Class 8

Between Class Sessions - Prep for Day 8

Please spend around 2 hours working between class sessions, focusing on the tasks below. Use any extra time to complete KnewtonAlta assignments and/or work on Project tasks.

Pick something from your prep today that you can share with your group in class. It might be something new that you learned. It might be questions you have that are still unanswered, or a question along with what helped you eventually answer it. It might be something tricky that you solved. It might be a review topic that helped you remember something. It might be a conversation you had with AI that was helpful. You will have a chance to share this with your peers during class. Come ready to articulate your thinking and questions.

Preparation

(1) Exponential Function Practice

Watch this video from mathispower4u, Graphing Exponential Functions (~6.5 minutes)

Now complete the following:

  1. Let \(f(x) = 4^x\) and complete the following exercises. Find the exact values.

    • Evaluate \(f(1)\)
    • Evaluate \(f(0)\)
    • Evaluate \(f(-1)\)
    • Evaluate \(f(-2)\)
    • Evaluate \(f(\frac{1}{2})\)
    • Evaluate \(f(3)\)
    • Are there any real numbers that would cause us a problem when trying to evaluate \(f\) at that number?
    • What do you notice about all the outputs of \(f\)?
    • Graph the function \(f(x) = 4^x\)
    • Is this function increasing or decreasing on the interval \((-\infty,\infty)\)?

  2. Graph the function \(g(x) = (1/2)^x\). Is this function increasing or decreasing on the interval \((-\infty,\infty)\)?

  3. If \(h(x) = a^x\) is an increasing function, what do you know about the parameter \(a\)?

(2) Project 1 Task 2

Work on project 1 task 2, with the goal of submitting it. Remember to complete the AI log.

Regular Reminders

Skill Practice (KA Homework)

Start working on

  • 1 – Definition of a Logarithm
  • 1 – Evaluate Exponential and Logarithmic Functions
  • 1 – Properties of Logarithms

Applied Practice (Project Work)

  • Work on Project 1 Task 2

During Class

Brain Gains

  1. Consider the function \(f(x; a,b) = a + bx^2\). If we know \(f(0) = 10\) and \(f(5) = 510\), what are the values of \(a\) and \(b\).
Answers

Since \(f(0) = a + b(0)^2 = 10\), we see \(a = 10\). Since \(f(5) = 10 + b(25) = 510\), we see \(b = 20\).

Note: We have used the method of substitution to solve the linear system of equations for the parameters \(a\) and \(b\). \(\left\{ \begin{array}{ll} a + b(0) &= 10 \\ a + b(25) &= 510. \end{array} \right.\)

  1. Construct a graph of \(f(x) = 3^{-x}\) for \(-2\leq x\leq 2\).
Solution

Making a table of 5 points (using \(x=-2,-1,0,1,2\)) is a simple way to complete this. We can also rapidly do this in R.

f <- function(x){3^(-x)}
x <- seq(-2,2,0.1)
plot(x,f(x),type = "l")
  1. Give a numerical approximation to \(e^{2}\) using R.
Solution

The function \(e^{ ( ) }\) in R is given using the exp() function.

exp(2)
#7.389056

What is \(e\)? It’s a number.

exp(1)
#2.718282

The number arises from calculus, and we’ll see in a few weeks why \(e\) is special.

Group Meeting

Start by giving each person a moment to share what they chose to prepare for class. Help each other address any questions. When each person has had a chance to share, move on the other activities.

Compare your Project 1 Task 2

Share with each other what you did for project 1 task 2. Help address any questions you have. After class today, please make any needed revisions and then resubmit.

Graph Exponential Functions

Graph each function below, showing at least 2 points, and any horizontal asymptote. End by stating the \(y\) intercept.

  • \(f(x) = 2^{x-3}\)
  • \(f(x) = (1/3)^{x}+4\)

Example Project Tasks 2 and 3

  • Open the Example Project Instructions, and briefly read Task 2 and Task 3.
  • Use the Desmos links to explore the three probability density models \(f_0\), \(f_1\), and \(f_2\).
    • Use the slider to alter the values of \(a\) and \(b\). Discuss how these parameters alter the graph of \(f_0\).
    • Use the slider to alter the values of \(h\) and \(a\). Discuss how these parameters alter the graph of \(f_1\).
    • Use the slider to alter the values of \(h\), \(a\), and \(b\). Discuss how these parameters alter the graph of \(f_2\).

Discussion

Deterministic Versus Probabilistic Models

We’ll have a high level discussion about where we are headed with the the projects.

  • What is a deterministic model?
Answer

A deterministic model gives a single, fixed output for each input — the same input always produces the same output. Examples include \(A = \pi r^2\) or \(d = v \cdot t\). If we model a quantity with \(f(x) = ax^2 + bx + c\), plugging in \(x = 3\) always yields the same number. Project Unit 2 is built around fitting deterministic models to data.

