BQ #7: Unit V Concept- Derivatives and The Area Problem

1. Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc.) Your post must include text and some sort of media to support your writing.

-The difference quotient is used in higher level math, aka Calculus. In Calculus, the difference quotient is used when we talk about the concept: derivatives. We find the difference quotient by finding f(x+h), the simplifying f(x+h)-f(x) and at the end divide everything by h. When we divide by h we may need to take out a h from the numerator. The difference quotient to us was just this formula in the beginning but now we know that the difference quotient is also known as finding the slope of the tangent line to a graph. So the answer we got from the difference quotient formula is actually the slope of a tangent line aka the derivative. But before we can call it a tangent line we must change it from a secant line to tangent line. The picture below is an example of a secant line because it touches the graph twice.

www.sagemath.org

We change it from a secant line to a tangent line by plugging in zero whereever there is an h. So all the h's cancel out and makes it a tangent line. Once it is a tangent line we notice in the picture below that a tangent line only tocuhes the graph as we sae above that a secant line touches it twice. Now we have the derivative. 

www.education.com

BQ #6: Unit U Concepts 1-7

1. What is continuity? What is discontinuity?

-Continuity has four different meanings. One of those meanings is that it is a predictable function which means it goes where it should go. Another thing is that it is a function that has no breaks, no jumps, or no holes in it, it must be continuous. The third thing is that we must be able to draw it without lifting our pencial. Lastly, continuity means that the value and limit are the same.

Discontinuity on the other hand is different. This means that the graph is not continuous due to the different types of discontinuities. (In order from left to right) Point, Jump, oscillating, and infinite discontinuity.

The first one on this picture is called Point discontinuity which is a removable discontinuiy. This one is also known as a hole in the function. The second on in the picture is called jump discontinuity. This means that there is different left and right numbers. The third one is called infinite disconinity which means there is a vertical asymptote in the graph which causes unbounded behavior. The last one in the picture is called oscillating behavior which is just a wiggly graph which menas there is not one single set point. 

2. What is a limit? When does a limit exisit? When does a limit not exsist? What is the difference between  a limit and a value?

- A Limit is the intended height of a function. The limit exisit when there is a continuous function. That means at all the discontunitues it only exisits at the removable one which is the point disconintuites. The limit of a function does not exisit when it is a nonremovable discontinuty due to the fact that there is different left and rights, unbounded behavior, or oscillating behavior. The difference between the limit and the value is that the limit is the intended height of the function where as the value is the actual height of the function. They do not always have to be the same but they can.

3. How do we evaluate limits numerically, graphically, amd algebracially?

- We evaluate limits numerically by setting up a table with the correct numbers. We plug in the function in our calculator and we just use the trace buttom to find the missing values. While doing so, we can see that the numbers are approaching closer and closer to some number but sometimes it can not be reaches. We write it out as " The limit as x approaches 'a number' of f(x) is equal to L'".

We evaluate limits graphically by getting two fingers or pencials and we put one of the left side of the function and one on the right, and then we just see if our fingers meet. If they met, then that means the limit exisits. If they did not meet, we must explain why and give a reason behind that. The reason must come from the disconintiuities.

We evaluate limits algebracially with three different ways. The first way we always try is direct substitution. This means we basically just plug the number into x and see what we get. If we get a numerical answer, we are done. If we get 0/# it is zero and we are done. If we get #/0 it is undefined and we are done. But if we get 0/0 this is indeterminate form which means we must try another way. The  next way we must try is the dividing out/ factoring method. This means we factor both the numerator and denominator and cancel terms to remove the zero in the denominator. After that we use direct substitution. If we cannot see anything that might factor out, we must use the last way which is rationalizing/ conjugate method. If it is a fraction we multiply the top and bottom by a conjugate and it should help us out. After that we are able to use direct substitution.

BQ #4 Unit T- Concept 3

4. Why is a "normal" tangent graph uphill, but a "normal" cotangent graph is downhill? Use unit circle ratios to explain.

From Mrs. Kirch's SSS packet

tangent: Quadrant 1 is postive, 2 is negative, 3 is postive, and 4 is negative. The ratio for tangent is Tan(x)= y/x. So as you can see when cosine equal zero that means that there is an asymptote. There is an asymptote because when it is zero, it is undefined which means there is an asymptote where ever x equals zero. Tangent has asymptotes at pi/2 and 3pi/2 because that is where x equals zero. So as you can see in the picture above pi/2 is before quadrant 2 which is negative, so the graph will start at the bottom and work its way up because the quadrant before 3pi/2 is postive.

cotangent: The quadrant have the same signs as tangent but the ratio for cotangent is cot(x)=x/y. So as you can see, now it is when sine equals zero where their is an asymptote. There is an asymptote at zero and pi for cotangent because that is where y equals zero. So as you can see in the picture above, the quadrant after 0 is postive and the quadrant before pi is negative. So the graph will start on the top and then go downhill.

