13.1: Vector Functions & Space Curves

00:34:15
https://www.youtube.com/watch?v=gqlVfXqhQMg

Ringkasan

TLDRLa vidéo présente un cours sur le calcul multivariable centré sur les fonctions vectorielles, évaluant principalement les dérivées et les intégrales associées aux courbes dans l'espace. Le professeur commence par réviser les courbes paramétriques qui décrivent les lignes en trois dimensions en utilisant des fonctions pour chaque coordonnée X, Y, Z en termes de T (un paramètre). La courbe créée a une orientation déterminée par l'ordre d'augmentation de T. Ensuite, le cours introduit les fonctions vectorielles notées généralement R(t), évoquant les aspects des fonctions composantes et l'importance de l'orientation dans le tracé des vecteurs. Les exemples sont donnés pour illustrer la manière de tracer ces vecteurs tant en deux qu'en trois dimensions, et comment les courbes vectorielles se manifestent sous forme d'hélices ou d'ellipses lorsqu'elles sont tracées dans l'espace. Enfin, des concepts comme les domaines, les limites, et la continuité des fonctions vectorielles sont examinés, concluant sur leur importance dans le calcul vectoriel global.

Takeaways

  • 🔄 Introduction aux courbes paramétriques et leur orientation.
  • 📐 Revue des fonctions vectorielles R(t) avec composantes vectorielles.
  • ✏️ Graphiques de fonctions vectorielles à travers des exemples.
  • 🔍 Discussion sur les domaines, limites et continuité des fonctions.
  • 📊 Importance de l'orientation dans le graphe des vecteurs.
  • 🧮 Application du calcul intégral et dérivatif aux vecteurs.
  • 📌 Exemple d'équations paramétriques comme "x = 2*cos(t)".
  • 📈 Visualisation des trajectoires en trois dimensions.
  • ❓ Comparaison et lien entre courbes paramétriques et fonctions vectorielles.
  • 📚 Concepts de base de fonctions, domaines et valeurs limites.

Garis waktu

  • 00:00:00 - 00:05:00

    Le chapitre 13 porte sur le calcul vectoriel des fonctions vectorielles, y compris les dérivées et les intégrales. On commence par réviser les courbes paramétriques, une notion déjà vue au chapitre précédent. Les courbes paramétriques sont définies par des fonctions de T et leur domaine est généralement l'ensemble des nombres réels. C'est important car cela nous aide à comprendre les fonctions vectorielles qui sont le sujet principal de cette section.

  • 00:05:00 - 00:10:00

    La fonction vectorielle, notée R(t), est un vecteur dont chaque composante est définie par une fonction. Elle représente un vecteur dont le point initial est à l'origine et le point terminal est fonction de T. La courbe tracée par la fonction vectorielle est obtenue en connectant les « pointes » des vecteurs de rayon, et elle possède une orientation. Un exemple en deux dimensions montre comment ces composants sont utilisés pour tracer des graphes.

  • 00:10:00 - 00:15:00

    Nous passons à l'exemple dans l'espace 3D pour tracer la courbe d'une fonction vectorielle. Ici, introduire une composante z de T transforme un cercle en une hélice, démontrant comment la fonction s'étend dans l'espace tridimensionnel. L'exemple montre comment identifier et positionner ces vecteurs dans un espace en trois dimensions pour construire la compréhension visuelle de fonctions vectorielles.

  • 00:15:00 - 00:20:00

    Ensuite, nous abordons comment trouver une fonction vectorielle pour la ligne reliant deux points donnés. La méthode nécessite de calculer un vecteur parallèle, puis d'utiliser les composantes pour construire des équations paramétriques. Cette technique est revue et utilisée pour compiler des fonctions vectorielles qui représentent des intersections de figures géométriques dans l'espace, en rationalisant les étapes et en démontrant le processus.

  • 00:20:00 - 00:25:00

    Nous examinons la détermination du domaine des fonctions vectorielles en tenant compte des contraintes individuelles de chaque composante. Le domaine global est l'intersection de tous les domaines possibles pour les composantes individuelles. Par la suite, une attention particulière est accordée aux limites des fonctions vectorielles, qui se calculent en évaluant la limite de chaque composante séparément, conformément à l'application des règles d'analyse classique.

