We're all familiar with the difference quotient, but where the heck does it come from and how does that affect derivatives? Origin of Difference Quotient: f(x+h) -f(x) all divided by h
In this graph, we have the visuals to aid us in our understanding of where the difference quotient is derived from. The first point is (x, f(x)). There is delta x (delta = change) distance between the first and second point meaning the second point is (x + delta x, f(x + delta x)). But for this class we will reference to delta x as 'h' so our second point can also be written as (x+h, f(x+h). The line connecting these two points is called the secant line, much different than the tangent line, which only touches the graph once and is depicted below. Now we want to find the slope of the secant line and for that we are going to use our favorite slope-finding formula: the slope formula! The slope formula is m = y2-y1 all divided by x2-x1. When we plug in our two points from the first graph we have f(x + delta x) - f(x) all over x + delta x - x. Or as we substituted h for delta x, we have f(x+h) - f(x) all over x + h -x. We use the additive process in our denominator which cancels out the 2 x's, leaving only h. This brings us to the difference quotient we all know and love: f(x+h) - f(x) divided by the letter h, that's the difference quotient!
This video by Mathbyfives named Difference Quotient.mov (which can be found here) verbally goes through the process in detail and can provide further explanation.
Continuity in this unit means a continuous functions, no jumps, breaks, strange behavior. A continuous function means a continuous graph -- meaning you are able to draw the graph without lifting your pencil from the paper. Sometimes a graph will show up that seems to demonstrate a discontinuity but it is also a change in function -- a change in function does not make it any less continuous. Discontinuities is much more vast and the main concern of this unit. There are two families of discontinuities. There is removable discontinuities such as point discontinuity, also known as a hole. Then there is non-removable discontinuities such as jump discontinuity, oscillating behavior, and infinite discontinuity, also known as unbounded behavior and occurs at vertical asymptotes.
2. Limits (vs. Values)
A limit is the INTENDED height of a function while a value is the ACTUAL height of a function. When the limit and value of a function are the same then the graph is continuous meaning the absence of of one of discontinuities. A limit does exist at continuous graphs and removable discontinuities such as point discontinuity (also known as a hole) because although the value is undefined, the limit INTENDED on reaching that height. A limit does not exist at non-removable discontinuities such as jump discontinuity because of different left/right behavior, oscillating behavior because it does not approach any single value, and infinite discontinuity because of unbounded behavior due to a vertical asymptote.
In this photo we can note the difference between a limit and value. In the first graph, there is a point discontinuity -- the limit exists, but the value is undefined. In the second graph, the limit exists at the point discontinuity as well but the value exists elsewhere (the black dot.) And the last example is one of a jump discontinuity where the limit does not exist but the value exists at one of the one-sided limits.
3. Evaluating Limits through VANG (minus the Verbally -- just reading the limit notation out as "The limit as x approaches 'a number' of f(x) is eqaul to 'L' "
There are three methods to evaluating limits algebraically. -Direct substitution method In which you plug in the number that is approaching the limit of f(x) and solve to see what you get. Your possible answers are numerical, 0/# which is 0, #/0 which is undefined so the limit does not exist because of the presence of a vertical asymptote which means unbounded behavior, or 0/0 which is indeterminate form meaning we must use factoring or rationalizing method. -Dividing out/Factoring method We use this method when we get indeterminate form. In this case, we factor both the numerator and denominator and cancel the common term -- removing the 0 from the fraction. Then we use direct substitution with the simplified expression. BUT always be sure to use direct substitution first!! -Rationalizing/Conjugate method This method is also helpful with a fraction, especially if it has radicals. You multiply by the conjugate of either numerator or denominator, FOIL the conjugates and DO NOT multiply out the non-conjugate part, leave it factored. Something ought to cancel and then you simplify and once again use direct substitution with the simplified expression.
Through this method we use a table that calculates how the limit approaches 'x' from the left and right by first subtracting 1/10 and then adding 1/10.
