m-the-vector-space-hoperatornamespanleft1-cos-2-t-cos-4-t-cos-6-tright-contains-at-least-two-interes

Question

[M] The vector space $H=\operatorname{Span}\left\{1, \cos ^{2} t, \cos ^{4} t, \cos ^{6} t\right\}$ contains at least two interesting functions that will be used in a later exercise:
$\mathbf{f}(t)=1-8 \cos ^{2} t+8 \cos ^{4} t$
$\mathbf{g}(t)=-1+18 \cos ^{2} t-48 \cos ^{4} t+32 \cos ^{6} t$
Study the graph of $\mathbf{f}$ for $0 \leq t \leq 2 \pi,$ and guess a simple formula for $\mathbf{f}(t) .$ Verify your conjecture by graphing the difference between $1+\mathbf{f}(t)$ and your formula for $\mathbf{f}(t) .$ (Hopefully, you will see the constant function $1 . )$ Repeat for $\mathbf{g} .$

Subhadeepta Sahoo · Accepted Answer

before the party of Talk Ocean 20 each equal to one f report 1.5 unteachable do effort people one point political three also won 104 to 1.5. So they make up a nectar that for BTO Porto 1 25 four or B new To prove that. So I want to be plus cost people left by the Spirit with Costea caused by by three. My Last Sigh Empty Find open. My three looks quit Must cost even after pilot, it is cost court to 53 Bless sci fi Sanctify Field Square This is a quarto course quality plus one. My four cost lefty Bless my four scrappy bless group lead by foot sign Costea plus one. My four course lefty last three by four times 20 minus adultery by what scientific? Musty When the put downs extensive and with one plus one right to less came by to science, this corn took 352 course Empty legal costs quickly percents What is 4 to 1

Heather Zimmers · Answer

all right and using a graphing utility for this problem. So here&#x27;s my graphing calculator. I&#x27;m going to check the mode and make sure I&#x27;m in radiance before I begin. And then we go into the y equals menu and you can see that I&#x27;ve already typed in half of X and G of X, and you&#x27;re going to need to be careful with parentheses so you can see that f of X has co sign Squared of something. And what I needed to do for my calculator was just put parentheses around the whole thing and with the squared on the outside and then for why? To that&#x27;s the function G. We have some extra parentheses in there as well. Okay, so I&#x27;m going to use my zoom trig graph graphing window. The calculator has a built in trig window, and so we&#x27;re seeing first in blue the graph of F of X and then next in red. We&#x27;re seeing the graph of G of X, and we see that these are actually identical graphs. And so that tells us that the functions are equivalent

Carlos Pinilla · Answer

even by who? Signed cube off the a longer iteration plus sign You d alone the generation. I mean, he is between cereal, my house. So this condition is sameness is like, but you can see there no sign We&#x27;ll get cereal in my house now something like these is for they sign Sino the this violently t when the sign gonna be something like that. So sign of the so in that interval who sign of tea and sign appear both positive. So let&#x27;s keep that in mind. Uh, we want to find the t. I didn&#x27;t normal curvature Monday. You need no no. So let&#x27;s keep it first. What? These bees and free who signed word de on green. That was the in order a little sign which is my no sign 50 about along the i direction and then 40 zone. We have one three sign Word of the that was the red evolve sign, which is Go sign off t the generation so you can fuck her here so you can write Dallas three times. Assign minus co sign a squirrel T sign off the I plus signed a squirrel e We&#x27;ll sign. Yeah, very civilized. the norm on the velocity we want to the square root of fighting those square several experts three i m school sign to the fourth power last signed no brown sine squared sent to the fourth that sign or the so the squared plus sign to the fourth power. You&#x27;re only no sign of the work. So that, uh, began doctor one Science squares of your vehicle. Well, you know, the trees beginning pounds this quote sign sward one the plane Because to the fourth Dana Plus, while we live to explain and they&#x27;re there, Sign square. I&#x27;m scores, squirt. Well, she have that on Dena. We&#x27;re gonna You can write that us three sees The sign is positive. So them saying off the and then luck during ah Assange Square does percent schools the from God times causes squared Los square. Mom. So these he&#x27;s one Monday in our well, it seems for Sandy&#x27;s was leave. He&#x27;s number is equal to the absolute value. Be the wrong is three sign off the Khost. So, um well, you want to compute What is the that is equal to be divided Lights norm tree? Yeah. Three minus causes with tea. That was a sign I plus Sinus Square goes time squared. Steve Perry on that. If I like you know you three times co sign them sign whether by three school sign the dubs sign of D so that for these son off mysteries goes away that it&#x27;s pretty clear these councils want power on councils that, well, councils want border councils so the teeth of people toe Italian. Normal ease minus goes I know he I glass sign of you Jing about his Italian Normal Uh, no copper. The curve of trouble People toe easy d start divided Why, you know we So it&#x27;s computerised. What is the key to be differentiating that? So they&#x27;re developed co science are minors Koscielny&#x27;s minus minus side of T high on the political science school Sign Teoh to your sorry. No, he all right, Plus, because being gay on its no people to school sign sine squared glass schools squared scores of bad These one, so does Norm one in the the normal level. Ah city is three sign over calls so that the perimeter coverage of the ego to three divided by one. Why anybody like being them Sign of the co signer thirties. Okay. And the normal? Well, our since these numbers, one is actually the normal. Because being normal is the key. Thank you. Divided by its norm. But he&#x27;s, uh, these numbers one. So this one is already being normal on what? You&#x27;ve been check, but he&#x27;s are thrown into the dungeon. It should be because it&#x27;s not a problem. Zeros old miners, cause I know signed them sign. Plus Sinus. Sign them. School science of use here. So the normal is normal. The unit dungeon on the curvature. Okay, he&#x27;s not.

