Today is Day 1
During Class
Welcome!
- The address for this Wiki is https://byuimath.com/bmw/all/119/.
- First Day Challenge: Exchange name and contact information with at least 1 other student in this class during our work today.
Brain Gains
1. Let $f(x,y) = 4x^2 -5y\sqrt{x+1}$. Find each of the following values.
- $f(3,2)$
- $f(0,-2.1)$
- $f(-2.1,5)$
Answers
- $f(3,2) = 16$
- $f(0,-2.1) = 10.5$
- $f(-2.1,5)$ is undefined
Note: The terms "undefined" and "no solution" have different meanings mathematically. In this case, $f(-2.1,5)$ is undefined because when we try to evaluate $f(-2.1,5)$, we see $f(-2.1,5) = 4(-2.1)^2 - 5(5)\sqrt{-2.1+1} = 17.64 - 25\sqrt{-1.1}$. Since there is no real number such that when you square the number you get -1.1, we see $\sqrt{-1.1}$ is undefined when we are working with real number. You can assume that we will always be working with real numbers in this class, unless otherwise specified.
2. Consider the piecewise function $$f(x) = \begin{cases} x & \quad x\geq 0 \\ \\ -x & \quad x<0 \end{cases}.$$ Find the following values.
- $f(-2)$
- $f(-27)$
- $f(0)$
- $f(\sqrt{3})$
Answers
- $f(-2) = 2$
- $f(-27) = 27$
- $f(0) = 0$
- $f(\sqrt{3}) = \sqrt{3} \approx 1.732051$
The number $f(3)$ an irrational number which means any decimal representation of this number is just an approximation. The exact number is represented by the symbol $\sqrt{3}$ which represents the positive real number that when you square it you get exactly 3.
Group Discussion
We'll split into randomized groups every day of class. Jot your name on the board to help us all learn each other's names. Then work through the following problems.
- Take turns acting as scribe for the group. When you are the scribe, it's fine to not know exactly how to proceed.
- The goal is to ask questions of each other and help each other.
- Leave up your work so that you can compare your answers with neighbors.
1. Let $f(x) = \frac{2x+4}{x^2}$. Find each of the following values.
- $f(-4)$
- $f(1)$
- $f(0)$
Answers
- $f(-4) = -0.25$
- $f(1) = 6$
- $f(0)$ is undefined
Note: The number $f(0)$ is undefined in this case because when we try to evaluate $f(0)$ we see $f(0) = \frac{2(0)+4}{0^2} = \frac{4}{0}$. What does it mean to divide the quantity 4 into 0 parts? This operation is not defined. Some of you may have had calculus before and tried to give an answer of infinity. Note that infinity is a concept, not a number.
Another way to think of division is that we are looking for a number $k$ so that $k = \frac{n}{d}$. Thus $k$ multiplied by $d$ would need to be equal to $n$, in other words $k d = n$. If the numerator is 4, $n=4$, and the denominator is 0, $d = 0$, this would mean $0 = 4$ ($k \cdot 0 = 4$) for any choice of $k$, which doesn't make any sense (we have a contradiction). So we say division by zero is undefined.
2. Let $f(x) = 8x^2 - 15$. Find each of the following values.
- $f(-2)$
- $f(1)$
Answers
- $f(-2) = 17$
- $f(1) = -7$
3.
Consider the piecewise function
$f(x) =
\begin{cases}
x^2 & \quad x\geq 0 \\
-x+4 & \quad x<0
\end{cases}.$
Find the following values.
- $f(-2)$
- $f(-27)$
- $f(0)$
- $f(\sqrt{3})$
Answers
- $f(-2) = 6$
- $f(-27) = 31$
- $f(0) = 0$
- $f(\sqrt{3}) = 3$
4.
Consider the piecewise function
$f(x) =
\begin{cases}
x^3 & \quad x < -1 \\
-2 & \quad -1 < x < 4 \\
\sqrt{x} & \quad x \geq 4
\end{cases}.$
Find the following values.
- $f(-2)$
- $f(-0.5)$
- $f(3)$
- $f(0)$
- $f(5.2)$
- $f(-1)$
- $f(4)$
Answers
- $f(-2) = -8$
- $f(-0.5) = -2$
- $f(3) = -2$
- $f(0) = -2$
- $f(5.2) \approx 2.280351$. The number $f(5.2)$ an irrational number which means any decimal representation of this number is just an approximation. The exact number is represented by the symbol $\sqrt{5.2}$ which represents the positive real number that when you square it you get exactly 5.2.
- $f(-1)$ is undefined. We have no rule that tells us what we the output will be when the input is -1. Generally in situations like this, we assume that -1 is not in the domain of $f$ rather than that the author just forgot to tell us the rule for the output connected to the input -1.