  • What is a probabilistic model?
Answer

A probabilistic model describes uncertainty. Instead of producing a single output, it tells us how likely different outputs are. The functions \(f_0, f_1, f_2\) are probability density models — they don’t say what a single light bulb’s intensity will be, they describe how the intensity of many bulbs is distributed. Probabilistic models are used when the outcome depends on many small, hard-to-predict factors. Project Unit 3 focuses on probabilistic models.

Same input, same output → deterministic. Same input, a distribution of possible outputs → probabilistic.

How does AI fit into this?

Function “Mad Libs”

Let’s list several quantities that we can measure. We’ll use these to construct madlibs with mathematics. Functions tell a story, and we need to learn to tell that story.

  • Quantity 1:
  • Quantity 2:

For the model \(y=f(x;a=10,b=4)=10-\sqrt{x+4}\), we’ll tell the story that occurs if we let \(x\) be quantity 1, and \(y\) be quantity 2.

Example
  • Quantity 1: tire pressure of a bicycle tire, measured in PSI
  • Quantity 2: distance from Earth to the Sun, measured in light-years

So our model becomes \[y = 10 - \sqrt{x+4}\] where \(x\) is the tire pressure and \(y\) is the distance to the Sun.

psi <- c(0, 5, 12, 21, 60, 96, 100)
distance <- 10 - sqrt(psi + 4)
data.frame(psi, distance)
  • Use the code below to plot this function in R, and then tell the story.
x <- seq(-4,20,0.1)
y <- 10-sqrt(x+4)
par(mar=c(2.5,2.5,0.5,0.5))
plot(x,y,type='l')
x <- seq(-4,200,0.1)
y <- 10-sqrt(x+4)
par(mar=c(2.5,2.5,0.5,0.5))
plot(x,y,type='l')
  • How could we interpret \(x=0\)?
  • How could we describe the meaning of \(x=-4\)?
  • What’s the story when \(x=21\)?
  • What does this story mean if \(x=117\)

Summary

A function tells a story about a relationship between two quantities, whether that relationship is realistic or not.

  • Telling the story given by a model is a skill different from but related to the skill of determining whether the model is a good description of a relationship being studied.
  • Telling the story given by a model is a skill different from but related to adjusting the model to better describe a relationship being studied.
  • Telling the story given by the model (no matter how ridiculous) is a skill that takes practice.
  • To use a model there are two translation steps required. (We focus on the second translation step in this class.)
    • Translate the story to mathematics
    • Use some mathematics to study the relationship (or process)
    • Translate the mathematics back to a story.
  • Can you tell the story given by the function, regardless of how absurd that story is?

Example Project Tasks 2 and 3

  • Open the Example Project Instructions, and briefly read Task 2 and Task 3.
  • Use the Desmos links to explore the three probability density models \(f_0\), \(f_1\), and \(f_2\).
    • Use the slider to alter the values of \(a\) and \(b\). Discuss how these parameters alter the graph of \(f_0\).
    • Use the slider to alter the values of \(h\) and \(a\). Discuss how these parameters alter the graph of \(f_1\).
    • Use the slider to alter the values of \(h\), \(a\), and \(b\). Discuss how these parameters alter the graph of \(f_2\).

We’ll now have a discussion in class about what you observed.

Our goal is to understand enough about the general functions \(f_0\), \(f_1\), and \(f_2\) and their parameters to be able to identify specific functions (so specified parameters) of the forms \(f_0\), \(f_1\), and \(f_2\) that looks like the density histogram of our light bulb data. The code below will regenerate this histogram for us.

library(data4led)
dist <- led_time(2100)
hist(dist$percent_intensity,probability = TRUE)
  • For each of the 3 functions, use the sliders in Desmos to try and pick reasonable values for the parameters to provide a reasonable visual fit to the probability histogram.

Summary Plots

We’ll spend some time today in class discussing the Example Project. Here’s a quick snippet of code to draw the histogram representing the percent intensity of 202 bulbs after about 2100 hours.

library(data4led)
dist <- led_time(2100)
hist(dist$percent_intensity,probability = TRUE)
  • What characteristics do you notice about the density histogram of our data? By identifying the characteristics we see in the data we know what behavior we would like the function curve to have.
    • What is the smallest value?
    • What is the largest value?
    • What is the location of the peak?
    • What is the width of the peak?

Let’s look at the first model \(f_0\).

  • What did you notice about the general model \(f_0\) and its parameters?
    • What is the domain of \(f_0\)?
    • What does \(a\) do?
    • What does \(b\) do?
    • Try to pick some values for the parameters that provide a visual fit to the data.
    • What story does this model tell, in the context of percent intensities of light bulbs?

