Beginning of Semester To Do List Math 1314

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1 Beginning of Semester To Do List Math Sign up for a CASA account in CourseWare at Read the "Departmental Policies for Math 13xx Face to Face Classes". You are responsible for knowing these policies. Read them carefully, and make sure you understand all of them. You will take a quiz over them. 3. Take the Course Policies Quiz on CourseWare. You must score 100% on this quiz before you can access Test 1 or the weekly quizzes. The answers to the Course Policies Quiz are all in the Departmental Policies for 13xx. You cannot take any further online quizzes or practice tests until you get a perfect score on the course policies quiz. 4. Take Practice Test 1 on CourseWare. This will count for bonus points, will help you get ready for Test 1 and will indicate what material is covered on Test 1. All material on Test 1 is pre-requisite material - material you are expected to have learned in a class you took before this one. This material will not be covered in class. Get help on anything you've forgotten. (You can seek help at the CASA Tutoring Center, Garrison Gym.) 5. Take Test 1 on CourseWare. Log on to CourseWare, click on Online Assignments, and you should see a link to the test. You may take the test two times. We allow ONLY two attempts at this test to allow for any technical difficulties. For example, if you get kicked offline while taking the test, you have used up one of your two attempts. If you score below 70% on Test 1, there is a good chance you are in the wrong class and are at risk of failing the course. If you score below 70% on Test 1, you should consider dropping Math 1313 and taking or auditing Math 1310 to build you basic skills up to the requisite level. 6. Memorize your ID and learn your course Section Number. 7. Buy your Access Code. You must purchase and enter an Access Code for CourseWare to have full semester access to the Website. Make sure you enter it right after you buy it, so that you don t misplace the Code! There are no refunds for lost Codes or for Codes purchased by students who drop the class. If you are taking two (or more) Math 13xx classes, you will need one Code for each Math 13xx class. 8. Buy Popper Bubbling Forms. You will need a packet of Bubbling Forms for your section of Math You can purchase a packet at the Bookstore. The forms are specific to each section of the course. You must have the correct form in order for your assignments to be scanned properly. The packets CANNOT be returned for a refund. If you change sections, you MUST purchase a new packet of bubbling forms for the new section.

2 Math 1314 Lesson 1: Prerequisites Prerequisites are topics you should have mastered before you enter this class. Because of the emphasis on technology in this course, there are few skills which you will have to do by hand. This lesson is intended as a quick review of these topics. 1. Exponents Recall a couple of Exponent Rules we ll use in this course: n 1 m n n m x = x = x n x Example 1: Simplify and write the answer without using negative exponents: a. 5 x b. ( x) 5 Example : Write as a radical: 3 4 x Example 3: Write using a rational exponent: 3 x 5. Analyzing a Polynomial and their Graphs To begin with, let s review the definition of a polynomial function. A polynomial is the sum and/or difference of terms that contain variables and/or real constants, with variables raised to whole number (0, 1,, 3, ) powers. Example 4: Which of the following are polynomial functions? 3x a. f ( x) = x + 0.5x + h. f ( x) = 3e b. f ( x) = 3x x i. f ( x ) = 5 1 c. f ( x) = 10x 4 x d. e. f. g. 5 3 f ( x) = x x + x f ( x) = 5x 0.5x 6 x e f ( x) = x + f ( x) = ln( x 3) Lesson 1 Prerequisites 1

3 The solution(s) (root(s), zero(s), x-intercept(s)) of a polynomial function f ( x ) is/are found by finding the values of the variable x when f ( x ) = 0. Example 5: Find the roots of the function: a. f ( x) = x x 3. b. ( ) = x 4 f x To find the y-intercept of a polynomial function, calculate f(0). In fact, to calculate the y- intercept of ANY function, find f(0). Example 6: Find the y-intercept of f ( x) = x x 3. Other times, we may need to evaluate a function at a given value. Example 7: Let 3 ( ) 10 f x = x + x, find f(-). Example 8: Suppose the total cost in dollars to produce x items is given by the function 3 C( x) = x x + 1x Find the total cost of producing 50 items. Lesson 1 Prerequisites