You may use your calculator by simply plugging in the limit's equation, hit TRACE and then trace to the value you are searching for. OR you can put your finger to the left and to the right of where you want to evaluate the limit --if your fingers meet, that is where the limit exists, if your fingers don't meet then the limit does not exist either due to different left/right behavior (jump discontinuity), it is interrupted by a vertical asymptote which leads to unbounded behavior (infinite discontinuity), or it does not approach any single value (oscillating behavior).
Let's go back to the unit circle - what appears to be the solid foundation of pre-calculus. In the unit circle tangent and cotangent were related by its ratios. Cotangent's ratio was the reciprocal of tangent's ratio. Tangent's ratio is y/x meaning cotangent's ratio would be x/y. Now in this unit we would read that as tangent's ratio being sine/cosine (according to Unit Q's identities) and cotangent's ratio would be cosine/sine. Now what these two graphs have (as well as secant and cosecant) are asymptotes. Remember that this occurs when dividing by 0 leading to undefined. Tangent's asymptotes would occur when cosine equals 0 which would be at pi/2 and 3pi/2. Cotangent's asymptotes would occur when sine equals 0, this would be at 0, pi, and 2pi. (Reflect back on the unit circle, think of these values.) Because their asymptotes are placed in different areas, this is going to affect the direction in which they go. Also remember that these graphs WILL NEVER TOUCH the asymptotes, EVER. They will get VERY close but NEVER TOUCH. Not only do the asymptotes affect its direction, but look at the different colored areas in the images. Red is the first quadrant of the unit circle, where all trig functions are positive. Green is the second quadrant, where tangent and cotangent will be negative (as well as cosine and secant, but not sine or cosecant.) Orange is the third quadrant, where tangent and cotangent will be positive (the rest of the trig functions will be negative.) Blue is the fourth quadrant, where tangent and cotangent will be negative (as well as sine and cosecant but not cosine or secant.) Re-using the images from BQ #3, all thanks again to the wonderful Mrs. Kirch. Her amazing help with these graphs can be found here.
Here we have two trig functions: sine and cosine. Because sine and cosine are always divided by r which is one they will never be divided by 0 hence never being undefined hence never needing an asymptote. There are five possible asymptotes which are depicted as dotted lines at the points of (0,0), (pi/2, 0), (pi, 0), (3pi/2, 0), and (2pi, 0). These points are were asymptotes can be found depending on the trig function selected. The four different colored sections represents one of the four quadrants from ASTC. The red is the first quadrant, green is the second, orange is third, and blue is the fourth. Remember that an asymptote results when a ratio is divided by zero, becoming an undefined ratio.
Tangent's ratio is y/x, meaning that there is a possiblity of an asymptote when x (cosine equals 0). Because of this we know cosine equals 0 at 90* and 270* so pi/2 and 3pi/2, this is where are asymptotes will be located. Now let's look at the first quadrant, remember that in the first quadrant all is positive so it goes in an uphill direction and will NEVER touch the asymptote of pi/2, it simply gets really, really, really close to it. In the second quadrant, both sine and cosine are heading in a downhill direction and in the second quadrant tangent is not positive so it heads downwards but in the third quadrant it is positive. The graph can continue in these two quadrants because there is no asymptote dividing them. And in the fourth quadrant, tangent is not negative hence it's downhill direction.
For cotangent our ratio is x/y, meaning we will have our asymptotes where y =0 (sine) and those locations would be at 0*, 180*, and 360*. On our graph these would be the values of 0, pi/2, 2pi. Because both sine and cosine are positive in the first quadrant as everything is, cotangent is positive as well. Yet in the second quadrant, sine is positive and cosine is negative leading contangent to continue in the negative direction crossing the x-axis when cosine does as well. The first and second quadrants already contain one period of cotangent. A similar process continues off from the asymptote of pi in the third and fourth quadrants. Because sine and cosine are negative in third quadrant, cotangent will be positive and because sine is negative and cosine is positive in the fourth quadrant, cotangent will be negative.
