vicho
DownloadTélécharger
Actions
Vote :
ScreenshotAperçu
Informations
Catégorie :Category: nCreator TI-Nspire
Auteur Author: joao.perrac
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 13.41 Ko KB
Mis en ligne Uploaded: 23/03/2025 - 11:54:41
Uploadeur Uploader: joao.perrac (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4543634
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 13.41 Ko KB
Mis en ligne Uploaded: 23/03/2025 - 11:54:41
Uploadeur Uploader: joao.perrac (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4543634
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
<<
1. Defining Variables and Understanding the Context 1. Identify your independent variable (often x ): " Typically represents time (e.g., year). " For the quoll data, x could be the number of years since 1995, or you can use the actual year as x . " For the smartphone data, x could be the number of years since the first data point (e.g., 2007), or again, use the actual year. 2. Identify your dependent variable (often y ): " This is the quantity being measured. " Quoll example: the number of quolls ( y ) in the population. " Smartphone example: the number (or millions) of smartphones sold ( y ). 3. Contextual meaning of variables: " For quolls, y is a population count , which cant be negative and is influenced by real-world factors like breeding, mortality, reintroduction programs, etc. " For smartphones, y is sales (in millions), which also cant be negative and typically grows with demand/technology changes. 4. Know your domain: " Years in which the data makes sense. Realistically, negative years or far-future years might not be as reliable in your model. " The domain is usually from the earliest year you have data for to the near future (unless you have a reason to extrapolate far). 2. Graphing the Data 1. Plot each data point: " For quolls, you might plot (1995, 255), (2000, 201), etc. " For smartphones, you might plot (2007, 122 million), (2010, 304 million), etc. 2. Visual inspection: " Is the graph trending up or down? " Is it roughly linear, exponential, quadratic, logistic, or something else? 3. What to look for visually: " Linear : Points roughly form a straight line. " Exponential : The rate of increase/decrease speeds up or slows down in a manner that looks multiplicative (e.g., faster and faster growth or a steady percentage decline). " Quadratic : A curved U-shape or )-shape. " Logistic (S-curve) : Growth rises quickly but then levels off near a carrying capacity. Or for a declining population, it might approach zero but never go negative. 3. Choosing the Model Type (Justification) A. When to choose an exponential model " Data that grows or declines by a constant percentage rate. " Example: If each years population or sales is some constant multiple of the previous years, or points clearly curve upward/downward in a multiplicative manner. B. When to choose a linear model " Data that changes by a constant difference each year. " Example: If the data points form an almost straight line (like +50 per year or -10 per year, consistently). C. When to choose a quadratic or polynomial model " Data that changes at a varying rate but has a parabolic shape (peaking or dipping). " Example: If the population might rise to a maximum and then fall, or sales accelerate but eventually slow. D. When to choose a logistic model " Data that grows quickly but then approaches some maximum limit (like a carrying capacity for a population, or a maximum potential market penetration for smartphone sales). " For a declining species with some minimum floor above 0, logistic can also model that it might not instantly go to zero, but asymptotically approaches an extremely small population. " This is often realistic for populations or for technology adoption, but also a bit more complex to work with. For the Quoll Example " Because the population is in decline (255 201 169 136 104 from 1995 to 2015), a decaying exponential can be quite plausible (populations often decline exponentially unless theres a threshold or outside intervention). " However, you might compare the fit of a linear model vs. an exponential model, or even a logistic model if you assume the population might not actually hit zero so quickly. For the Smartphone Example " Sales go from 122 million in 2007 to 1500 million (1.5 billion) by 2016an explosive growth that often suggests an exponential or logistic model. " However, because real-world sales eventually slow down (a market can saturate), a logistic model can also make sense. " Often, teachers allow you to pick either exponential or logistic, then defend your choice with the context (exponential might overpredict if you go too far out, logistic may be more realistic if you think there is an upper limit). 4. Defining the Model (Function Form) A. General forms youll likely use 1. Linear model : y = ax + b " a : slope (change per year), b : y -intercept (value at x = 0 ). 2. Exponential model (common for growth/decay): y = A cdot b^xquad text{or} quady = A cdot e^{kx} " A is the initial value (the value when x = 0 ), " b is the growth/decay factor (if b>1 , growth; if 0<b<1 , decay), " or k is the continuous growth/decay rate in the e^{kx} form. 3. Logistic model : y = frac{L}{1 + Ae^{-kx}} " L is the carrying capacity (the maximum limit the function approaches). " More complex to get parameters from basic regression, but many graphing calculators or software can do logistic regression. 4. Polynomial model (like quadratic): y = ax^2 + bx + c
[...]
