Precalculus Labs (ENTIRE COURSE) Mr. Johnston
This is a series of Labs I am in the process of creating to accompany my Precalculus course. Each lab has an accompanying Google Document. It follows along with Precalculus, Larson/Hostetler, 6th Edition. Feel free to email me at eric1221cire@gmail.com with any questions. The Google Docs questions are found [url]https://drive.google.com/folderview?id=0B1WUbennl4FDVFprMXJ2Tjk0dzA&usp=sharing[/url].
Sommario
- Functions and their Graphs
- Lab01 Exploring GeoGebra and Circles
- Lab04 Function Transformations, Compositions and Inverses
- Lab05 GeoGebra and Function Graphs
- Polynomial and Rational Functions
- Lab06 Polynomial Function Graphs; Long and Synthetic Division
- Lab07 Polynomial Graphs and Complex Numbers
- Exponential and Logarithmic Functions
- Lab09 Exponential and Logarithmic Functions
- Lab10 Sound and Earthquake Simulations
- Trigonometry
- Lab01 Right Triangle Trig and The Unit Circle
- Law of Sines (& Area)
- Analytic Trigonometry
- Additional Topics in Trigonometry
- Systems of Equations and Inequalities
- Lab11 System of Equations - Multivariable
- Lab 12 Multimedia Presentations
- Matrices and Determinants
- Lab13 Welcome to the Matrix
- Lab14 This is so Matrice-sy!
- Sequences, Series, and Probability
- Topics in Analytic Geometry
Lab01 Exploring GeoGebra and Circles
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Lab06 Polynomial Function Graphs; Long and Synthetic Division
1) End Behavior of Polynomials
In the following applet, investigate the degree and leading coefficient of each polynomial and compare it to the graphs end behavior.
2) Zeros of a Polynomial Function
Use the slides to investigate the behavior of a zero based on the degree of the factor.
3) Intermediate Value Theorem
The Intermediate Value Theorem is an Existence Theorem. it can either confirm the existence of a value or it can not confirm. We will use this theorem to narrow down zeros of our polynomial functions.[br][br]Verify the first table entry in the Lab06.[br][br]Move the sliders along the axis so that N = 0, a = 1 and b = 3.[br][br]Complete the table in the lab.
4) Quadratic Modeling
On pg 115 of the textbook, the path of the baseball can be modeled using the function [math]f\left(x\right)=-0.0032x^2+x+3[/math]. We have already determined the distance and maximum height of the ball (vertex) algebraically but this can also be performed on GeoGebra. In general, when you know how to determine something 100% algebraically, you can then explore using technology.[br][br]Determine the x-intercepts algebraically (quadratic formula) and fill in the lab.[br][br]Enter the function for the path of the ball in the applet below.[br][br]A synonymous term for x-intercept is root. We can confirm our Quadratic Formula by typing the command [b]root(f(x)) [/b] typing "f(x)" and not the entire function[b].[/b] Watch for the two roots to be listed as points A and B. *If your algebra does not match up, fix your work before submitting the snapshot. [br][br]The roots give us the practical [b]domain[/b] [A,B] on which our function takes on practical [b]range[/b] values ( positive values [0, MAXIMUM] for the height of a ball). [br][br]Use the x-intercepts as left and right bounds to find the maximum function value by typing the command: [b]max(f(x), the x-value from point A,the x-value from point B)[/b].[b] [/b]This should show up as point "C".
5) Use the GeoGebra Applet Above
[br]Reset the applet above and use the applet to complete pg 118 # 79 (The number of fixtures produced to yield a minimum cost). This function has no roots so we need to create a really wide range for the minimum. Type the [b]min[/b] command and use the x-values 0 and 1000000.[br][br]No algebraic work is necessary, all work should be done on GeoGebra.
Lab09 Exponential and Logarithmic Functions
1.) Compounding Interest - Simple Spreadsheets
Open the spreadsheet [url=https://docs.google.com/spreadsheets/d/1eHm7MbiRWi1BIOhzll9XZf7RXsi2Rs9bXVot_xC0kgc/edit?usp=sharing]Investment Calculator[/url] and save a copy to your drive so that you may edit it. Use the spreadsheet to complete question 1. [br][br]A time saving feature and one that allows exploration of "what-if" scenarios is the ability to reference cells in a formula. When you change the value of a cell that is being referenced, the formula automatically updates. For ease of use, the yellow cells are where the user should enter the varying input values and the green cells represent the desired result.[br][br]a) Double click in cell B7. The formula (function) will appear in multiple colored fonts in the header bar after fx. Analyze the function.[br][br]b) Use the "Simple" sheet to explore the resulting balance (A) for a principal investment of $2500 at an interest rate of 8% for 10 years. Complete the table in the lab. As you enter the new value in each of the yellow cells, what as cell B7 automatically updates. [br][br]c) To compute continuous interest, the [i]e[/i] in the formula [math]A=Pe^{rt}[/math] would be entered as EXP([i]exponent[/i]). In cell B8, enter a function that will compute the balance when the interest is compounded continuously ([i]notice that when you type in EXP( Sheets will show a completion and what you need to type in the parenthesis to use the EXP function).[/i]
2) Compounding Interest - Click n Drag Tables
Another time saving feature of using a spreadsheet is the ability to create a table (multiple applications of a formula) so that you can copy a formula without retyping the same formula over and over. [br][br]a) Analyze the formulas in cells B7 and E7 on Sheet "Click n Drag". [br][br]b) Select cell B7 on "Click n Drag" and it should be outlined with a black box on the bottom right corner. Click and drag the box on the bottom right corner of the cell and drag it so that B7 to B12 (B7:B12) are covered. When you release the clicker, the boxes should be filled with the formula and the balance should appear.[br][br]c) Instead of clicking and dragging, you can also copy and paste. Select cell E7. Press Ctrl + c to copy. Select cell E8:E12 and press Ctrl + v to paste the formula. Again, analyze the formulas in cells E8:E12.