Remember that secant is the reciprocal of cosine's ratio which will be r/x meaning that there will be asymptotes where cosine is equal to 0, similar to tangent's asymptotes. A similar pattern also follows here. In the first quadrant, both sine and cosine are positive and so will secant. But however in the second quadrant, sine and cosine are both negative, as will secant and will continue to be negative because although cosine is positive in the third quadrant, sine is negative and a negative and positive will result in a negative. In the fourth quadrant, both sine and cosine are moving in an uphill direction and secant will also be positive. Once again notice how none of secant's graph is touching the asymptotes and how they develop at the mountains and valleys of cosine's graph.
Cosecant is the inverse of sine meaning its ratio will be r/y. This being said, cosecant will have asymptotes wherever sine equals 0, also similar to cotangent (oooh connections!) In the first quadrant secant will remain positive because all functions are positive in the first quadrant. Yet although cosine is negative and sine is positive in the second quadrant, cosecant will still be positive because it is positive in the sine quadrant of the unit circle. The graph continues into the third and fourth quadrant even after having gone through its period and it's direction is enforced by the unit circle's positive or negative values of the cosecant function. However more importantly because cosecant is the inverse of sine, it relies on the sine graph to be drawn because once again notice, like secant, it is drawn on the mountains and valleys of its corresponding reciprocal of the sine function.
All images made available thanks to the amazing and wonderful Mrs. Kirch on Desmos, you can view and animate as well here.
Image found here. (thank you Google!) Sine and cosine do not have asymptotes because they are always divided by 1 (by "r")
Yet once we move away from being divided by "r" we start reaching asymptote area.
If you reference to the photos you will notice undefined under tangent. Do not forget about their reciprocal ratios of cosecant (r/y), secant (r/x) and tangent (x/y).
Remember that we get asymptotes when our circle ratios equal undefined which results when you divide by 0.
For sine and cosine you will never be dividing by 0 as you may for all other four circle ratios. But because you'll never be dividing by 0, you will never reach undefined, so you'll never have asymptotes!
Sine and Cosine In these images you see how the Unit Circle translates and unravels itself into a sine curve graph. Period: The period for sine and cosine is 2pi because it takes four quadrants (ASTC) to repeat the pattern. Sine's pattern according to ASTC from the unit circle is + + - -
While cosine's pattern is + - - + It merely takes all of the unit circle (which is 2pi at 360* to complete sine and cosine's patterns. Graphs are merely snapshot of the graph, these graphs are infinite as circles are - no ends, no beginning but for the sake of this class we will only be graphing one single period.
Amplitude: Remember that sine equals y/r and cosine equals x/r and that r=1. In the unit circle, the values cannot be bigger or smaller than 1, thus x and y always equal 1. 1 divided by 1 equals *drumroll please* ONE!!! Meanwhile all the other trig functions have asymptotes because they do not have "r" such as tangent and cotangent (y/x and x/y respectively.)
**here we go, but continue scrolling just in case**
Meanwhile with tangent and cotangent..... Period: Tangent and cotangent have a period of just pi. According to ASTC from the Unit Circle, the pattern this trig function has is + - + -. Meaning the pattern is completed in the first two quadrants which is 180* aka pi aka half of the circle. Just remember again to keep in mind that a period is one time through their cycle, their pattern.
This SP #7 was made in collaboration with Anthony Lopez. Please visit the other awesome and non-cheesebuckety posts on their blog by going here. **Also, both Anthony (cheesebucket) and I worked together on this post but someone *cough*Anthony*cough* has poor penmanship so in order to spare you the trouble of trying to decipher it, here it is in my writing.