Eric Mockensturm · Answer

you were given the curve co sign T 50 in X direction plus sign. Why direction for tea between zero and pi over too. Uh, vector in the tangent direction is minus the scientific co sign squared team in the extraction plus t coastline T science. Quite a t in my direction. He can get the magnitude of this guy, and that is three scientific coast. If we divide, we divide this by this we get the unique tangent factor, which is actually just minus coastline tea in the direction plus sign Jay direction. Take a derivative of that, get a director in the normal direction and we get scientist E scientific I plus co sign TJ, which actually is a unit vector. So that is our unit, Donald Becker. We can then figure out that our curvature is the magnitude of this which is one divided by the magnitude of this, which is three sign post. And if we plot these curves, this is, uh this is why ex on cap and you And so we get a curve that looks like this actually come continues on like this, and it&#x27;s kind of a four appointments star shaped if he were the planet for a bigger range of keys. And here we see that the curvature. There&#x27;s a kink here. And then I came here. So it goes to insanity at zero and pi over, too. It&#x27;s a minimum value, but not zero at the big point here. And so we get this shape here for the curvature.

Carlos Pinilla · Answer

So you have, uh, possesion given by Cruz do you? To us. He sign e along the i component plus sign de in Meyers T course t over the Jacob moment on. So drink a, um pre k over the secret bonus. So the velocity when we already appreciate each so they will call sign. That&#x27;s minus sign. They&#x27;re your design. Be differentiating the first. We have signed t know appreciate science of the lost ecos de that along I component. Plus, they&#x27;re the most Sinus cold sign course d been differentiate integration that minus that book, he&#x27;s one. So that goes D Now we differentiate science of minus d. How is the article co sign minus society over the J warrant, There will three, zero. But aunt something convenient hovels this cost using with that 50 that scientist was with that. Thank you. So that what was going to completely ski a couple. So let&#x27;s compute first, what is be so t course t along the i component. This is minus and minus. So want in waas Ah di scientific. Okay, sign so that our deejay component on the normal b not be square root but dont warrants his word at it because the score plus he signed squared. From there, we can see that these will be squared plus the square plus B squared sign squared. So they become factor a team out of those people to see square of the plane that now we cease one. So we go toe square square, which is thing. So that team she wanted me divided like it&#x27;s norm one day. You know, the, uh Is these 50? Do you most, Do you? I plus the sign in the normal is just eat so bad If I&#x27;d like tea by seeing exit you away So he s coast ei last scientific. So this is theme. Noah was a couple couple people to the norm. Oh, dear. You guys fighting hard. So all the ticking we&#x27;ll be able to first meeting that was being deficient in. So there is a good sign off the along the i component plus a sign of the come on a j component. So, uh, here for Cassandra, we had minor sign I plus the reverse course, I inti j. So the computer it&#x27;s from this is a victory. You did? It seems yummy. Square root. Well, the norm Riddle that squared marina science courses with same squared law school sign squared, and that is equal to one. So these one I learned that normal me. Who&#x27;s there? Oh, you are Teoh or teach positive. De positive is, uh show me that. Well, I should is gonna be for energy is a lot of the ultimate value. Lt So car should be one over the absolute value And no, um, always the normal is between you divided by its norm. But we already computed that this is one so divided right down people to just that there were developed to you there speculative because it&#x27;s your house already known mine, which is a bad so been norm normal unit. That would be my society by plus sign G J Onda. Uh, discover on there. They and d she won&#x27;t go. Do you see looking Ghostie? I last sign J

Problem

[M] Repeat Exercise 37 for the functions $\mathb…

Problem 37 Easy Difficulty

Answer

Observed that most of the function values of the function $1+\mathbf{g}(t)-\cos (6 t)$ are moving very
closed to the $y=1$ line. Therefore, our conjecture is verified.

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Anna Marie V.

Caleb E.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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Observed that most of the function values of the function $1+\mathbf{g}(t)-\cos (6 t)$ are moving veryclosed to the $y=1$ line. Therefore, our conjecture is verified.

Linear Algebra and Its Applications

Anna Marie V.

Caleb E.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Anna Marie V.

Caleb E.

Samuel H.

Joseph L.

Observed that most of the function values of the function $1+\mathbf{g}(t)-\cos (6 t)$ are moving very
closed to the $y=1$ line. Therefore, our conjecture is verified.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications