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Catégorie :Category: nCreator TI-Nspire
Auteur Author: joao.perrac
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 8.56 Ko KB
Mis en ligne Uploaded: 23/03/2025 - 12:21:29
Uploadeur Uploader: joao.perrac (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4543643
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 8.56 Ko KB
Mis en ligne Uploaded: 23/03/2025 - 12:21:29
Uploadeur Uploader: joao.perrac (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4543643
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
<<
Below is a thoroughly expanded guide on how to approach a real-world data modeling problem (for example, where you have time vs. measured quantity data). This includes why each step is necessary, what you might write in your explanation, and how to find an exponential equation by hand and using a calculator (without just letting a program spit out the entire result). 1. Organize & Understand the Data Why this step? You need a clear picture of what exactly your data points representthis sets the stage for everything else. Having disorganized or misunderstood data makes it impossible to choose the right model or interpret results correctly. What to do: List all the data points in a clear table. For example: x x (Time) y y (Measured Quantity) 1 40 2 55 3 72 & & Define your variables precisely: Example explanation : Let x x = the number of years since 2010 (so x = 0 x=0 corresponds to 2010, x = 1 x=1 to 2011, etc.). Let y y = the population of a certain species at year x x . Note any special events or data oddities: Did something happen that year (e.g., an intervention, a market shift)? Are there missing values or outliers? Example text you might write : We collected data over several years, tabulating the measured quantity (our dependent variable) alongside the corresponding year (our independent variable). To keep the numbers manageable, we define x x as the number of years since 2010. This means our data table is as follows: & 2. Plot the Data (On Paper) Why this step? Visualizing the data helps you spot patterns like a roughly straight line (linear), a J-shaped curve (exponential), a U- or )-shaped pattern (quadratic), or other possibilities (logistic, etc.). A quick scatterplot can save you a lot of time deciding on the function type. What to do: Draw an x-axis and y-axis on your paper. Mark each data point : The x x -coordinate is your time variable (or whatever your independent variable is). The y y -coordinate is the measured quantity. Observe the shape : If it looks like its increasing at an increasing rate, that might suggest exponential growth. If its decreasing at a (roughly) constant percentage, that suggests exponential decay. If its a straight-line pattern, it might be linear. Example text you might write : Plotting each data pair ( x , y ) (x,y) on a scatterplot, I notice the points form a curve that rises more steeply each year rather than forming a straight line. This visual suggests an exponential trend rather than a linear one. 3. Choose & Justify the Model Why this step? You must confirm which mathematical function family fits best with your data. Teachers expect you to explain why you chose it (not just I guessed. ). What to do: Recall typical shapes : Linear : changes by a constant difference each time step. Exponential : changes by a constant factor (or percentage) each time step. Quadratic : a single peak or trough (U-shape or )-shape). Logarithmic / Logistic : specialized patterns (very rapid initial change, then slowing or leveling off). Match your datas shape from the scatterplot to one of these patterns. Explain your reasoning clearly. Example text you might write : The data shows that the quantity grows more each yeara hallmark of exponential growth. Thus, we will model it with an exponential function of the form y = A Å b x y = A cdot b^x . A linear model would not capture the accelerating increases we observe. 4. Formulating the Exponential Model Now that youve decided on an exponential form ( y = A Å b x y = A cdot b^x or y = A e k x y = A e^{kx} ), you need to find the parameters A A and b b (or A A and k k ). Why this step? You need a numerical equation that can be used to make predictions. Simply saying it looks exponential is incomplete. What to do (Two Methods): Method A: By Hand (If You Have Two Key Points) Select two representative data points : ( x 1 , y 1 ) (x_1, y_1) and ( x 2 , y 2 ) (x_2, y_2) . Solve for b b : b = ( y 2 y 1 ) 1 x 2 x 1 . b = left(frac{y_2}{y_1}right)^{frac{1}{x_2 - x_1}}. This step uses the fact that y 2 y 1 = A Å b x 2 A Å b x 1 = b ( x 2 x 1 ) . frac{y_2}{y_1} = frac{A cdot b^{x_2}}{A cdot b^{x_1}} = b^{, (x_2 - x_1)}. Solve for A A using one of the points (like ( x 1 , y 1 ) (x_1, y_1) ): A = y 1 b x 1 . A = frac{y_1}{b^{x_1}}. Example : Using points (1, 40) and (4, 100), b = ( 100 40 ) 1 4 1 H 1.35 , A = 40 1.35 1 H 29.6. b = left(frac{100}{40}right)^{frac{1}{4-1}} approx 1.35, quadA = frac{40}{1.35^{,1}} approx 29.6. Hence, y H 29.6 × ( 1.35 ) x . y approx 29.6 times (1.35)^x. Method B: Using a Calculator (When You Have Several Points) Enter your data into the calculators lists: x x -values in L1, y y -values in L2 (or similar). Choose ExpReg from the regression menu (in many graphing calculators, its under [STAT] [CALC]). The calculator computes best-fit parameters a a and b b that minimize the sum of squared erro
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Compatible OS 3.0 et ultérieurs.