So tangent goes uphill and cotangent goes downhill because of the location of their asymptotes.

BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.

a. Tangent?

-Sine and cosine relate to cosine because of their signs on the unit cicrle. The ratio for tangent is tan(x)=Sin(x0/cos(x). So since the tangent ratio includes sine and cosine, their signs affect the tangent graph. So if sine is postive and cosine is negative, then the tangent group will be negative and go downhill. If sine and cosine is postive then tangent will be postive and going uphill.

b. Cotangent?

-Cotangent is just the reciporcal of tangent. Cotangent's ratio is cot(x)=cos(x)/sin(x). So cosine and sine's signs depend on what way the graph is going. So if sin is negative and cosine is postive then the graph will go downhill because it will make cotangent negative. We get these signs from the unit circle.  Plus it has diffferent asymptotes than tangent. 

c. Secant? 

-For secant, sine does not affect this graph at all. The only one that affects it is cosine. Cosine affects this graph because the ratio for secant is sec(x)=1/cos(x). So this means that cosine determine where the asymptotes go because cosine can equal 0. If cosine is 0 then it is undefined which means there are asymptotes. So cosine affectets secant because of the asymptotes.

d. Cosecant?

                            

                                     

-For cosecant, cosine does not affect this graph at all. The only one that affects it is sine. Sine affects this graph because the ratio for cosecant is csc(x)=1/sin(x). So this means that sine determines where the asymptotes go because sine can equal 0. If sine is 0 then it is undefined which means there are asymptotes. So sine affects cosecant because of the asymptotes. 

BQ #5: Unit T Concepts 1-3

Mrs. Kirch's SSS packet

5. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use Unit Circle ratios to explain. 

-Sine and Cosine are the only two trig functions that do NOT have asymptotes. The reason behind this is because asymptotes happen when you get undefined. The only way you can get undefined is when you divide by zero. According the the trig ratios, sine is y/r and cosine is x/r. So as you can see, they do not divide by zero because r is equal to one for the Unit Cicrle. Cosecant is r/y and cotangent is x/y so they always have the same asymptotes because sine can equal zero. Secant is r/x and tangent is y/x so they have the same asymptote because they both divide by cosine, and cosine can be zero so that means there is an asymptote present. 

BQ #2- Unit T Concept Intro

From Mrs. Kirch's awesome SSS packet 

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

-Trig Graphs relate to the Unit Circle because they are basically the same thing. We just unwrap the unit cirlce and make it a line and it turns into a trig graph. It has the same pie values in the same four quadrants. Another reason why they relate is by the signs. The four quadrants stay the same as well. All the signs for each trig functions are the same, so we must remember ALL STUDENTS TAKE CALCULUS. As seen in the picture above you can see how the signs correlate with the unit circle.

a. Period? Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

-The period for sine and cosine is 2pie. The reason behind this is because the pattern for sine is postive postive negative negative. The pattern for cosine is postive negative negative postive. It takes 2pie for this pattern to repeat. So this makes it the period because these graphs go on forever and to repeat the period it takes 2pie. You can see a visual of the sine cosine trig graphs below.

The period for tangent and cotangent is pie. The pattern is postive negative postive negative. As you can see, the pattern repeats half way through the graph/ unit circle so that means that it is only pie and not 2pie because it repeats half way through the revelation. You can see a visial of the tangent trig graph below.

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

-The fact that sine and cosine have amplitudes of one relate to the unit circle big time. On the Unit Circle we remember that sine and cosine could not be smaller than -1 and larger than 1. It relates to this because their amplitudes must be 1 as well. If it is greater than 1 or less than 1 it is considered undefined just like it was on the Unit Circle.

Reflection #1: Unit Q- Verifying Trig Functions

1. In this Unit, Unit Q, we learned how to verify a trig function. This actually means that we must prove that both sides are equal to each other because we know our answer already. So we must use our identities, which we all memorized to make the left side equal to the right side. WE NEVER TOUCH THE RIGHT SIDE. We are just verifying that is proven to be right. 

