BQ#7: Unit V: Derivatives

1. Explain in detail where the formula for the different quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc).

The formula of the different quotient comes from the plots of a graph. Since the formula helps us find the slope of a straight line, we use the slope formula; but the different quotient helps you find the slope of any curve or line at any single point. Both the different quotient and the slope formula is simply the change in "y" divided by the change in "x"; but the difference is that the slope formula's "y" is used as the y-axis, while in the different quotient the change of "y" is called f(x). The formula for the different quotient helps us find the derivative. The derivative is the slope of the tangent line to a graph f(x), and is usually denoted f '(x). A tangent line is just a line that touches the graph once. A function can have different derivatives at different values of "x". To solve the different quotient, you first have to find f(x+h), then simplify f(x+h)-f(x), you then divide the result from step 2 by "h" (or multiply the result by 1/h). 

Slope Formula

Difference Quotient

Tangent Line

Secant Line

References:
Mrs. Kirch's Unit V SSS Packet
http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg
http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/img002.GIF
http://clas.sa.ucsb.edu/staff/lee/Tangent%20and%20Derivative.gif
http://clas.sa.ucsb.edu/staff/lee/Secant%20and%20Tangent%20lines.gif

BQ#6: Unit U: Continuities and Limits

1. What is continuity? What is discontinuity?
A continuous function is predictable (goes where you think it should go).  It also has no holes, no jumps, and it can be drawn without lifting your pencil/no breaks in the graph. A discontinuity is made up of two families, which consist of removable discontinuities and non-removable discontinuities. A removable discontinuity has a point of discontinuity, which is also known as having a hole. A non-removable discontinuity has jumps, is unpredictable, you lift your pencil when you draw it, has vertical asymptotes, has different limits from values. A jump discontinuity has the right and left different. An oscillating behavior is wiggly. An infinite discontinuity, which is also known as unbounded behavior, occurs where there is vertical asymptote, which increases or decreases without bound of infinity or negative infinity.

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 at a point of discontinuity, which is also known as a removable discontinuity or hole. A limit exists as long as you reach the same height from both the left and right. In order for a limit to exist, both the right hand limit and left hand limit must be the same, so the limit does not exist at a jump discontinuity (left&right different), oscillating (doesn't have a certain point/wiggly), and infinite discontinuity (has a vertical asymptote). A limit is the intended height, but a value is the actual height. 

3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically: You set up a table to help evaluate limits. You first draw a table and you add 1/10 on one side, and subtract 1/10 to the other side; with this, you slower get closer to the middle number.

Graphically: To help evaluate limits, you can draw a graph to help you pain a picture to where your limits are and it helps you explains when why you get undefined for your values. Or, you can use your fingers to help you. You start on one side and the other finger on the other side, and you try to make them come together to see if it's a removable or non-removable discontinuity.
Algebraically: To help evaluate limits, you do problem solving by using three methods: direct substitution method, factoring method, and conjugate method. When you use the direct substitution method, all you do is plug in the number that x approaches and you're done. But if you get 0/0 or indeterminate form, you keep on going to the other two methods. When you use the factoring method, you factor the denominator and numerator, and you then cancel what you can, but remember to not combine when you can cancel something from it, but remember, to always try direct substitution first! When using the conjugate method, you multiply to where the radical is and multiply by "one" and after you cancel, you get your answer.

References:
Mrs. Kirch's Unit U SSS Packet

BQ#4: Unit T Concept 3: Difference Between a Tangent and Cotangent Graph

4. Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill? Use unit circle ratios to explain.
Asymptotes are based on the Unit Circle, so since tangent is sin/cos and cotangent is cos/sin, their denominators differs and so do their asymptotes. For tangent, cosine is its denominator, so we have to see where cosine is zero (0,1) (0,-1) and that leads to undefined which is an asymptote. For cotangent, sine is its denominator, so we have to see where sine is zero (1,0) (-1,0) and that will lead for your answer to be undefines which is an asymptote. Both tangent and cotangent are positive in the first and third quadrant, and negative in the second and fourth quadrant. Because we know this, we understand that in a "normal" tangent graph it goes uphill because of its asymptotes, at pi/2 or 90 degrees, and 3pi/2 or 270 degrees. On the other hand, a "normal" cotangent graph goes downhill because of its asymptotes at zero and pi or 180 degrees.

BQ#3: Unit T Concepts 1-3: Relations to Sine and Cosine Graphs

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

Since Sine (y/r) and Cosine (x/r) have “r” as their denominator which equals one; they do not have to follow the same rules as Cot, Tan, Sec, and Csc because they will not be able to get zero as their denominator, so they will never get undefined. Cot, Tan, Sec, and Csc have “x” or “y” as their denominator; and because of this, they are able to get undefined as their answer because they are able to get zero as their denominator. Cotangent and Cosecant both have “y” as their denominator, so when their denominator equals zero, they will become undefined (1,0) (-1,0). Tangent and Secant both have “x” as their denominator, so when their denominator equals zero, they will become undefined (0,1) (0,-1). Because those trig functions can get undefined, they will have asymptotes, which are the boundaries (line) that can never be touched even they get really close to them. On the other hand, Sine and Cosine don’t play by these rules, because their denominators will always be one, so they can never be undefined.