  • 00:25:00 - 00:34:15

    Nous abordons la continuité des fonctions vectorielles. Une fonction vectorielle est continue à un point donné si la limite de la fonction en ce point est égale à la valeur de la fonction en ce point. Des exemples sont fournis pour illustrer comment déterminer la continuité en évaluant les limites et en appliquant une définition conceptuelle de la continuité à des cas concrets, indiquant des situations où elle échoue, comme démontré par l'utilisation de la règle de l'Hôpital.

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Pertanyaan yang Sering Diajukan

  • Qu'est-ce qu'une courbe paramétrique?

    Une courbe paramétrique est une courbe définie par des équations où les coordonnées sont données en fonction d'un paramètre commun, souvent noté t.

  • Qu'est-ce que l'orientation dans le contexte des courbes?

    L'orientation est la direction dans laquelle une courbe est tracée lorsque le paramètre augmente.

  • Pourquoi besoin-t-on des courbes paramétriques pour les fonctions vectorielles?

    Les courbes paramétriques aident à visualiser la trajectoire d'un vecteur en trois dimensions, ce qui est essentiel pour comprendre les fonctions vectorielles.

  • Qu'est-ce qu'une fonction vectorielle?

    Une fonction vectorielle est une fonction qui a des vecteurs comme valeurs.

  • Comment représente-t-on graphiquement une fonction vectorielle?

    On connecte les extrémités des vecteurs tracés pour certaines valeurs de t afin de représenter la courbe de la fonction.

  • Quelle est la relation entre les fonctions vectorielles et les courbes paramétriques?

    Une fonction vectorielle peut être considérée comme une courbe paramétrique avec des vecteurs comme paramètres.

  • Comment trouve-t-on le domaine d'une fonction vectorielle?

    Le domaine d'une fonction vectorielle est l'intersection des domaines de ses fonctions composantes.

  • Comment calculer la limite d'une fonction vectorielle?

    On calcule les limites pour chaque fonction composante individuellement.

  • Quelle condition rend une fonction vectorielle continue?

    Une fonction vectorielle est continue si la limite de la fonction en un point est égale à la valeur de la fonction en ce point.

