oblique triangles all comes from the formula derived from the area of a triangle
A = 1/2bh. Using the picture below, we know that sinC= h/a, so to derive it to equal to the h we therefore get h = asinC. Next we substitute in our regular area equation for h and we get A= 1/2b(asinC). When solving for these triangles we are most likely going to be given two sides and one angle. The two sides are imputed into the equation as a and b. The angle that is given is always found with the use of sin. 

(https://www.google.com/search?hl=en&site=imghp&tbm=isch&source=hp&biw=1680&bih=949&q=non+right+triangles&oq=non+right+triangles&gs_l=img.3..0.1425.6210.0.6666.23.21.2.0.0.0.150.1641.19j2.21.0....0...1ac.1.37.img..2.21.1450.ngevANF7olE#hl=en&q=oblique%20triangle&tbm=isch&facrc=_&imgdii=_&imgrc=ffiBKZSnnuP3lM%253A%3B9k91py1yIcyOVM%3Bhttp%253A%252F%252Fwww.compuhigh.com%252Fdemo%252Flesson07_files%252Foblique.gif%3Bhttp%253A%252F%252Fwww.compuhigh.com%252Fdemo%252Ftriglesson07.html%3B780%3B355)

WPP #12: Unit O Concept 10: Angles of Elevation and Depression

Problem: 

Solution:

I/D #2: Unit O Concept 7-8: How can we derive the patterns for our Special Right Triangles?

Inquiry Activity Summary 

30, 60, 90 Degree Triangle

In the 60, 60, 60 degrees triangle we use the rule that the triangles sides all equal one, and to get a  30, 60, 90 degrees triangle we must cut the original triangle in half, leaving us with one 30 degree angle, one 90 degree angle and last one remains 60 degrees. When dividing the triangle in half, we were left with one side equaling 1/2, the side across from the 30 degrees angle. The side across from the 90 degree angle is the hypotenuse which always equals 1. The next side is the one we search for. In searching for that side we use the Pythagorean Theorem (a^2 + b^2 = c^2) to give us the accurate answer, but to do that we use a variable that represents any number and  eliminates the fraction 1/2. plugging in the variable n, and equaling any number, such as 2, the 1/2 cancels to n, and the hypotenuse to a 2n. Next we plug those numbers and variables into the Pythagorean Theorem and solve. we are left with n radical 3; giving us our last side value.

45, 45, 90 Degree Triangle 

In a  90, 90, 90, 90 degrees square all sides are equaled to one; so in order to get a 45, 45, 90 degree triangle, we must divide the square into two triangles in half, diagonally. when dividing the square, we were left with two sides still equaling to 1, while the hypotenuse is undetermined so far. So we must use the Pythagorean Theorem (a^2 + b^2 = c^2) to solve for the "r" value; leaving us with radical 2. Once again the input of the "n" only means that it can be any number, as for the value does not always equal 1.

Inquiry Activity Reflection: 

1) Something I never noticed before about special right triangles is that the values that come with the sides and angles make sense in that they are not just random values, but are calculated to make a special right triangle.

2) Being able to derive these patterns myself aids in my learning because if i get lost and somehow forget my derivatives, i can use this method of finding the values of both the angles and the sides, by either using Pythagorean Theorem or simply remembering the steps and recalling the values.

ID #1: Unit N Concept 7: Knowing all Degrees and Radians Around the Unit Circle, Knowing all the Ordered Pairs Around the Unit Circle, Understanding and Applying ASTC to the Unit Circle

INQUIRY ACTIVITY SUMMARY

Many parts of this triangle have a specific label based on the Special Right Triangles rule. The first rule includes that the hypotenuse must equal 1 (ALWAYS). According to the 30, 60, 90 angles rules, the adjacent side to 30 (the side labeled x) has the value of x radical 3; the opposite side from the 30 degree angle (the side labeled y) has the value of 1/2; the hypotenuse of a 30 degree angle (the side labeled r) has the value of 2x . If this triangle were to be put in the unit circle across in the x- axis the origin (0,0) would be (radical 3/ 2, 0), and right above the previous point would be (radical 3/2, 1/2).