2. There are many tips and tricks that I have found helpful in this unit to help me understand it way more. One huge tip that I learned that if you can, change EVERYTHING into sine and cosine. By doing this it will make our lives easier because they will either cancel out or be equal to one. Another tip is to always work on the complicated side first that has a lot going on. By doing this first we are able to make it easier and get the hard side out of the way. A third tip is if you can, look for the greatest common factor. This will allow us to make our problems simplier and closer to our answer. The biggest tip I can give is to ALWAYS identify what you did in each step and make it neat. This will help you see where you went wrong if you did not get the answer right. It will also look nicer as well. 

3. The first step in my thought process is to see which identies I have in my problem. I always look to see if I have sine and cosine together, secant and tangent together, or cosecants and cotangents together because they pair up well with their idenities. Next I look for to see if anything cancel or if I have a greatest common factor so I can simplify it even more. If none of that works, the next step I do is look for a common denominator. Once I have done one or all of those steps, usually the identies work out the way I want it to and give me the same answer as the right side. Concepts 1 and 5 are tricky, but if you practice it will all make sense. Just memorize the identities. 

SP #7: Unit Q Concept 2

This SP#7 was made in collaboration with Kelsea Del Campo please visit their awesome blog by clicking here.

Using Identities:

Using SOCAHTOA: 

 In this student problem me and my partner made our own example from Unit Q Concept 2. In this first picture we showed how to find the values using Ratio Identities, Reciprocal Identities, and Pythagorean Identities. In the second picture we showed how to find the values usng SOCAHTOA. As you can see, we can find the values using both ways. In the first picture, using identities, you can see that we found all our values using different Identities and just substituting them into the problem. We used one identity to find another and so on. For the second picture, using SOCAHTOA, we just used the Unit Circle ratios from the previous unit. By doing these two pictures we have proven that identities can be associated with SOCAHTOA. 

I/D #3: Unit Q Concept 1- Using Fundamental Idenities to Simplify or Verify Expressions

1. An "identitiy" are proven facts and formulas that are ALWAYS true. The Pythagorean Theorem is a type of "identitiy" because we get proven facts and formulas that end up being true for any numbers we plug into it. With using the Pythagorean Theorem we use for this unit we use x, y, and r. We use these instead of a, b, and c because on a graph it when we graph it becomes x, y, and r.

As you see in the picture above all we had to do to get the Pythagorean Theorem equal to 1 is divide by r^2. Once we had r^2 it became equal to 1. From dividing by r^2 I noticed that we got x/r which is the ratio for cosine we used for the unit circle. I also noticed that we got y/r which is the ratio for sine we used for the unit circle. If we subsitiute those two for cos and sin we get cos^2x + sin^2x = 1. We can conclude that cos^2x + sin^2x = 1 beigins with the Pythagorean Theorem. It is referred to the Pythagorean Ideinity because we use the Pythagorean Theorem to get it.

As you can see in the picture above, I chose one of the "Magic 3" ordered pairs from the unit circle to show that it is true. The ordred pair used was the 60* angle one which is (1/2, rad3/2).

2.

The picture above shows that how to derive the identity with Secant and Tangent. First we had to divde everything by cos^2ø. Then that would cancel out the cos^2ø to make that be 1. Then we got sin^2ø/cos^2ø which we subtitute with tan^2ø. Lastly we have 1/cos^2ø and we substitute that with sec^2ø. Our final answer will look like tan^2ø + 1 = sec^2ø. 

The picture above shows that how to derive the idenity Cosecant and Cotangent. First we had to divdie everything by sin^2ø. Then that would cancel and make it be 1. After we got cos^2ø/sin^2ø which we substitute with cot^2ø. Lastly we have 1/sin^2ø which turns into csc^2ø. Our final answer will look like 1 + cot^2ø = csc^2ø.

WPP#13-14: Unit P: Concept 6 and 7

Please see my WPP 13-14, made in callboration with Kelsea Del Campo, by visiting their blog here. Also be sure to check out the other awesome posts on their blog!

BQ #1: Unit P Concept 2 and 4: Law of Sines SSA and Oblique Triangles

2. Law of Sines- Side Side Angle (SSA) is an ambigious case not like AAS or ASA. When dealing with SSA the three angles are not all known as easily as the others because we only know ONE angle out of the three. It is amigious because it can be three different types: one triangle, two triangles, or no triangle at all.

One Triangle 

As seen in the picture above there is only one possible triangle for this problem. We know there is not a second triangle because angle A and Angle C add up to 325.9 which is way past 180 degrees so therefore there is no second triangle because it is greater than 180.

Two Traingles 

As seen in the picture above this example has two possible triangles. We know there are two triangles because the law of sines means there is one angle in the first quadrant and another angle in the second quadrant. We find the second angle (the prime angle) by subtracting the first angle we got by 180.