- $f(4) = 2$
5. Let $f(x) = x^2$. Determine whether or not $f(a+b) = f(a) + f(b)$. Carefully explain your answer. While an example is not an explanation, sometimes examples can be a helpful part of an explanation.
Possible Explanation
Given $f(x) = x^2$, we will show $f(a+b) = f(a) + f(b)$ is not true for all values of $a$ and $b$ by providing a pair of number $a$ and $b$ such that $(a+b)^2 \neq a^2 + b^2$ (a counter example). Let $a=1$ and $b=7$. We see $(a+b)^2 = (1+7)^2 = 8^2 = 64$ and $a^2 + b^2 = 1^2 + 7^2 = 1 + 49 = 50$. We see $(a+b)^2 = 64 \neq 50 = a^2 + b^2$ or $(a+b)^2 \neq a^2 + b^2$.
We can also show that $f(a+b) = f(a) + f(b)$ is not a true statement by showing that $f(a+b) \neq f(a) + f(b)$ in general when $f(x) = x^2$. We see $f(a) = a^2$ and $f(b) = b^2$. Now we compute $f(a+b) = (a+b)^2 = (a+b)(a+b) = a^2 + 2ab + b^2 \neq a^2 + b^2 = f(a) + f(b)$ when $2ab \neq 0$. Thus we see that $f(a+b) \neq f(a) + f(b)$ unless $2ab = 0$.
Important take away:
- The square of a sum is NOT the sum of the squares, $(a + b)^2 \neq a^2 + b^2$.
- The operation of addition must happen before the power. The order of operations is grouping and then exponents.
6. Construct a plot of as many of the functions above as you have time. Use whatever software package you are comfortable with, helping each other to replicate the plot.
Class Discussion
Mastery Learning - Specifications Grading
- What is mastery learning?
- What is specifications grading?
- Course Introduction and Syllabus.
- Are you in the right class?
- Weekly rhythm, what should I be doing between class?
- Attendance & Preparation - Between Class Sessions
- Knewton-Alta
- Projects
Between Class Sessions - Prep for Day 2
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) Familiarize yourself with class
- Read the Syllabus
- Complete the Background Survey
- Open a Knewton Alta assignment and answer some questions (for example you could open up "1 -- Function Evaluation"). The point is to make sure you have gotten into the system, and seen how it works. Feel free to complete one of the assignments.
- Skim read the instructions for Project 1.
(2) Graphing Functions
Our first Unit is focused on expanding our knowledge and understanding of functions. Plot the following functions by hand. Select five or more input values, calculate the corresponding output values, plot the points, and connect the points with a line. Check your work by plotting the function with Desmos (or another graphing calculator).
- $f(x) = 1-4x$
| $x$ | $f(x)$ | $ (x, f(x)) $ |
| -2 | 9 | (-2,9) |
| -1 | ||
| 0 | ||
| 1 | ||
| 2 |
- $g(x) = |2x+4|$
| $x$ | $g(x)$ | $ (x, g(x)) $ |
| -4 | 4 | (-4,4) |
| -3 | ||
| -2 | ||
| -1 | ||
| 0 |
- $h(x) = \frac{1}{2}x^2+3$
| $x$ | $h(x)$ | $ (x, h(x)) $ |
| -2 | 5 | (-2,5) |
| -1 | ||
| 0 | ||
| 1 | ||
| 2 |
(3) Start working with R and RStudio
- Install R - https://www.r-project.org/
- Install RStudio - https://www.rstudio.com/products/rstudio/download/
- Open a new file in RStudio and use it to perform a few computations. One option is to repeat some of the computations you did during class in groups, but this time using R to perform the computations. Or you may pick something else of interest. Feel free to search the web for a tutorial on performing computations with R in RStudio.
- Construct the plot of a function (either one from part 2 above, from in class, or one of your choice).
You could play with and adjust the code below as a starting point.
x <- c(1,2,3,4,5) y <- c(1,4,9,16,25) plot(x,y,type='l')
(4) Pick something to share
Remember to pick something from the above that you can share with your peers tomorrow. What could that be?
- Did you encounter any issues with KnewtonAlta or have questions? Was there a problem in what you attempted that helped you remember something.
- As you plotted functions from (2), were there any tricky parts? Did you use Desmos or something else? Do you have any tips for plotting?
- How did installing R and RStudio go? If there were any hiccups along the way, what did you do to resolve them? Did you plot a function and are you willing to share the code you used? Were there any bits of code that didn't make sense? Did you ask AI for help to plot a function and have a record of your chat history you can share?
- Something else. Remember that questions can be just as valuable as solutions. If you worked on something and know it's wrong, sometimes the most valuable thing you can share is that wrong attempt. This will generate discussion and help all grow.
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