Along with the previous triangle, may rules apply to the sides and angles of the 45 degree triangle. The same rule about the hypotenuse applies to this triangle that it must equal 1 (labeled r) with the value of x radical 2. The adjacent side (labeled x) has the value of radical 2/2. the side opposite (labeled x) was valued the same of radical 2/2. The values are found when equaling the x's to the hypotenuse (x radical 2) leaving both of the sides equaling radical 2/2. If this triangle was inserted onto the unit circle, across from the origin (0,0) would lie the point (radical 2/2, 0), and above the previous point would be the point (radical 2/2, radical 2/2).

Just as the previous triangles, this 60 degree triangle includes rules to its sides and angles. The rule about the hypotenuse (labeled r) applies to this triangle with the value of 2x (the same as the 30 degree triangle). The side adjacent to the hypotenuse (labeled x) has the value of 1/2, and opposite from that side (labeled y) is valued radical 3/2. If this triangle were to be placed into the unit circle, across from the origin (0,0) on the x-axis would be point (1/2, 0), and right above would be point (1/2, radical 3/ 2).

                                                                                                               
All depending on where the ASTC is found on the circle, we will know which sin, cos, or cot is positive or negative. The ASTC starts from up right to the bottom from the left. Quadrant A (I) has ALL positive and NONE negative. Quadrant S (II), has csc/sin positive, and sec/cos, tan/cot negative. In quadrant T (III) tan/cot is positive, and csc/sin, cos/sec negative. Finally quadrant C (IV) has cos/sec positive and csc/sin, tan/cot negative. 

1. The coolest thing I learned from this activity was realizing that all along there were triangle involved that made the unit circle easier to understand. 
2. This activity will help me in this unit because it allowed me to find what i need to find using prior knowledge instead of just memorizing the entire unit circle. 
3. Something I never realized before about special right triangles and the unit circle is that they are connected through the point and radians values.

RWA #1: Unit M Concept 4:Graphing parabolas given equation

RWA Conic Section: Parabola

1. Definition: "The set of all points the same distance from a point and a line"

2.Properties:  

•                    Algebraically:  

•                            Formula/Equation: 

("www.mathwords.com")

•                        Graphically: The graph consist of many key factors that belong to a parabola. The center of the parabola is called the VERTEX and it focuses around the FOCUS. The AXIS OF SYMMETRY goes straight through the parabola, which is also perpendicular to the DIRECTRIX.  

•  The directrix is the line outside the parabola that you can identify by subtracting the value from the vertex (h or k) by p. The notation for the directrix is x=# and y=#. "p" is a point that is determined by setting the term outside of the non-squared portion of the formula equal to 4p and solving. "p" is the value that determines how far away the focus and the directrix are from the vertex.To put the equation into standard form,  complete the square and be sure that only one term is squared. If the x term of the equation is squared and the value of p is negative the graph will go down and if the value of p is positive the graph will go up. In contrast,  if the y term is squared and the value of p is positive the graph will go right, if p is negative the graph will go left. The vertex is an ordered pair that is the center of the parabola. The values of h,k is the vertex as an ordered pair. Remember that h always goes with x and y always goes with k. 
• The axis of symmetry is a line that lies in the middle of the parabola that is written as x=#  or y=#.The value for the axis of symmetry is the number from the vertex that isn't changing.  The focus is an ordered pair found inside the parabola that is identified by adding the value of p with the term that isn't changing (h or k).
• The distance that the focus is from the vertex determines how skinny or how fat the parabola is. In addition, the distance from the focus to any point on the parabola to the directrix is always equal and that is called the eccentricity. A parabola's eccentricity is equal to 1 which is why the two distances are equal.

4. RWA: Parabolic Antennas 

The  Parabolic Antennas are constructed as parabolas because that allows any sound and waves to hit it faster and easier than as a regular square would. he shape of he Parabola is used to attract waves. 

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