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Teks
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Gulir Otomatis:
  • 00:00:00
    okay so as you guys saw from the packet
  • 00:00:02
    chapter 13 is all vectors calculus of
  • 00:00:18
    vector valued functions like derivatives
  • 00:00:20
    and integrals and stuff like that so I'm
  • 00:00:24
    good I mean you guys knew that was gonna
  • 00:00:26
    come right like it's a calculus class
  • 00:00:30
    first thing I want to review and this is
  • 00:00:33
    something we talked about last chapter
  • 00:00:34
    is parametric curves so last chapter we
  • 00:00:42
    looked at parametric curves specifically
  • 00:00:44
    lines in three space and two space
  • 00:00:48
    parametric curves are gonna look like
  • 00:00:49
    this X is going to be some function of T
  • 00:00:52
    Y is gonna be some function of T and Z
  • 00:00:56
    is gonna be some function of T these are
  • 00:01:00
    gonna represent a space curve in three
  • 00:01:02
    space
  • 00:01:09
    and of course you could do the same
  • 00:01:11
    thing in 2-space well we're gonna be
  • 00:01:14
    looking at three space now those
  • 00:01:18
    equations represent a path in space that
  • 00:01:29
    is traced in a specific direction as T
  • 00:01:41
    increases that direction is often
  • 00:01:46
    referred to as orientation
  • 00:01:49
    so we didn't focus too much on that but
  • 00:01:51
    as you plugged in your values of T you
  • 00:01:53
    got your curve in a specific direction
  • 00:01:56
    if you remember talking about the
  • 00:01:58
    equations of lines when you chose T
  • 00:02:00
    values greater than zero you were going
  • 00:02:02
    to the right whereas T values less than
  • 00:02:05
    zero we're gonna move you to the left on
  • 00:02:08
    your curve last thing I want to review
  • 00:02:12
    before we move on is domain your domain
  • 00:02:16
    is gonna be all real numbers unless
  • 00:02:18
    specified otherwise
  • 00:02:28
    okay so the whole reason we need to
  • 00:02:31
    review parametric curves is that's going
  • 00:02:33
    to help us with this first section which
  • 00:02:36
    is vector valued functions so next thing
  • 00:02:41
    we're going to talk about is vector
  • 00:02:42
    valued functions we are not gonna write
  • 00:02:51
    out a whole long definition I'm just
  • 00:02:54
    gonna show you what's most important
  • 00:02:56
    your vector valued function is going to
  • 00:02:59
    be called R sometimes it'll be written
  • 00:03:02
    as R of T it's going to be a vector
  • 00:03:09
    where the the X component is defined by
  • 00:03:13
    some function the Y component is defined
  • 00:03:16
    by some function and the Z component is
  • 00:03:18
    defined by some function all three of
  • 00:03:22
    these are called component functions
  • 00:03:30
    of our so here's what's important at any
  • 00:03:37
    given T value R represents a vector
  • 00:03:51
    whose initial point is at the origin and
  • 00:04:01
    then the terminal point is going to be
  • 00:04:08
    the point F of T G of T H of T
  • 00:04:19
    [Music]
  • 00:04:21
    so first thing that we're gonna be
  • 00:04:23
    interested in is graphs so anytime you
  • 00:04:27
    talked about graphs of course you're
  • 00:04:28
    gonna look at the domain and range
  • 00:04:29
    domain for these vector valued functions
  • 00:04:37
    is gonna be all real numbers range is
  • 00:04:42
    gonna be a set of vectors most of the
  • 00:04:46
    time your domain is all real numbers
  • 00:04:47
    we're gonna look at some cases where
  • 00:04:49
    it's not so in terms of a graph of a
  • 00:04:58
    vector-valued function so from now on
  • 00:05:00
    I'm going to use V V F that means vector
  • 00:05:04
    value function the graph of a vector
  • 00:05:07
    value function is gonna be a curve that
  • 00:05:11
    is traced by connecting what I'm going
  • 00:05:18
    to call the tips of the radius vectors
  • 00:05:30
    so here is an example so with that R of
  • 00:05:36
    T what you're gonna be doing is you're
  • 00:05:37
    gonna be plugging in different values of