What is this problem about? This quadrant discusses how to find ALL trig. functions when you are given one trig. functions and quadrant, using the identities from concept 1. It allows us to apply these fundamental identities to more difficult concepts such as this one. What must the viewer pay special attention to in order to understand? You first have to decide the quadrant in which the values lie in by using ASTC quadrants, this will only determine the signs (very important!!) But also please remember you must using reciprocal, ratio, or Pythagorean identities to find the remaining values (don't rely on just SOHCAHTOA, we will use this to find our values.) AND you can only have ONE unknown trig function within solving the identity no matter what identities you use. Also make sure that you properly rationalize all your answers!!
SOHCAHTOA Work (Verification)
We use SOHCAHTOA to verify our answers from the identities we had previously used. We found the hypotenuse value by using the Pythagorean theorem and then simply used the values from the triangle in our ratios to verify that what we had originally gotten with the identities was correct.
Let's go back to oh like maybe three weeks?? (I'm awful with time) where we first learned about the Unit Circle. In the Unit Circle we learned that the ratio for cosine is (x/r) and the ratio of sine is (y/r). Now let's reflect back on Pythagorean Theorem which is a^2 + b^2 = c^2, but in context of the Unit Circle it would be x^2 + y^2 = r^2, but in the Unit Circle r always equaled 1 and in order to make this true and correct, we would divide by r^2 on both sides, leaving us with (x/r)^2 + (y/r)^2 = 1. But wait...(x/r) and (y/r) look similar...wait....isn't that cosine and sine? Why yes, yes it is. But they're squared. Because they are squared, we have resulted towards a Pythagorean Identity. Which cannot be "powered up"or "powered down", because it is a Pythagorean Identity it must always be squared and no power greater and no power less. Alsobecause the Pythagorean Theorem is a proven fact and formula that is always true, it is called an identity. We can prove this by demonstrating one of the "Magic 3" ordered pair from the Unit Circle (30*, 45*, 60*). We'll use 60* now theta of 60* is (1/2, radical 3/2) with 1/2 being x and radical 3/2 being y. Since we want to prove our derivation of x^2 + y^2 = r^2 we're going to use cos^2 +sin^2 = 1, so (1/2)^2 + (radical 3/2) ^2 =1.
In order to derive the remaining two Pythogorean Identities we must divide the whole thing by cos^2x to find the tangent derivation which will lead us to tan^2x + 1 = sec^2x
And then we divide the original by sin^2x to find the cotangent derivation leading us to 1 + cot^2x = csc^2x.
Now you might be wondering how did we know some of this information? Well this information was found from our Unit Q SSS packet, information we should have, say it with me, MEMORIZED!!!
Inquiry Activity Reflection
1. The connections that I see between Units N, O, P, and Q so far are...
The most obvious one would be the role of sine and cosine in this Unit Q from the Unit Circle and its properties which can be seen in the reciprocal identities which are similar to the inverse of these trig. functions from Unit O and P.
In Unit P Concept 3 when using the distance formula for Law of Cosines with SSS or SAS, one of the formulas is a^2 = c^2(sin^2A + cos^2A) - 2bcCosA + b^2 and we know that cos^2A + sin^2A = 1.
2. If I had to describe trignometry in THREE words, they would be...
mentally-draining (this counts as one word), stressful, and insightful.
This WPP was made in collaboration with Sandibel Ramirez. Please visit the other spectacular, non-cheesebuckety posts on their blog by going here! Well Phil has become a lonesome person with no friends after having invested so much time on his record store. BUT then his great friend Scott from freshman years comes back home to San Francisco and they quickly reconnect through spontaneous adventures through great 'ole San Fran.
a) Law of Sines
Problem: Scott and Phil agreed to meet at the record store so Phil can show Scott what he's been up to in life. They realize they're 15 feet apart. Scott heads to the store at an angle of N25E while Phil is heading off at an angle of N55W. Considering that Phil is a slowpoke, how far is he from the record store?
b) Law of Cosines
Problem: Phil and Scott decide to go on a boat ride to relax and just ride the waves out after such a long day. Phil gets to the harbor before Scott however and told him to amount the SS Kirch at 5pm. At 5pm, Scott gets aboard the SS Cheesbucket. Phil's boat, the SS Kirch, is traveling at80 mph and is at a bearing of 220*while Scott's boat is traveling at 115 mph at a bearing of 335*. It takes four hours for both "men" to realize they aren't aboard the same boat (they're huge ships and cell reception isn't exactly great out in the waters.) How far apart are the boats when Phil and Scott finally realize they're aboard different ships?
Remember that bearing always starts from the top, so once we draw both angles according to our bearings, we extend the lines and connect them with a line we'll call 'x'. In order to find our angle we use what we have, if the bearing is 220* and we subtract 180* we're left with 40* and if we subtract 335* from 360* we are left with 115* as our angle. Now Scott's boat, the SS Cheesebucket is traveling at 115mph for 4 hours then it has traveled a distance of 460 miles. Phil's boat, SS Kirch travels at 80 mph for 4 hours so really has traveled 320 miles.
Our triangle is SAS, making sure our included angle is between two sides. Now that we have this information we can plug it into our equation of Law of Cosines, a^2 = b^2 + c^2 - 2bcCosA, which we plug straight into our calculator and then square root to find that the distance between Scott and Phil is a tragic 662.13 miles apart...someone better start swimming now.
In an SSA triangle, our three angles are not carved in stone as AAS or ASA triangles where even though we had two angles, in reality we had all three because all one had to do was add the two and subtract it from 180 to get the third angle's values, meaning there was no ambiguity there.
With SSA we only know one angle's value! (and two sides) There is a possibility however that with only this information that there may be no triangle at all! We always start the problem assuming there is two possible triangles, until you "hit a wall" which will lead to only one possible triangle or no possible triangle. A "wall" can either be if SinA=1.1 or 2.8, something that is not possible. Remember from our unit circle trig functions, we know that sin and cos are not compatible and do not work with numbers greater than one, (-1 less than or equal to sin theta less than or equal to 1), or if the angles add up to be something that is greater than 180. In this triangle, we are solving for angle B and C, and side b. We have information for both angle A and side a, our magic pair, and we have information for side C but not side c, which is our bridge. We set our magic pair and bridge in proportion to one another to solve for the angle of C and use inverse sin to find it. But because we are assuming we have two triangles, and we are using sin (remember that with sin, we have positive answers in the first and second quadrant but our calculator only gives us the positive answer in the first quadrant), we use the reference angle of the first quadrant and subtract it from 180 to find the measurement of angle C'. In this case the angle of C' of 171 added with angle A of 155 was much bigger than 180 meaning only one triangle is possible. Had it been an acute angle like angle C of 9, there could have been a possibility of two triangles.
4. Area Formulas
In a normal triangle where the area could be found with the formula of A = 1/2bh, we know that the perpendicular height of the triangle is h and b is the base. While the oblique triangles (where all sides are different lengths) area is one half of the product of two sides and the sine of their included angle <- very important to realize, it must be in between the two sides.
We know that sinC=h/a from concept 1, so once you mulitply 'a' on both sides h (height) = asinC, and plugging 'h' into that A=1/2bh formula we really have A=1/2b(asinC)
Our dear friend Phil has been struck with the wish to travel around the country, he's been bitten by the wander bug.
a) Phil packs his bags and takes a roadtrip up to Seattle. There he finds Seattle's Space Needle of 605 feet casting a 410 foot shadow at which Phil is standing at the end of. If Phil looks at the top of the building (and avoids getting his eyes burnt by the sun) what is the angle of Phil's eyes to the top of the building? (to the nearest hundredth of a degree AND assume Phil's eyes are 5 feet above ground level since he's a short little fellow.(http://upload.wikimedia.org/wikipedia/commons/3/38/BMX_aloft_and_Space_Needle_03.jpg)
b) Once Phil gets tired of all the Space Needle this and Space Needle that, and the city overall in general, he heads over to Yosemite where Moro Rock lies standing tall at 300 feetwhere he can enjoy the panoramic view. But the mighty rock is slanted at its highest point, Phil estimates the angle of depression from where he is standing (at 300 feet) to the bone-crushing bottom to be 53*. Should Phil fall, like the klutz he is, how long is his path to certain doom? (Round to the nearest foot.)