4 The graph of a polynomial function looks similar to one of the four graphs below. Notice that the graphs are nice smooth curves, no sharp corners, no holes and no asymptotes. The domain (set of valid inputs for the function) of any polynomial function is (, ), which we can see clearly from the graphs (the set of x-values on the graph). The range (set of outputs after evaluating with valid inputs) is best found by observing the graph (the set of y-values on the graph). The end behavior of a graph is the behavior to the far left and far right of the graph. If we are given only the polynomial function and not its graph, we can determine the end behavior by simply looking at its leading term (term with the highest power on the variable x). End Behavior of a Polynomial Function An even-degree polynomial s end behavior will be if its leading coefficient is negative. An odd-degree polynomial s end behavior will be if its leading coefficient is negative. if its leading coefficient is positive and if its leading coefficient is positive or Example 9: Determine the end behavior of Example 10: Determine the end behavior of 5 f ( x) 3x x = +. f ( x) x (x 1) =. Description of the Behavior at Each x-intercept 1. Even Multiplicity: The graph touches the x-axis, but does not cross it (looks like a parabola there).. Odd Multiplicity of 1: The graph crosses the x-axis (looks like a line there). 3. Odd Multiplicity greater than or equal to 3: The graph crosses the x-axis and it looks like a cubic there. Example 11: Given graph. Leading Term: End Behavior: f ( x) x( x 1) ( x 1) 3 = + the zeros are 0, 1, and -1, respectively. Sketch its Lesson 1 Prerequisites 3

5 Other times we re given the graph of a polynomial function and wish to find certain function values or when the function is positive or negative. Example 1: Given the following graph of a polynomial function, a. For which x-value(s) is the function equal to 0. b. Find f(-1). c. Find f(). Example 13: Given the following graph of a polynomial function, a. Give the inteval(s) over which the function is negative. b. Over how many intervals is the funciton positive? c. Give the interval(s) over which the function is positive. d. Find the domain and range. Lesson 1 Prerequisites 4

6 3. Simplifying an Algebraic Expression x 4 3x 4x + x Example 14: Simplify ( ) ( 3 ) Example 15: Multiply ( x 1)( x + 3) 4. Rational Functions If we form the ratio of two polynomials we obtain a rational function. Graphs of rational function may look like: These types of graphs may have zeros, a y-intercept, vertical and/or horizontal asymptotes, and/or holes. The domain of such a function will be the set of all real numbers except those x- values at any holes and at any vertical aysmptotes. A vertical line is a vertical asymptote of a rational funciton if its graph approaches that line at the far top and far bottom of the graph. The graph can never cross these lines. A horizontal line is a horizontal asymptote of a rational funciton if its graph approaches that line at the far left and far right of the graph. The graph may cross this line. Example 16: For the graph above, find: a. Any roots. b. Any vertical asymptote. c. Any horizontal asymptote. Lesson 1 Prerequisites 5

7 We can algebraically find a rational function s domain, range, roots, y-intercept, asymptotes, and holes. Recall that we stated before, a rational funciton can never cross its vertical asymptote(s). We shall now see why. Vertical Asymptotes To find a rational function s roots, vertical asymptotes and holes you must first factor the numerator and denominator as much as possible and simplify. Then, LOOK at the denominator. If a factor cancels with a factor in the numerator, then there is a hole where that factor equals zero. If a factor does not cancel, then there is a vertical asymptote where that factor equals zero. The roots are found by setting the numerator equal to zero and solving for x. The y-intercept is found by calculating f(0), if possible. The domain of a rational function excludes those values of x where holes and vertical asymptotes occur. (Or set the original function s denominator equal to zero and solve for x.) Example 17: Let g( x) = x + x 48 x 36 Find: a. Any vertical asymptote. b. Any roots. c. Any y-intercept. Example 18: Find the domain of f ( x) =. x 3x Lesson 1 Prerequisites 6

8 Horizontal Asymptotes p( x) Let f ( x) =, q( x) Shorthand: degree of f = deg(f), numerator = N, denominator = D 1. If deg(n) > deg(d) then there is no horizontal asymptote.. If deg(n) < deg(d) then there is a horizontal asymptote and it is y = 0 (x-axis). 3. If deg(n) = deg(d) then there is a horizontal asymptote and it is y = b a, where a is the leading coefficient of the numerator. b is the leading coefficient of the denominator. Example 19: Find any horizontal asymptote given: 5x + 5 a. g( x) = x 7x + 1 b. g( x) = 8x + 1 6x 3 5. Simplifying More Algebraic Expressions Example 0: Simplify x x x 1 x 4 a. ( + + ) b. ( 9x 6x 4) 4 3x 4 Lesson 1 Prerequisites 7