  • 00:05:39
    T so let's say you plug in T equals zero
  • 00:05:42
    that might give you this little vector
  • 00:05:45
    right here so let's say that that's R of
  • 00:05:48
    0 so that's what I get when I plug in T
  • 00:05:50
    equals zero that I might plug in T
  • 00:05:53
    equals one and I might get this vector
  • 00:05:55
    right here and then maybe I plug in two
  • 00:05:59
    and I get this vector over here okay so
  • 00:06:05
    then the graph of the vector value
  • 00:06:06
    function is when you connect all of
  • 00:06:08
    these tips so it might be a curve like
  • 00:06:13
    this so we call that R of T or just R
  • 00:06:20
    then you put the arrows to show the
  • 00:06:22
    orientation so graph of a vector-valued
  • 00:06:27
    function is really similar to a
  • 00:06:29
    parametric curve it's still a curve and
  • 00:06:32
    it still has orientation just the way
  • 00:06:34
    you graph it is a little bit different
  • 00:06:39
    how we doing so far okay
  • 00:06:47
    we're gonna do a few graphing examples
  • 00:06:50
    and then we'll move on to our next idea
  • 00:06:53
    first example this one is going to be in
  • 00:06:56
    two space R of T is gonna be two cosine
  • 00:07:06
    of T I subtract 3 sine of TJ and I'm
  • 00:07:15
    telling you in this case that T is
  • 00:07:17
    between 0 and 2pi ok so any time you've
  • 00:07:25
    ever learned graphene before or you
  • 00:07:27
    haven't known how to graph something
  • 00:07:28
    you've plugged in some points so that's
  • 00:07:31
    what we're gonna do now you're gonna
  • 00:07:32
    plug in some values of T those will give
  • 00:07:36
    you an X component and a y component so
  • 00:07:42
    obviously you want to plug in 0 because
  • 00:07:44
    that's where you start some good ones
  • 00:07:46
    would be PI over 2 pi 3 PI over 2 and 2
  • 00:07:51
    pi if you plug in 0 cosine of 0 is going
  • 00:07:57
    to give us 1 times 2 is 2 sine of 0 is
  • 00:08:03
    going to be 0
  • 00:08:03
    so that gives us the first vector we
  • 00:08:07
    plugged in PI over 2 cosine a cosine of
  • 00:08:09
    PI over 2 is going to be 0 sine of PI
  • 00:08:12
    over 2 is 1 so that gives me negative 3
  • 00:08:14
    so these are the components of the
  • 00:08:17
    vectors that we'll be graphing we plug
  • 00:08:19
    in PI we're gonna get negative 2 0 we
  • 00:08:23
    plug in 3 PI over 2 we'll get 0 3 and
  • 00:08:26
    then 2 pi will give us 2 0
  • 00:08:34
    okay so now we have to graph all of
  • 00:08:36
    these vectors first vector is 2 0 that's
  • 00:08:43
    that vector 0 negative 3 negative 2 0 0
  • 00:08:58
    3 so what is this gonna form no lips so
  • 00:09:08
    this is where we started so we're gonna
  • 00:09:10
    connect the points of our curves or the
  • 00:09:14
    points of our radius vectors rather you
  • 00:09:18
    have to show the orientation so we
  • 00:09:20
    started here heading this direction now
  • 00:09:27
    remember that those are just the radius
  • 00:09:29
    vectors that we chose if we plugged in
  • 00:09:31
    other values there might be a radius
  • 00:09:33
    vector there and there and there etc so
  • 00:09:40
    do we get how this works
  • 00:09:41
    if you wanted to think of it
  • 00:09:43
    parametrically it would be x equals 2
  • 00:09:45
    cosine of T and y equals negative 3 sine
  • 00:09:49
    of T so that's the same function just
  • 00:09:53
    represented parametrically ready for our
  • 00:09:58
    next graphing one ok good this one's in
  • 00:10:01
    3-space so you ready for that
  • 00:10:07
    okay next example R of T is gonna be for
  • 00:10:13
    cosine T comma for sine of T comma T now
  • 00:10:23
    we should have some idea of what this is
  • 00:10:25
    gonna look like if we do it in two space
  • 00:10:27
    for cosine T for sine T based on the
  • 00:10:31
    last example that we did what do we
  • 00:10:33
    think this will give us circle what
  • 00:10:36
    happens when you introduce that Z of T
  • 00:10:42
    it's kind of a cylinder it's like a
  • 00:10:44
    circle that's opening up this will graph
  • 00:10:48
    it but that's a circle but it's gonna be