a) Because it is from Phil's eye level of 5 feet we subtract 5 from 605 giving us 600. Its height is now 600 and the length is 410 which gives us the opposite (600) and adj (410) side. So we are looking for the angle (x). We use tan because of TOA and we use the inverse to undo tan to find our angle so it is tan -1 x (600/410) which is 55.65 degrees.
b) We know that the height is 300ft and the angle of depression to be 53*. We have the opposite side of 300 and we are looking for x which is the hypotenuse. Opposite and hypotenuse, SOH - Sin. So this time it's sine of 53. So sin 53 = 300/x, we multiply by x on both sides which leads to us dividing 300 by sin 53 which equals approximately 758 feet.
First of all, make sure you recognize that this is an equilateral triangle, whose angles sum is 180*.
We cut straight down the triangle because if 180* is the total sum of the triangle's angle we are left with 30* at the top, 60* at the side, and 90* at the bottom of the triangle from 30*.
And since it is an equilateral triangle, all of the sides are of equal length of 1. But since we cut the triangle in half, the bottom side's length is cut to 1/2. With this information, we're able to find the height since the hypotenuse (c) equals 1 is already given to us and we know that one of the values is 1/2, let's call 1/2 (a) for now. If we plug these values into the Pythagorean Theorem of a^2 + b^2 = c^2, **refer to the first picture for clarification** we realize our height is radical 3/ 2.
If we give 1 a variable instead, like n, we are left with 30* - 1/2n, 60* - radical 3/ 2, and 90* - 1n. But we use the variable2n so our general pattern doesn't have fractions. By doing this, we have the pattern we all know, love, and recognize of 30* - n, 60* - n radical 3, 90* - 2n. The ratio of n : 2n is to show the relationships between the sides (and like I mentioned, using the variable 2n only makes deriving easier since it takes those pesty fractions away); it only expands as the ratio remains; such as 2: 4, 3 : 6, 4: 8, etc.
2. 45-45-90 Triangle
This time we are given an equilateral square whose sides' lengths are 1. Now a squares' total angle sum is 360* meaning each corner has an angle of 90*. We cut diagonallyin half because by doing so, we cut 90* corner angle into 2 which gives 45* angles - exactly what we want! We have two sides that equal 1 and we are looking for hypotenuse's value (which is that line by which we cut the square diagonally half by.) In order to find our hypotenuse, we shall use the Pythagorean Theorem of a^2 + b^2 = c^2. Our two equal sides of 1 shall be a and b. **look over photo for clarification** Now by inputting 1 for a and b, our c value equals radical 2.
To find our pattern, we use a variable - which represents any number. Let's use 'n' for now. By using n to represent our two equal sides of 1, our hypotenuse now equals n radical 2; the variable does allow for the relationship to stay consistent and allows us to expand it should the case ever be that the side length equal a number greater than one (this also applies to the 30-60-90 triangle.)
Inquiry Activity Reflection
Something I never noticed before about right triangles is how we are able to find these two triangles in equilateral shapes which just allows the whole derivation to be much easier; but something that I had forgotten and was brought back to my attention was the total angle sum of these equilateral shapes that allows us to create the triangles in the first place by cutting these shapes up either diagonally or straight down the half. Being able to derive these triangles myself aids in my learning because heaven forbid that I should ever have a brain fart on these concepts, I'll be able to fully comprehend special rights triangle and remember how to solve problems such as concept 7-8 by going back to these basics. And also being able to understand why a 45* side value is n and why a 60* side's value is n radical 3 and not because of that cute little memory trick of the 60's being radical times, which can also be considered with the 45-45-90 triangle with the 90* side value of n radical 2 because the 90's were also radical times.