During Class
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 following 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.
Mastery Learning - Specifications Grading
- What is mastery learning?
- What is specifications grading?
- How do I get an A? Let's check the Silly Bus.
- What should I be doing between class?
- Engagement - Between Class Sessions
- Knewton-Alta
- Projects
- What should I be doing between class?
- Remember The Power of Yet.
Team Discussion
We'll split into teams. Jot you name on the board to help everyone remember names. Then work through the following problems, having someone act as scribe for the group. Remember to Pass The Chalk between each problem, giving everyone a chance to act as scribe.
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: Infinity is a concept, not a number. 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.
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 following 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.
4.
Consider the following 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 came 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 all in your team to replicate the plot.
Before Next Class Session
Work through the following Tasks and come ready to teach something from what you did in the prep to your team during our next class.
Start working with R and RStudio
- Install R - https://www.r-project.org/ (Use 4.1.1) - Take screenshots of anything you have questions on, and post them to the Prep Upload page.
- 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 Teams, 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 in class, or one of your choice).
Here are some examples of meaningful work you could upload.
- Copy/pasting some R code along with the output from R for your computations and/or graph.
- Pictures of error messages that you get when trying to get things installed, or questions you were unable to find answer to after working towards completing this task.
- Links to a tutorial that helped you complete parts of this task.
Familiarize yourself with class
- Read the syllabus - https://byui.instructure.com/courses/152636/assignments/syllabus. If you have any questions, jot them down to discuss tomorrow.
- Open up a Knewton Alta assignment and work on a problem. When you complete one, get stuck on one, or encounter any issues logging in, please take a screenshot of what happened to share. Hopefully as a team you'll be able to troubleshoot together any issues you encounter.
- Skim read the instructions for Project 1. If you have any questions, jot them down to discuss tomorrow.
Here are some examples of meaningful work you could upload.
- Questions you have from the reading.
- A screenshot of completed problem from KnewtonAlta that you enjoyed, showing what it looks like to complete a problem.
- A screenshot of a problem that stumped you from KnewtonAlta that you would like help with.
- Error messages that may have appeared as you tried to interact with KnewtonAlta.
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