  • 00:10:51
    moving also okay so same thing from
  • 00:10:56
    before plug in values of T we're gonna
  • 00:11:03
    plug in all the same ones we did before
  • 00:11:05
    so 0 PI over 2 pi 3 PI over 2 and 2 pi
  • 00:11:13
    ok can I trust that you guys can plug
  • 00:11:15
    these in so can I just write them all
  • 00:11:18
    down for you I'm gonna read you the X
  • 00:11:20
    column going down for 0 negative 4 0 for
  • 00:11:26
    the Y column is gonna be 0 for 0
  • 00:11:31
    negative 4 0 Z column is gonna be 0 PI
  • 00:11:37
    over 2 pi 3 PI over 2 2 pi
  • 00:11:43
    okay you all know from our last chapter
  • 00:11:47
    that I am NOT a good artist so go with
  • 00:11:50
    me a little bit on this one okay
  • 00:11:52
    I don't want to hear any comments about
  • 00:11:53
    how my drawing sucks okay okay so first
  • 00:12:07
    point four zero zero it's gonna be that
  • 00:12:12
    one right there then we have zero 4 and
  • 00:12:16
    PI over 2
  • 00:12:29
    ok so 0 4 PI over 2 is about there then
  • 00:12:39
    we have negative 4 0 and PI ok here's my
  • 00:12:43
    suggestion here PI 3.14 is about here I
  • 00:12:49
    would move another dotted line out so
  • 00:12:51
    that you get the perspective right so 0
  • 00:12:54
    or rather or negative 4 is 0 and then PI
  • 00:12:58
    that's about there bless you okay and
  • 00:13:04
    then we have 0 negative 4 3 PI over 2 3
  • 00:13:09
    PI over 2 is about 4.7 1 just about here
  • 00:13:21
    okay and then let's start by connecting
  • 00:13:25
    those
  • 00:13:34
    then there will be another point about
  • 00:13:36
    there
  • 00:13:42
    so do you see how it's a curved wrapping
  • 00:13:45
    around a cylinder so it's helpful maybe
  • 00:13:49
    to I really feel like this drawings not
  • 00:13:54
    that bad it's not guys yeah it's like it
  • 00:13:57
    slinky
  • 00:13:57
    yeah it's a slinky so it's like this
  • 00:14:01
    look if you draw on a cylinder the
  • 00:14:03
    cylinder is not part of the graph but
  • 00:14:06
    the graph is wrapping around the
  • 00:14:08
    cylinder good okay if you want to see
  • 00:14:24
    the drawing on my notes which is a
  • 00:14:25
    little bit better looks like that so we
  • 00:14:32
    have the cylinder we're wrapping around
  • 00:14:33
    it
  • 00:14:41
    good news for you is that WebAssign
  • 00:14:44
    obviously can't force you to sketch
  • 00:14:46
    anything so on WebAssign it's gonna be
  • 00:14:48
    more of here's the vector valued
  • 00:14:50
    function pick the correct graph in your
  • 00:14:53
    packet dollars make you sketch questions
  • 00:14:57
    on graphs before we move on no great
  • 00:15:01
    here's our next example and this is
  • 00:15:06
    something that we've done before so this
  • 00:15:08
    is not going to be new this next one
  • 00:15:09
    find a vector and parametric equations
  • 00:15:23
    for the line segment that joins a which
  • 00:15:34
    is the point 1 negative 3 for 2 B which
  • 00:15:39
    is negative 5 1 7 okay so we did this
  • 00:15:46
    last chapter we talked about if you're
  • 00:15:48
    writing the equation of a line this is a
  • 00:15:50
    segment but it's the same idea you need
  • 00:15:51
    a vector parallel to your line and you
  • 00:15:53
    need a point so we obviously have a
  • 00:15:55
    point we have to to choose from how we
  • 00:15:56
    can find the vector yeah find the vector
  • 00:16:02
    joining A to B I'm gonna call that
  • 00:16:04
    vector R so really what I mean is AP
  • 00:16:10
    negative 5 minus 1 will give us negative
  • 00:16:13
    6 1 subtract negative 3 is gonna give us
  • 00:16:16
    4 7 subtract 4 is gonna give us 3 so
  • 00:16:23
    then our R of T these will be the slopes
  • 00:16:27
    I decided to use the first point so we
  • 00:16:31
    get 1 subtract 60 negative 3 at 40
  • 00:16:37
    for at 3t so that's the way to write it
  • 00:16:43
    as a vector function or a vector valued
  • 00:16:45
    function we decided to write it
  • 00:16:49
    parametrically X would be 1 minus 6t y
  • 00:16:52
    would be negative 3 plus 4 T and Z would
  • 00:16:56
    be 4 plus 3t now that's a line how do we