9 6. Piecewise Defined Functions A function that is defined by two (or more) equations over a specified domain is called a piecewise function. Example 1: Let Find: a. f (5). 3 x 1, x < 1 f ( x) = x + 5, 1 x < 3. x+ e, x 3 b. f ( ). The graphs of piecewise functions look similar to the following graph: Example : Use the graph above to find each of the following. a. f(0) = b. f(3) = c. f(6) = d. f() = e. For which x-value(s) is f(x) = 3. Lesson 1 Prerequisites 8

10 The last type of function whose domain we need to review is the square root function. Recall that over real numbers we cannot take an even root of a negative number. Hence, to find its domain, exclude real numbers that result in an even root of a negative number. Example 3: Find the domain and write the answer in interval notation: f ( x) = 3x + 4 Now take Practice Test 1 (0 attempts), then Test 1 ( attempts, from anywhere online, no CASA reservation needed). Lesson 1 Prerequisites 9

11 Math 1314 Lesson : An Introduction to Geogebra (GGB) Course Overview What is calculus? Calculus is the study of change, particularly, how things change over time. It gives us a framework for measuring change using some fairly simple models. The formal body of information that makes up a calculus course was developed in the 17 th century by two mathematicians, each working separately: Isaac Newton and Gottfried Wilheim Leibniz. Newton method of fluxions studied how things change. Leibniz did similar work at about the same time, and contributed additionally some of the notation we use to this day. We will address two basic questions in this course. 1. How can we find the slope of a line that is tangent to an arbitrary curve at a given point? The graph below shows a cubic function and the line tangent to the function at the point where x = -1 We are interested in finding the slope of this line, because it gives the rate at which the y value of the function is changing. You already have some experience working with slope. You have y y1 found the slope of a line in earlier coursework using a formula, m =. This gives a rate of x x 1 change: it gives the change in y over the change in x. It tells us how much y changes for a computed change in x. We will use this idea when we learn how to find the slope of a tangent line. Differential calculus treats problems of this type. Lesson An Introduction to GGB 1

12 . How can we find the area of a region such as the one pictured below? The shape of this region does not conform to the formulas we know, such as for a triangle, rectangle or circle, so we will need another method for finding this area. We will use integral calculus to solve problems of this type. Both of these areas of inquiry rely on the concept of a limit. We will begin our study of calculus by looking at limits. Then we will investigate the types of problems we can solve using differential calculus and integral calculus. A major focus of this course will be the use of technology in the teaching and learning of calculus. We will be less concerned with paper and pencil computations in this course, and more interested in solving problems. Our main tool will be GeoGebra, a free software, that you will need to download and use on a daily basis in this course. An Introduction to GeoGebra GeoGebra (GGB) is a free software package that we will use throughout the semester. The program can be used as a simple calculator, and it can also be used to perform some fairly sophisticated calculus operations. In this lesson, you will learn your way around the GGB display, and you will also learn to do some basic graphing. You will need to download GGB onto your computer. Go to and follow the instructions for downloading the program. If you are using a computer where you cannot download the program, choose the Applet option on the download page. Lesson An Introduction to GGB

13 This is what you will see when you open GGB. The GGB display consists of four parts: AT TOP: the menus/icons ON RIGHT: graphics window ON LEFT: algebra window AT BOTTOM: input line TO ADJUST FONT: Go to Options, then click on Font and select the size that you d like. TO RESIZE ALGEBRA AND GRAPHICS WINDOWS: Put cursor on the vertical line between algebra and graphics windows until you see a double-headed arrow. Then drag the line to the left or the right depending on which part you wish to enlarge. TO SHOW GRID: Put cursor on graphics window. Right-click and choose Grid. To undo follow the same steps. TO ADJUST DECIMAL PLACES: Go to Options, then click on Rounding and select 4 Decimal Places. Occasionally, we ll need to change this setting. TO SAVE CHANGES TO SETTINGS: Go to Options, then click on Settings. In the popup window, at the bottom, choose Save Settings. You will enter functions or expressions using the input line. Put the cursor in the input line and start typing. USE THE CARET KEY ( ^ ) to raise a number or variable to a power. USE SQRT for the square root To enter a cube root such as, 3, type: ^(1/3) Example 1: Use GGB to compute ( ) 8 Command: Answer: Lesson An Introduction to GGB 3