Let’s summarize our observations with a few plots. Run this code in your Console to plot representative curves for the parameter \(a\) in function \(f_0\).

rm(list=ls())
f0 <- function(L,a=0,b=1){
  # Make sure a < b when using this function.
  ifelse(L < a,NaN, ifelse(L <= b, 1/(b-a), NaN))
}

a <- [your selected value]
b <- [your selected value]
L <- seq(a,b,0.1)
y <- f0(L,a,b)

par(mfrow=c(1,2),mar=c(2.5,2.5,1,0.25))
plot(L,y,type='l',xlim=c(90,110), ylim = c(0,1))
mtext('For f0: a=, b=', side = 3, line = 0)

a <- [another selected value]
L <- seq(a,b,0.1)
y <- f0(L,a,b)

plot(L,y,type='l',xlim=c(90,110), ylim = c(0,1))
mtext('change a= (keep b=)', side = 3, line = 0)

Now let’s look at the second model \(f_1\).

  • What did you notice about the general model \(f_1\) and its parameters?
    • What is the domain of \(f_1\)?
    • What does \(h\) do?
    • What does \(a\) do?
    • Try to pick some values for the parameters that provide a visual fit to the data.
    • What story does this model tell, in the context of percent intensities of light bulbs?

Let’s summarize our observations with a few plots. Run this code in your Console to plot representative curves for the parameter \(h\) in function \(f_1\).

rm(list=ls())
f1 <- function(L,h=0,a=1){
#Make sure h > 0 and a > 0.
1/sqrt(2*pi*a)*exp(-(L-h)^2/(2*a))
}

a <- [your selected value]
L <- seq(80,120,0.1)
h <- [your selected value]
y <- f1(L,h,a)

par(mfrow=c(1,2),mar=c(2.5,2.5,1,0.25))
plot(L,y,type='l',xlim=c(80,120))
mtext('plot f1 with h= and a=', side = 3, line = 0)

h <- [another selected value]
y <- f1(L,h,a)

plot(L,y,type='l',xlim=c(80,120))
mtext('change h=, keep a=', side = 3, line = 0)

Run this code in your Console to plot representative curves for the parameter \(a\) in function \(f_1\).

h <- [your selected value]
L <- seq(80,120,0.1)
a1 <- [your selected value]
y3 <- f1(L,h,a1)

par(mfrow=c(1,2),mar=c(2.5,2.5,1,0.25))
plot(L,y3,type='l',xlim=c(80,120))

a2 <- [your selected value]
y4 <- f1(L,h,a2)

plot(L,y4,type='l',xlim=c(80,120))
mtext('change h=, keep a=', side = 3, line = 0)
  • What did you notice about the general model \(f_2\) and its parameters?
    • What is the domain of \(f_2\)?
    • How do the parameters of \(f_2\) change the behavior of \(f_2\)?
    • Try to pick some values for the parameters that provide a visual fit to the data.
    • What story does this model tell, in the context of percent intensities of light bulbs?

Let’s summarize our observations with a few plots.

Run this code in your Console to plot representative curves for the parameter \(h\) in function \(f_2\).

rm(list=ls())
f2 <- function(L,h=0,a=1,b=5){
# Make sure a > 0 and b > 0.

out <- rep(-1,length(L))
out[(L < h)] <- 0*L[(L < h)]
out[(L >= h)] <- b^a/gamma(a)*(L[(L >= h)]-h)^(a-1)*exp(-b*(L[(L >= h)]-h))

return(out)
}

a <- [your selected value]
b <- [your selected value]
h <- [your selected value]
L <- seq(h,120,0.1)
y <- f2(L,h,a,b)

par(mfrow=c(1,2),mar=c(2.5,2.5,1,0.25))
plot(L,y,type='l',xlim=c(80,120))

h <- [your selected value]
L <- seq(h,120,0.1)
y <- f2(L,h,a,b)

plot(L,y,type='l',xlim=c(80,120))
mtext('Changing h in f2', side = 3, line = 0, outer = TRUE)

Run this code in your Console to plot representative curves for the parameter \(a\) in function \(f_2\).

h <- [your selected value]
b <- [your selected value]
L <- seq(h,120,0.1)
a <- [your selected value]
y <- f2(L,h,a,b)

par(mfrow=c(1,2),mar=c(2.5,2.5,1,0.25))
plot(L,y,type='l',xlim=c(80,120))

a <- [your selected value]
y <- f2(L,h,a,b)

plot(L,y,type='l',xlim=c(80,120))
mtext('Changing a in f2', side = 3, line = 0, outer = TRUE)

Run this code in your Console to plot representative curves for the parameter \(b\) in function \(f_2\).

h <- [your selected value]
a <- [your selected value]
L <- seq(h,120,0.1)
b <- [your selected value]
y <- f2(L,h,a,b)

par(mfrow=c(1,2),mar=c(2.5,2.5,1,0.25))
plot(L,y,type='l',xlim=c(80,120))

b <- [your selected value]
y <- f2(L,h,a,b)

plot(L,y,type='l',xlim=c(80,120))
mtext('Changing b in f2', side = 3, line = 0, outer = TRUE)

Source: Class.8 on byuimath.com