<<
Below is a thoroughly expanded guide on how to approach a real-world data modeling problem (for example, where you have time vs. measured quantity data). This includes why each step is necessary, what you might write in your explanation, and how to find an exponential equation by hand and using a calculator (without just letting a program spit out the entire result). 1. Organize & Understand the Data Why this step? You need a clear picture of what exactly your data points representthis sets the stage for everything else. Having disorganized or misunderstood data makes it impossible to choose the right model or interpret results correctly. What to do: List all the data points in a clear table. For example: x x (Time) y y (Measured Quantity) 1 40 2 55 3 72 & & Define your variables precisely: Example explanation : Let x x = the number of years since 2010 (so x = 0 x=0 corresponds to 2010, x = 1 x=1 to 2011, etc.). Let y y = the population of a certain species at year x x . Note any special events or data oddities: Did something happen that year (e.g., an intervention, a market shift)? Are there missing values or outliers? Example text you might write : We collected data over several years, tabulating the measured quantity (our dependent variable) alongside the corresponding year (our independent variable). To keep the numbers manageable, we define x x as the number of years since 2010. This means our data table is as follows: & 2. Plot the Data (On Paper) Why this step? Visualizing the data helps you spot patterns like a roughly straight line (linear), a J-shaped curve (exponential), a U- or )-shaped pattern (quadratic), or other possibilities (logistic, etc.). A quick scatterplot can save you a lot of time deciding on the function type. What to do: Draw an x-axis and y-axis on your paper. Mark each data point : The x x -coordinate is your time variable (or whatever your independent variable is). The y y -coordinate is the measured quantity. Observe the shape : If it looks like its increasing at an increasing rate, that might suggest exponential growth. If its decreasing at a (roughly) constant percentage, that suggests exponential decay. If its a straight-line pattern, it might be linear. Example text you might write : Plotting each data pair ( x , y ) (x,y) on a scatterplot, I notice the points form a curve that rises more steeply each year rather than forming a straight line. This visual suggests an exponential trend rather than a linear one. 3. Choose & Justify the Model Why this step? You must confirm which mathematical function family fits best with your data. Teachers expect you to explain why you chose it (not just I guessed. ). What to do: Recall typical shapes : Linear : changes by a constant difference each time step. Exponential : changes by a constant factor (or percentage) each time step. Quadratic : a single peak or trough (U-shape or )-shape). Logarithmic / Logistic : specialized patterns (very rapid initial change, then slowing or leveling off). Match your datas shape from the scatterplot to one of these patterns. Explain your reasoning clearly. Example text you might write : The data shows that the quantity grows more each yeara hallmark of exponential growth. Thus, we will model it with an exponential function of the form y = A Å b x y = A cdot b^x . A linear model would not capture the accelerating increases we observe. 4. Formulating the Exponential Model Now that youve decided on an exponential form ( y = A Å b x y = A cdot b^x or y = A e k x y = A e^{kx} ), you need to find the parameters A A and b b (or A A and k k ). Why this step? You need a numerical equation that can be used to make predictions. Simply saying it looks exponential is incomplete. What to do (Two Methods): Method A: By Hand (If You Have Two Key Points) Select two representative data points : ( x 1 , y 1 ) (x_1, y_1) and ( x 2 , y 2 ) (x_2, y_2) . Solve for b b : b = ( y 2 y 1 ) 1 x 2 x 1 . b = left(frac{y_2}{y_1}right)^{frac{1}{x_2 - x_1}}. This step uses the fact that y 2 y 1 = A Å b x 2 A Å b x 1 = b ( x 2 x 1 ) . frac{y_2}{y_1} = frac{A cdot b^{x_2}}{A cdot b^{x_1}} = b^{, (x_2 - x_1)}. Solve for A A using one of the points (like ( x 1 , y 1 ) (x_1, y_1) ): A = y 1 b x 1 . A = frac{y_1}{b^{x_1}}. Example : Using points (1, 40) and (4, 100), b = ( 100 40 ) 1 4 1 H 1.35 , A = 40 1.35 1 H 29.6. b = left(frac{100}{40}right)^{frac{1}{4-1}} approx 1.35, quadA = frac{40}{1.35^{,1}} approx 29.6. Hence, y H 29.6 × ( 1.35 ) x . y approx 29.6 times (1.35)^x. Method B: Using a Calculator (When You Have Several Points) Enter your data into the calculators lists: x x -values in L1, y y -values in L2 (or similar). Choose ExpReg from the regression menu (in many graphing calculators, its under [STAT] [CALC]). The calculator computes best-fit parameters a a and b b that minimize the sum of squared erro
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