No Triangle 

As seen in the picture above this example has no possible triangles. We know there are no possible triangles because once we used the law of sines with SSA, we got sinC: 1.75 and we know from the previous unit that sin can not be greater than 1 so therefore that leads to no solution. Another reason we know a triangle has no solution is when there is more than one obtuse angle.

4. Area Formulas- The area of an oblique triangle is derived from the formula for the area of a trianlge which is A=1/2bh. It area of an oblique triangle is one-half of the product of two sides and the sine of the angle the problem gives you. So basically the three types of equations can be A=1/2bcSinA, A=1/2acSinB, and A=1/2abSinC. It relates to the formula we are familar with by substituting in our h in the normal equation with the a side and sine of an angle given. We just have to make sure when we have our equation, all the letters are different.

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

My new puppy Lady 

A. From a dog named Lady on the grass, the angle of elevation to the top of a tree is 22*25'. If the base of the tree is 625 feet from Lady, how high is the tree? (to the nearest foot). 

B. Lady the dog is just about to dig down underground. She estimates the angle of depression from where she is now to the bone in the dirt to be 35*. Lady knows she is 250 feet higher than the bone, how long is the path she will dig? (to the nearest foot).

I/D #2: Unit O Concept 7 & 8- Special Right Triangles

Inquiry Activity Summary 

1.  We get a 30-60-90 triangle by cutting an equilateral triangle in half. We start off by using the Pythagorean theorem, which is a^2 + b^2 = c^2. By using this we know that our a is going to be 1 and our b is still unknown. Our c is going to be 2 as seen in the picture above.  Since we know our a and c we can plug those numbers into the Pythagorean theorem for our b. As seen in the picture above after plugging in those two values we find our b which is rad3.

As seen in this picture above we found a pattern with the 30-60-90 triangles. We added a variable at the end of each value side because we noticed that it can work for any number possible. We just start off with n to get a, then times it by rad3 to get b, and by 2 to get c. We multiplied each variable by 2 to get rid of the fractions so it can be simpler as well. 

2. We get a 45-45-90 triangle by drawing a slanted line through a square. We start off using the Pythagorean theorem, which is a^2 + b^2 = c^2. Since it is a square all sides are the same. For this example our sides are going to be 1. So we must plug in 1 into a and b to find c. As seen in the picture above, the work shows that our c is rad2.

As seen in this other picture above, we found a pattern with the 45-45-90 triangles. We added a variable, which is n to each value because the sides will not always be 1. This can work for any number because it is derived from the square. We put “n” in the pattern because it can be any number if it follows the 45-45-90 pattern. 

Inquiry Activity Reflection 

1. Something I never noticed before about special right triangles is where these patterns for them actually came from. After doing this activity I found out the origins of the patterns, which makes sense to me now.  I never knew the 45-45-90 came from a square and the 30-60-90 came from an equilateral triangle.

2. Being able to derive these patterns myself aids to my learning because it kind of ties everything together now. It even ties in together to unit circle because of the vales of the triangles. It also helps me remember the patterns now thanks to the Pythagorean theorem.

I/D #1: Unit N Concept 7- Unit Circle Derivation

Inquiry Activity Summary 
                              
1. The 30* right triangle is one of the special types of right triangles. This triangle has a special way to find the lengths of all three sides. The first thing we need to do it memorize the equations we need to get each side. The formula for the hypotenuse is 2x. The formula for the side where 30* is opening towards is x. The formula for the side where 60* is opening towards is xrad3. For this unit the hypotenuse is r and r is ALWAYS going to be 1. So that is our starting point to find the other two sides. Since 1 is the length of where the 2x formula is we need to find a way to find just x for the other side. We do that by just diving 1 by 2 and that will give us x. For the last side we just plug in x into the xrad3 and it will give us rad3/2. This is the first 30 degree angle in the unit circle.

2. The 45* right triangle is another type of special right triangle. This triangle has a different way of finding the sides of the triangle. The first thing we need to do for this one is memorize different formulas. This one is easier because the two 45* angles sides are both x. The 90* one is xrad2. The hypotenuse is 1 again because of the value of r. So we need to find a way of getting 1 into xrad2. Once you figure out how to get the 1 into xrad2 form you shoukld be able to find x which is rad2/2. This is the first  45* angle in the unit circle.