Also, since the graph goes on forever (cyclical), we show a period so it can fit in the window so we can see how the graph develops in a rotation. We also don’t have to use radians as our x-axis, but we like to use it because it helps us see the whole graph, while if we use degrees, we can’t see the whole graph. When the graph starts to go up the x-axis, it shows that it is positive in that designated part of the Unit Circle, and if the graph goes below the negative x-axis, then it demonstrates that it is negative in the designated part of the Unit Circle.

BQ#5: Unit T Concepts 1-3: Graphing Sine, Cosine, Secant, Cosecant, Tangent, and Cotangent

5. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use the unit circle ratios to explain.
Sine (y/r) and Cosine (x/r) do not have asymptotes, because their denominators will never equal to zero, since their denominator is "r", which equals to one. Since Cosecant (r/y), Secant (r/x), Tangent (y/x), and Cotangent (x/y) don't have "r" as their denominator, they are able to get zero as their denominator which will be undefined which turns into an asymptote. Since Secant and Tangent both have the asymptote in the same location; the points they share is (0,1) and (0,-1). Cosecant and Cotangent both have the asymptote in the same location; the points they share is (1,0) and (-1,0).

BQ#2: Unit T Concept Intro: Trigonometric Graphs

2. How do the trig graphs relate to the Unit Circle?
The trig graphs relate to the Unit Circle in that it helps define where and why the graph is negative or positive. For example, since Sine is positive in the first quadrant and the second quadrant, the graph will be up and positive when the degrees is between those two quadrants. But since Sine is negative in the third and fourth quadrant, the graph will do down and become negative. In Cosine's case, it is positive in the first and fourth quadrant, and it is negative in the second and third quadrant; so Cosine's graph will go up and be positive because it's in the first quadrant, then go down to negative because it is in the second and third quadrant, and then go up to be positive when it goes to the fourth quadrant. For Tangent/Cotangent, it is positive in the first and third quadrant, and negative in the second and fourth quadrant, so the graph will go up to be positive in the first quadrant, then go down to negative in the second quadrant, then go up again and be positive in the third quadrant, and then go down to be negative in the fourth quadrant.

A.) Period?- Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
A period is when the graph cycles one time through their cycle. The period for Sine and Cosine is 2pi, whereas the period for Tangent and Cotangent is pi, because that's how long it takes them to repeat the pattern.

B.) Amplitude?- How does the fact that Sine and Cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?
"Sine and Cosine graphs have an amplitude. Amplitudes are half the distance between the highest and lowest points on the graph. They can be found by looking at the equation at the values of the absolute value of 'a'." The fact that Sine and Cosine have amplitudes of one relate to what we know about the Unit Circle is that Sine and Cosine can't be greater than 1 or less than -1. We can explain this since Sine is "y/r" and Cosine is "x/r", and "r=1", so it creates a restriction since that is what an amplitude is.

References:
Unit T SSS Packet

Reflection #1: Unit Q: Verifying Trig Identities

1. The meaning of verifying a trig identity is to prove that the equation is true and we do this by showing that both sides are equal to each other.

2. Tips and tricks that I found helpful are to try to always change the identities to sin(x) and cos(x) when possible. Another tip I would use is to try to convert the equation into a Pythagorean identity when it applies. Another important tip is to NEVER TOUCH THE RIGHT SIDE OF THE EQUATION! Always start with the left because you are verifying if it does equal to the right side. We love having our equations to have squares, so if your answers can not go further, you then can square both sides to get a Pythagorean identity out of it. A very important tip to always do when squaring both sides is to check your answer in the end to make sure it works, because the answer can be extraneous. A trick I would give you is to use the "u-substitution" because it is helpful when factoring trigonometric expressions. And it makes it easier during the factoring process because after you factor, you just plug everything back in! So easy!

3. My thought process through the steps when verifying trig identities is to look at the equation in general. I first check to see if we want the left and right side of the equation to equal to each other or if we are looking for a specific answer (degrees/radians). If we are looking for an answer, than the first thing I would do is to try to change the equation into sin(x) and cos(x). I would then look for a Pythagorean identity in order to have the same identity throughout the problem. I would also check to see what quadrants it lands on because it helps us determine where our answer will lay in. I would also see if I can use the "u-substitution" or "m-substitution" to make my life easier. If we were looking to see if the equations on both sides equal to each other, I would start with the left side and leave the right side alone and then bring it back down when in the end. I will also see if I can separate the equation, get a common denominator, FOIL, combine, cancel, replace, convert, or change identities. I need to have my options open to how I want to approach the problem. I also use my right side equation as guidance to where I want to be. My right side equation can tell me if a should combine and cancel or change and convert to a different Pythagorean identity. What I recommend to do is to look at the equation first and analyze the starting point and then see where we want to end. In my opinion, that is the key. Analyze the equation before making a move.