  • 00:17:01
    make it a segment restriction and how do
  • 00:17:05
    we restrict T 0 to 1 and this applies to
  • 00:17:13
    both the vector valued function and
  • 00:17:15
    parametrically you plug in 0 you're
  • 00:17:17
    gonna get point a since that's when we
  • 00:17:19
    started if you plug in 1 you'll get
  • 00:17:20
    point B ok questions before we do
  • 00:17:27
    another example know
  • 00:17:37
    find a vector function that represents
  • 00:17:51
    the curve of the intersection of the
  • 00:17:59
    following two equations first one is
  • 00:18:02
    going to be x squared ad y squared
  • 00:18:04
    equals one second one is y plus Z equals
  • 00:18:09
    two okay if we were to graph this first
  • 00:18:17
    one the x squared plus y squared equals
  • 00:18:19
    one what is that going to look like
  • 00:18:22
    cylinder so just want to be a cylinder y
  • 00:18:25
    plus Z equals two
  • 00:18:29
    that one's a plane
  • 00:18:31
    okay so we're taking a plane we're
  • 00:18:33
    taking in a cylinder we're gonna
  • 00:18:34
    intersect them first thing that we're
  • 00:18:36
    gonna do is we are going to start with
  • 00:18:37
    that x squared plus y squared equals 1
  • 00:18:40
    well what we want to do is write that
  • 00:18:43
    parametrically do you guys know how to
  • 00:18:44
    write x squared plus y squared equals 1
  • 00:18:46
    parametrically
  • 00:18:53
    I think back to this example that we did
  • 00:19:00
    we talked about if you take away the T
  • 00:19:03
    what are you left with me fewer just to
  • 00:19:05
    graph that yeah remember how we talked
  • 00:19:11
    about this as a circle when you put the
  • 00:19:13
    T in though it makes the circle open up
  • 00:19:16
    it's not really a circle anymore but
  • 00:19:18
    that's the idea okay so for example for
  • 00:19:20
    then X is gonna be cosine T y is gonna
  • 00:19:26
    be sine of T parametrically that give us
  • 00:19:30
    a circle T is gonna be between 0 and 2pi
  • 00:19:38
    ok so that takes care of x and y we need
  • 00:19:43
    Z now
  • 00:19:49
    why plus C so if I solve for Z I get to
  • 00:19:54
    subtract Y Y is what I have up here that
  • 00:20:01
    sine of T so now I have a parametric
  • 00:20:07
    expression for x y&z that takes both
  • 00:20:09
    figures into account so our curve then
  • 00:20:13
    we want to write it as a vector value
  • 00:20:15
    function X is cosine of T so we get
  • 00:20:18
    cosine of T I and sign of TJ J add to
  • 00:20:27
    subtract sine of T with T being between
  • 00:20:33
    0 & 2 pi
  • 00:20:46
    questions okay that's not the only
  • 00:20:50
    option what you could have done is you
  • 00:20:53
    could have solved for X or Y either one
  • 00:20:56
    and you would have gotten X in terms of
  • 00:20:59
    Y and you could have then plugged it in
  • 00:21:02
    does that make sense what I'm saying
  • 00:21:04
    without writing it down okay
  • 00:21:06
    so sometimes what people do like another
  • 00:21:09
    option if we did it over here maybe you
  • 00:21:11
    choose X to be T so then if you solved
  • 00:21:15
    for y you would get y equals plus or
  • 00:21:18
    minus the square root of one minus x
  • 00:21:19
    squared from here and you could then
  • 00:21:23
    plug in T same thing here you would
  • 00:21:26
    solve for Z you would get two minus y
  • 00:21:28
    and then you would have to plug in your
  • 00:21:29
    expression for y with me okay that just
  • 00:21:36
    does not look as nice so you can do it
  • 00:21:38
    that way totally fine it's not gonna
  • 00:21:40
    look as nice there's some that that's
  • 00:21:42
    gonna be the best option if you're
  • 00:21:44
    curious this is what those figures look
  • 00:21:48
    like so here's the cylinder here's that
  • 00:21:50
    plane so you're left with an ellipse so
  • 00:21:54
    this is the equation that we just wrote
  • 00:21:57
    kind of cool right
  • 00:22:01
    like don't have too much energy so they
  • 00:22:04
    do
  • 00:22:09
    are you guys more excited by the storm
  • 00:22:12
    than by my notes right now the lightning