14 Example : Use GGB to compute 5e 3( ) Command: Answer: Example 3: Use GGB to evaluate Command: Answer: x 1 y + if x = and y = 10. Tables You can make a table, similar to a table in Excel. TO VIEW SPREADSHEET VIEW: Click on View and enable Spreadsheet. Example 4: Suppose f ( x) = 1.875x + 1.3x Use GGB to create a table of values that starts at x = 1.63 and has an increment of 0.6. A. Create the list of x values. Result: Lesson An Introduction to GGB 4

15 B. Enter the function in the input line. If you make an error in entering the function, in the algebra window, double click on the function and make any corrections. Command: C. Create the list of y values. Result: D. Create the list of ordered pairs. Lesson An Introduction to GGB 5

16 Result: Once the ordered pairs are formed AND highlighted, find the icon the GGB window and click it. at the top left area of Lesson An Introduction to GGB 6

17 Result: Good things to know when working with GGB: NEW WINDOW: Control N or click on the File menu and select new. DEACTIVATE OR ACTIVATE EQUATION OR VALUE: In the Algebra Window, click on the circle to the left of the item. Example 5: The path of a small rocket is modeled by the function h( t) = 16t + 18t + 1 where initial velocity is 18 feet per second and initial height is 1 feet. The model gives the height of the rocket in feet, t seconds after launch. Find the height of the rocket: A. seconds after launch. B. 4 seconds after launch. C. 5 seconds after launch. D. 8 seconds after launch. Enter the function in the input line. Command: Evaluate the function at each t value given above. Command for A: Answer: Command for B: Answer: Command for C: Answer: Command for D: Answer: Lesson An Introduction to GGB 7

18 Graphing Example 6: Use GGB to graph the function f ( x) = x 3. Enter the function in the input line. Resize the graphics window, if needed, to get a good view of the function. TO MOVE GRID: Make sure that is selected then put the cursor anywhere on the grid, and move it. TO VIEW STANDARD VIEW: Ctrl+M or right-click on the grid and select Standard View. TO ADJUST X- Y- AXES: Put the cursor on the respective axis and move up, down, right or left. Example 7: Graph the function window. 3 ( ) h x = x + x + x and find an appropriate viewing Lesson An Introduction to GGB 8

19 Finding Some Features of a Graphed Function You can find the zeros (also called roots or x intercepts) of a function using GGB. Example 8: Suppose 3 g( x) = x x 9x Find the zeros of the polynomial function. A. Enter the function in the input line. Graph and resize if needed. B. Find the zeros of the polynomial function. (When you begin to enter the command, a list will appear.) Command: Answer: Lesson An Introduction to GGB 9

20 Example 9: Suppose f ( x) = x 3x 5. Find the zeros of the function. x 3 A. Enter the function in the input line. Graph and resize if needed. B. Find the zeros of the function. The command in the input line will be the same as in Part B of the previous example; however, this function is not a polynomial so we ll need to choose the command: Roots[<function>,<start value>,<end value>] Make sure to choose the command ROOTS NOT ROOT. Command: Answer: The relative extrema of a function are the high points and low points of the graph of a function, when compared to other points that are close to the relative extremum. A relative maximum will be higher than the points near it, and a relative minimum will be lower than the points near it. Lesson An Introduction to GGB 10

21 GGB will help you find these points. Example 10: Suppose 3 g( x) = x + 5x x + 3. Find any relative extrema. A. Enter the function in the input line. Graph and resize, if needed. B. Find the relative extrema. Command: Relative Max: Relative Min: Lesson An Introduction to GGB 11

22 Example 11: Suppose 1 3 h( x) = x 3x. Find any relative extrema. A. Enter the function in the input line. Graph and resize, if needed. B. Find the relative extrema. Since the function is not a polynomial, the command is: Extremum[<Function>,<Start x-value>,<end x-value>] Command: Relative Max: Relative Min: Lesson An Introduction to GGB 1

23 Intersection of Two Functions Example 1: Find any points where intersect. f ( x) = 1.45x 7.x 1.6 and g( x) =.84x 1.9 A. Enter the functions one by one in the input line. Graph and resize if necessary to view all points of intersection. B. Find intersection points. Command: Answer: Lesson An Introduction to GGB 13

24 6 x 1 Example 13: Find any points where f ( x) = x and g( x) = 4e + x intersect. x A. Enter the functions one by one in the input line. Graph and resize if necessary to view all points of intersection. B. Find intersection points. Since the function is not a polynomial, the command is: Intersect[<Function>,<Function>,<Start x-value>,<end x-value>] Command: Answer: Lesson An Introduction to GGB 14