3) Exponential Functions
In the following applet, explore the affects of changing a, b, h and k.
Exploring Exponential Functions
4) Exploring Logarithmic Functions
5) The mathematical Constant e
* Read about [i]e[/i] onWikipedia.[br][br]* One place where e comes up is when we calculate continuously compounded interest. If we take the [math]\left(1+\frac{1}{n}\right)^n[/math] part out of the compound interest formula, continuously compounded interest occurs when [math]n\longrightarrow\infty[/math] (n approaches infinity) or the frequency at which interest is compounded at all times.[br][br][br][br][br]Graph [math]y=\left(1+\frac{1}{x}\right)^x[/math]on your calculator. Determine the equation of the horizontal asymptote by looking an infinitely higher x values in the table OR using the Trace feature on the graph.[br][br]BONUS:[br]Make a "Click n Drag" table in a spreadsheet and paste the table in you lab. The table should show n =1, 2, 5, 10, 50, 100, 10000, 1000000000000 and the resulting value form your equation.
Lab01 Right Triangle Trig and The Unit Circle
Lab11 System of Equations - Multivariable
1) Carl Friedrich Gauss
Like most top mathematicians, Carl Friedrich Gauss was an interesting person...to say the least. [br] [img]http://www.rugusavay.com/wp-content/uploads/2012/12/Carl-Friedrich-Gauss-Quotes-5.jpg[/img][br] (quotesgram.com)[br][br]A popular anecdote of his:[br] "According to [url=https://en.wikipedia.org/wiki/Isaac_Asimov]Isaac Asimov[/url], Gauss was once interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment till I'm done." "(wikipedia)[br][br]Research the mathematician that brought us the method of Gaussian Elimination and the Gaussian Distributions (normal curves). Determine where he was born, when he lived, at what age and how he was recognized as a math genius and how he is commemorated. Include one interesting story about Gauss as well.
2) Gaussian Elimination Applet
Use the [url=http://math.mercyhurst.edu/~lwilliams/Applets/GaussianElimination.html]Gaussian Elimination Applet[/url] to solve three different systems of equations (they are generated at random for you). For each:[br][br]a) Use the row operations for Gauss-Jordan Elimination to get the system in Reduced Row Echelon Form.[br][br]b) "Take a partial screen shot" of the original problem. Paste it in the lab (see lab for what should be included in screenshot)[br] [img]http://8c58e64aa1a7874c55b4-be3e3aa1b17f89aab5ecca1936c616df.r83.cf3.rackcdn.com/wp-content/uploads/2015/03/Chromebook-Keyboard-Ctrl-Shift-and-Windows-Switcher1-e1426580459983.jpg[/img] (head4space.com)[br][br]Click and drag to highlight the area to capture. [br]Click "copy to clipboard" and paste in your lab using Ctrl + v.[br][br]c) On GeoGebra,plot the three original planes using "Start Creating" a "3-D Graphics" or editing the applet below.[br] Make each plane a different color[br] Plot the solution point on the graph[br] Adjust the graph to clearly show the solution point.[br] Take a screenshot of the entire applet below (one you change it for your problem).
Lab13 Welcome to the Matrix
1) Geometry
Begin this lab by cutting out the outline of your geometric shape. If you were not given a paper, print one [url=https://docs.google.com/document/d/13rYLFLgHRa5JOVgY10SRBUYoFg--R4I4x5yznKgxsB8/edit?usp=sharing]here[/url]. Fold the edge flaps and tape them together.
2) Computer Science
Ever wonder how to take that 3-D object in your hand and put it into a computer? Use your geometric figure to follow along with the Computerphiles video. Jot down the connection between the various matrix operations and the effect on the graph.
3) Math
Explore the GeoGebra applets below. Homer Simpson would be encoded into a computer using a massive matrix to plot each point on his face as well as a corresponding color code. To explore the effect of multiplying by the transformation matrix, change some of the numbers in the matrix and see how Homer changes. Can you make his image reflect over the x-axis, y-axis, rotate, dilate?
4) Create
Create a new 3-D graphics page using GeoGebra. Plot your pyramid in the x,y,z coordinate plane. To do this, the five vertices in (x,y,z) form and then connect them with using [icon]/images/ggb/toolbar/mode_segment.png[/icon]the line segment tool OR typing the command Segment[<Point>,<Point>]. Paste a screenshot in your lab (from a good angle).