  • 00:22:17
    is cool math is cool too okay two other
  • 00:22:24
    things we need to talk about we're gonna
  • 00:22:26
    talk about well three things domain
  • 00:22:28
    limits continuity so our domain example
  • 00:22:34
    we are going to find the domain of R of
  • 00:22:42
    T R of T is going to be in three space
  • 00:22:46
    the first component is the natural log
  • 00:22:49
    of the absolute value of 2t minus 1 and
  • 00:22:52
    then e to the T is the second component
  • 00:22:55
    third component is the square root of T
  • 00:23:00
    okay here's what I mean by domain of a
  • 00:23:03
    vector valued function domain includes
  • 00:23:06
    all values of T for which R is defined
  • 00:23:18
    so the basic idea is you're gonna have
  • 00:23:20
    to treat all three domains separately or
  • 00:23:23
    all three components separately so x y
  • 00:23:27
    and z okay y is the easiest e 2t for e
  • 00:23:33
    2t what are the values of T for which
  • 00:23:35
    that function is defined all real
  • 00:23:40
    numbers so based on YT has to be a real
  • 00:23:45
    number based on Z so the square root of
  • 00:23:49
    T what values is T for what values of T
  • 00:23:52
    is that function defined greater than or
  • 00:23:55
    equal to zero
  • 00:23:59
    okay how do we approach the natural log
  • 00:24:02
    of the absolute value of t minus one I
  • 00:24:09
    don't want to know the domain right off
  • 00:24:11
    the bat I just want to know what do we
  • 00:24:13
    need to consider what do we need to
  • 00:24:19
    think about first okay
  • 00:24:23
    so we know that natural log you only
  • 00:24:25
    have positive numbers so anything bigger
  • 00:24:27
    than zero so we need the absolute value
  • 00:24:29
    of t minus one to be greater than zero
  • 00:24:36
    do you remember how to approach problems
  • 00:24:38
    like this
  • 00:24:56
    okay when are we gonna be equal to zero
  • 00:25:03
    let's consider that when T equals one so
  • 00:25:07
    this is what I would do is I would
  • 00:25:08
    consider a number line when T equals one
  • 00:25:11
    we get zero which we don't want when T
  • 00:25:15
    is greater than one is that good is that
  • 00:25:17
    part of our domain yes when T is less
  • 00:25:21
    than one is that part of our domain yes
  • 00:25:26
    so any number is okay except T equals
  • 00:25:29
    one the domain here is going to be
  • 00:25:31
    negative infinity to 1 Union 1 to
  • 00:25:34
    infinity so then the overall domain is
  • 00:25:38
    going to be the intersection of all
  • 00:25:41
    three of those so that one's not helpful
  • 00:25:47
    so we can ignore that so we need this to
  • 00:25:50
    be true and T to be greater than or
  • 00:25:52
    equal to zero so our hotel name then is
  • 00:25:54
    gonna be 0 to 1 Union 1 to infinity
  • 00:26:11
    does that make sense so you consider it
  • 00:26:14
    all three domains separately and then
  • 00:26:15
    find where their intersection is okay we
  • 00:26:20
    got one more page notes like probably
  • 00:26:22
    ten minutes last thing we need to talk
  • 00:26:25
    about is limits and continuity starting
  • 00:26:35
    with a limit consider R of T to be a
  • 00:26:40
    vector valued function with the
  • 00:26:43
    components or component functions f of T
  • 00:26:45
    G of T and H of T then the limit as T
  • 00:26:56
    approaches a of R of T where you find
  • 00:27:02
    that limit as you find the limit of each
  • 00:27:04
    of the component functions so you're
  • 00:27:07
    going to have the limit as T approaches
  • 00:27:09
    a of f of T the limit as T approaches a
  • 00:27:14
    G of T and then the limit as T
  • 00:27:17
    approaches a of H of T and that is as
  • 00:27:24
    long as all three limits exist
  • 00:27:44
    so here's an example we are going to let
  • 00:27:50
    R of T be T squared I add e to the TJ
  • 00:27:58
    subtract 2 cosine of PI T ok we are
  • 00:28:06
    going to find the limit as T approaches
  • 00:28:10
    0 of R of T
  • 00:28:19
    okay so if I want to find the limit as T
  • 00:28:22
    approaches 0 of R of T I have to take
  • 00:28:26