3. The 60* right triangle is another special right triangle but it is very similar to the 30 degrees triangle. This first thing we need to do is memorize the equations which is 2x for the side where 90* is opening up. The side where 60* is opening up it is xrad3. The side where 30* is opening up is x. Since they gave us 1 for the hypotenuse again we just work our way back wards like we did in the first questions. That will give us 1/2 for x and rad3/2 for xrad3. This is the 60* for the uncle cirlce, which means the second 30*. 

4. This activity helped me derive the unit circle in many different ways. After completing this activity I recognized these numbers from somewhere. The place I remember them from are from the unit circle! These four triangles make up the first quadrant of the unit circle. It made me realize where the numbers of the unit circle came from. 

5. The quadrant this activity was drawn in was quadrant 1. The values do not change when they go around the unit circle. As seen in the picture above, it is all the same numbers according to the reference angle of 30*, 45*, or 60*. If it is the 30* reference angle it is always going to be (rad3/2, 1/2). If it is 45* it is always going to be (rad2/2, rad2/2). If it is the 60* reference angle it is always going to be (1/2, rad3/2). They just change signs when they move quadrants. In quadrant 1 they are all postive. When they are in quadrant 2 the x value is negative and y value is potive. In the third quadrant the x value is negative and the y value is negative. In the 4th quadrant the x value is postive and the y value is negative. Same numbers just different signs. 

Inquiry Activity Reflection 

1. The coolest thing I learned in this activity was where all these numbers came from. I thought it was really cool to know how to get them because in my last class we just had to memorize where they went not where they came from. 
2. This activity will help me in this unit because it is bascially telling me how to get the first quadrant of the unit circle. This will help because this unit is basically all about the unit circle. If we do not know the unit circle then we will do terrible. The unit circle helps us find everything and makes everything easier to find. 
3. Something I never realized before about the special right triangles and the unit circle is that they are both related. If we know the special right triangles we know the unit circle. If we know the unit circle then we know the speical right triangles. I did not know they connected somehow. 

RWA #1: Unit M Concepts 4-6- Conic Sections in Real Life

http://www.purplemath.com/modules/ellipse.htm

1. The mathimatical defintiion of an ellipse is "the set of all points such that the sum of the distance from two points is a constant." 

2.           The first way to descibe an ellipse is algebraically. The equation of an ellipse looks something like this    .  From this information we will be able to find the center of the graph which is always (h, k). To find the vertices and co-vertices from the equation we must find a nd b. We find a and b from the equation, all we need to do is sqaure root a^2 and b^2. By finding a we know how far each vertices are away from the center. By finding B we know how far each co-vertices are from the center. To find c we need to use a^2-b^2=c^2. After sqaure rooting that answer we now have found c agebrically and we know how far away the 2 foci are from the center. Lastly if the major axis is vertical we know a, which is the bigger number will be under y^2. If the major axis is horizontalm a will be under x^2. 

               Another way to describe an ellipse is graphically. First we find the center of the ellipse by just looking at the equation. After that we can identify the major axis by looking at the x^2 and y^2. If a is under the x^2 then it is horinzontal. If a is under the y^2 then it is vertical. Next we find a and b to be able to find the vertices and co-vertices. For plotting the vertices we move a times to the left and right if a is under the x^2 and a units up and down if it is under the y^2. For plotting the co-vertices we got b units up and down if b under y^2 and goes left and right when it is under x^2. If the foci of the ellipse are closer to the center, the ellipse will become more ciruclar. If it is farther away from the center, then it will become less circular. We find the eccentricity of an ellipse by doing c/a and it has to be between 0 and 1. 

3.        The real life application I found for ellipse's are the oribits of the planets in our solar system. The orbits of the plants are ellipses because "the Sun is one focus of the ellipse."(http://csep10.phys.utk.edu/astr161/lect/history/kepler.html). So according to this website, the sun is one of the two foci in an ellipse. The other focus is probably another planet. Scientist usually say planets have an eccentric orbit. When they say eccentric orbit they mean far from being circular. So pluto, even though it is not a planet anymore, still oribits around eccentrically. 

           Another thing i found out that the planets are the vertices and co vertices. I say this because the planets are the objects going around like an ellipse so they are the ones touching the ellipse. If a planet is orbiting farther away, then it is on the major axis. It is on the major axis because that is the bigger axis. Once it is rotating and it closer to the center, then it is on the minor axis. It is on the minor axis because it got closer from the major axis. 

4. Work Cited: 

• http://www.purplemath.com/modules/ellipse.htm
• http://csep10.phys.utk.edu/astr161/lect/history/kepler.html
• http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet 
• http://www.lessonpaths.com/learn/i/unit-m-conic-sections-in-real-life/conic-sections-in-real-life