>>
Compatible OS 3.0 et ultérieurs.
<<
1. Defining Variables and Understanding the Context 1. Identify your independent variable (often x ): " Typically represents time (e.g., year). " For the quoll data, x could be the number of years since 1995, or you can use the actual year as x . " For the smartphone data, x could be the number of years since the first data point (e.g., 2007), or again, use the actual year. 2. Identify your dependent variable (often y ): " This is the quantity being measured. " Quoll example: the number of quolls ( y ) in the population. " Smartphone example: the number (or millions) of smartphones sold ( y ). 3. Contextual meaning of variables: " For quolls, y is a population count , which cant be negative and is influenced by real-world factors like breeding, mortality, reintroduction programs, etc. " For smartphones, y is sales (in millions), which also cant be negative and typically grows with demand/technology changes. 4. Know your domain: " Years in which the data makes sense. Realistically, negative years or far-future years might not be as reliable in your model. " The domain is usually from the earliest year you have data for to the near future (unless you have a reason to extrapolate far). 2. Graphing the Data 1. Plot each data point: " For quolls, you might plot (1995, 255), (2000, 201), etc. " For smartphones, you might plot (2007, 122 million), (2010, 304 million), etc. 2. Visual inspection: " Is the graph trending up or down? " Is it roughly linear, exponential, quadratic, logistic, or something else? 3. What to look for visually: " Linear : Points roughly form a straight line. " Exponential : The rate of increase/decrease speeds up or slows down in a manner that looks multiplicative (e.g., faster and faster growth or a steady percentage decline). " Quadratic : A curved U-shape or )-shape. " Logistic (S-curve) : Growth rises quickly but then levels off near a carrying capacity. Or for a declining population, it might approach zero but never go negative. 3. Choosing the Model Type (Justification) A. When to choose an exponential model " Data that grows or declines by a constant percentage rate. " Example: If each years population or sales is some constant multiple of the previous years, or points clearly curve upward/downward in a multiplicative manner. B. When to choose a linear model " Data that changes by a constant difference each year. " Example: If the data points form an almost straight line (like +50 per year or -10 per year, consistently). C. When to choose a quadratic or polynomial model " Data that changes at a varying rate but has a parabolic shape (peaking or dipping). " Example: If the population might rise to a maximum and then fall, or sales accelerate but eventually slow. D. When to choose a logistic model " Data that grows quickly but then approaches some maximum limit (like a carrying capacity for a population, or a maximum potential market penetration for smartphone sales). " For a declining species with some minimum floor above 0, logistic can also model that it might not instantly go to zero, but asymptotically approaches an extremely small population. " This is often realistic for populations or for technology adoption, but also a bit more complex to work with. For the Quoll Example " Because the population is in decline (255 201 169 136 104 from 1995 to 2015), a decaying exponential can be quite plausible (populations often decline exponentially unless theres a threshold or outside intervention). " However, you might compare the fit of a linear model vs. an exponential model, or even a logistic model if you assume the population might not actually hit zero so quickly. For the Smartphone Example " Sales go from 122 million in 2007 to 1500 million (1.5 billion) by 2016an explosive growth that often suggests an exponential or logistic model. " However, because real-world sales eventually slow down (a market can saturate), a logistic model can also make sense. " Often, teachers allow you to pick either exponential or logistic, then defend your choice with the context (exponential might overpredict if you go too far out, logistic may be more realistic if you think there is an upper limit). 4. Defining the Model (Function Form) A. General forms youll likely use 1. Linear model : y = ax + b " a : slope (change per year), b : y -intercept (value at x = 0 ). 2. Exponential model (common for growth/decay): y = A cdot b^xquad text{or} quady = A cdot e^{kx} " A is the initial value (the value when x = 0 ), " b is the growth/decay factor (if b>1 , growth; if 0<b<1 , decay), " or k is the continuous growth/decay rate in the e^{kx} form. 3. Logistic model : y = frac{L}{1 + Ae^{-kx}} " L is the carrying capacity (the maximum limit the function approaches). " More complex to get parameters from basic regression, but many graphing calculators or software can do logistic regression. 4. Polynomial model (like quadratic): y = ax^2 + bx + c
[...]
>>