    the limit as T approaches 0 for each of
  • 00:28:29
    the component functions so this is gonna
  • 00:28:33
    be alright differently the limit as T
  • 00:28:39
    approaches 0 of T squared whatever that
  • 00:28:43
    is multiplied by I the limit as T
  • 00:28:47
    approaches 0 of e to the T multiplied by
  • 00:28:50
    J and then the limit as T approaches 0
  • 00:28:56
    to cosine PI T all of that times K limit
  • 00:29:04
    as T approaches 0 of T squared that's
  • 00:29:06
    just 0 so we get 0 I I plug in 0 that
  • 00:29:11
    ends up being 1 so plus 1 J I plug in 0
  • 00:29:15
    cosine of 0 0 times 2 will give me
  • 00:29:17
    negative 2 okay you don't need to write
  • 00:29:24
    the 0 I I just wrote it to help you know
  • 00:29:27
    where it came from
  • 00:29:37
    questions on limits before we do one
  • 00:29:39
    more example am I going too quickly for
  • 00:29:44
    us
  • 00:29:45
    no we're okay okay next example
  • 00:29:53
    R of T the first function is 4t cubed
  • 00:30:00
    plus 5 divided by 3t cubed plus one
  • 00:30:05
    second function is 1 subtract cosine of
  • 00:30:08
    T over T third component function is the
  • 00:30:11
    natural log T plus 1 that quantity
  • 00:30:14
    divided by T we're going to find the
  • 00:30:20
    limit as T approaches 0 of R of T
  • 00:30:30
    okay so the limit as T approaches 0 of R
  • 00:30:34
    of T I'm not gonna write out all of that
  • 00:30:39
    again is that okay you just know that
  • 00:30:41
    you're looking for the limit of each
  • 00:30:43
    component function if we plug in 0 here
  • 00:30:46
    we're gonna get 5 over 1 which is 5 if
  • 00:30:51
    we plug in 0 here that'll give me a 1 so
  • 00:30:55
    1 minus 1 will give us 0 divided by 0
  • 00:31:00
    plug in 0 here oh look we get 0/0 again
  • 00:31:05
    what do we have to do l'hopital's rule
  • 00:31:08
    remember alofi tiles is also when you
  • 00:31:10
    have infinity over infinity or plus or
  • 00:31:12
    minus infinity over infinity okay so
  • 00:31:16
    take the derivative of 1 minus cosine t
  • 00:31:18
    what is that sine T so we get sine T
  • 00:31:24
    over 1 derivative of natural log of t
  • 00:31:29
    plus 101 over T plus 1 and then the
  • 00:31:38
    derivative again gives us 1 now we can
  • 00:31:42
    plug in 0 again if we plug in 0 out here
  • 00:31:44
    we get 0 over 1 which is 0 if we plug in
  • 00:31:48
    0 here we get 1 over 1 which is 1
  • 00:31:57
    hey how do we feel about limits good all
  • 00:32:02
    of this should be fairly intuitive
  • 00:32:04
    because it's stuff that you've done
  • 00:32:05
    before just apply in a different manner
  • 00:32:07
    last thing we need to talk about is
  • 00:32:08
    continuity a vector function when I say
  • 00:32:19
    a vector function that's the same as a
  • 00:32:20
    vector valued function so just know
  • 00:32:22
    they're the same sometimes we leave out
  • 00:32:25
    the word valued a vector function R of T
  • 00:32:29
    is continuous at T equals a if and only
  • 00:32:38
    if you guys remember once
  • 00:32:43
    [Music]
  • 00:33:07
    yes I'm gonna write it differently if
  • 00:33:18
    the limit as T approaches a of our R of
  • 00:33:21
    T is equal to R they so Alex was saying
  • 00:33:24
    as the limit as T approaches a from the
  • 00:33:26
    left and the right are the same that
  • 00:33:28
    means that the limit exists at that
  • 00:33:30
    point so that's the other way of writing
  • 00:33:32
    it okay looking at example six is R of T
  • 00:33:39
    continuous at 0 so as the is the limit
  • 00:33:47
    as T approaches 0 of R of T the same as
  • 00:33:49
    R of 0 yeah right we just plugged in 0
  • 00:33:52
    for all 3 functions so this example is
  • 00:33:56
    continuous at T equals 0
  • 00:33:57
    example 7 are we continuous at T equals
  • 00:33:59
    0 no no and that's because of the
  • 00:34:03
    l'hopital's rule that we had to do
  • 00:34:07
    questions on vector valued functions
Tags
  • Calcul vectoriel
  • Fonctions vectorielles
  • Courbes paramétriques
  • Dérivées
  • Intégrales
  • Orientation
  • Domaines
  • Limites
  • Continuité
  • Graphes