math116 Discussion questions
math116 Discussion questions
CSU - Dominguez hills
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Date Created: 11/16/15
Week 1 Discussion Question 1 An expression is a variable, a number, or a combination of both connected by operation signs, while an equation is two expressions separated by an equal sign which means they are equivalent. The difference between an expression and an equation is in the way they are solved. Precisely, an expression cannot be solved; it only can be evaluated by substituting values for the variables. Example: 4x + 10 If x = 2, the value of the expression is 18, and it changes according to the value of the variable. So, finding the value of an expression, we are looking for the result we get from the expression evaluation which can vary in many ways. An equation though can be solved and there is a single solution to it. Example: 5x + 5 = 10 5x = 10 5 5x = 5 x = 5/5 x = 1 Only when x equals 1, both parts of the equation become equal. Example for my classmates: Translate. Let x represent the unknown number. Two more than four times a number is the same as six. Week 1 Discussion Question 2 There are rules we apply when several operations should be done. The following is the correct order of operations: 1. Doing calculations within grouping symbols 2. Evaluating exponents 3. Multiplying and dividing in order from left to right 4. Adding and subtracting in order from left to right It is important to follow the correct order to get the right answer. Otherwise it will result in a different answer. The reason why it should be done this way is that this is the rule set by mathematicians to get the only correct answer and to avoid confusion. Example: 9 + 6 x (8 5) Week 3 Discussion Question # 1 Remember, in an *inequality,* one side is BIGGER than the other side, right? some number (let's say x) > some other number (let's say y) so if it's an inequality it's an equation that looks like either y > x or x > y If you remember what you know about positive and negative numbers, you know that, the bigger the NUMERAL, the smaller the NUMBER, right? So, for example, -9 is smaller, or less than, -3, right? -9 < -3 but--! 9 > 3 See what happened? When we made both sides positive, the inequality sign changed! This will always happen when you multiply or divide both sides by a negative number, because when you multiply or divide by a negative number you will change the signs of the numbers (either from positive to negative or from negative to positive) on both sides of the inequality. This will make the inequality sign change. On the other hand, an equation is different. It is like a perfectly balanced scale. Everything on both sides of the equation is the SAME. So whenever you do the same thing to both sides of an equation (think of adding 5 pounds to each side of a balance), it's STILL balanced. No change. Both sides are still equal, even when you multiply or divide by a negative number. Week 3 Discussion Question # 2 Well, an inequality shows two statements that are true if and only if one statement differs from the other by being greater than, less than, or equal to the other. For example: x + 1 > 3 If you change the > to =, then we get x to be: x = 2 Now we re-substitute the > x > 2 Meaning that x + 1 > 3 is true only for all x > 2. Therefore x = 2 is the solution of the equation, but 2 is not a solution of the inequality. Inequalities have multiple solutions, while equations have unique solutions. If the solution is a solution of the inequality, it will not be a solution of the equation formed from the inequality unless the inequality has a greater than or equal to sign or a less than or equal to sign, in which case, x = the same value that x >=, and that value is the UNIQUE solution to the equation formed from the inequality.. For example: x + 3 > 4 x > 1 Let's just choose 2 for the value, since it is the closest whole number above 1: 2 + 3 > 4 5 > 4 TRUE 2 + 3 = 4 5 = 4 FALSE And if you look here, if we chose x = 1 1 + 3 > 4 FALSE 1 + 3 = 4 4 = 4 TRUE Week 5 Discussion Question # 1 A function is almost always confused with an equation and most students of Math may not appreciate a very clear distinction between the two. An equation has no special notation to identify it, whereas a function is denoted by letters such as f, g, h, , F etc. [If we have only one function to discuss, we usually use the notation “f” meaning “function”] Consider f(x) = 8x + 5 This is a linear function. You can think of a function as a machine that takes in a number, does some sort of work on it, and puts it out as another number at the other end of the machine. In this case, you put in some number x; the function multiplies it by 8, then adds 5 to the result, and puts out this new number, which is called f(x). If you give that new number (the output of the function) the name y, then you have your familiar linear equation in slopeintercept form: y = 8x + 5. A function is only allowed to put out one number for each number that goes in. If you think about a vertical line such as x = 3, you can't make this into a function. You can only put in the number 3, and any number can come out. [The point (5, y) is on the line for any value of y.] A function isn't allowed to do this. Therefore, a linear function is never graphed by a vertical line. The slopeintercept form can describe any linear function. A function always outputs a value. Suppose the function f(x) = 10x + 20, what this means is that this function outputs different values for different input values of variable 'x'. For example: f(10) = 10 10 + 20 = 120, f(20) = 10 20 + 20 = 220. i.e. we substitute different values into x and get different outputs. The important point is that every function outputs a value. Equations define the relationship between one or more variables. For example: x² 10x = 0 is a single variable equation, whereas 2x + 3y = 10 is a multi variable equation. An equation doesn’t output a value. We can solve the equation and find out the values of the variables. The equation would be valid only for certain values of the variables. A function is always an equation, but not all equations are functions. For an equation, each abscissa (xcoordinate) may have one or more distinct ordinates (y coordinates)... For a function, each abscissa may have only one ordinate. For example, x = 10 is an equation but not a function For a function, if you use a vertical line test, the line will cross the graph of a function in exactly one point...For an equation, the line will cross the graph of an equation in one or more points. Week 5 Discussion Question # 2 Domain is the set of all values that an independent variable can take. In other words, domain is the set of values of the independent variable for which the dependent variable or the function is defined. Range is the set of all the values that the dependent variable or the function can possibly take. Here is an example of a linear function in real life: Kelly opens a bank account by depositing $100. Every month he deposits $25 in his account. What is the amount of money Kelly has after 8 months? The linear equation is y = f(x) = 25x + 100 x is the number of months for which the deposits are made and y is the total amount in the account after x months. When x = 2, y = 25(2) + 100 = $150 When x = 5, y = 25(5) + 100 = $225 When x = 8, y = 25(8) + 100 = $300 This means Kelly has $150, $225 and $300 in his account after 2 months, 5 months and 8 months respectively. In the above example, x = 1 lies within the domain of the function f(x) but we can't take x = 1 since we can't have a bank deposit for negative number of years. Week 7 Discussion Question 1 Graphing: Pro: you can see where is solution(s) on graph, where the equations intersect together is solution of your system. Con: if you graph not accurate enough, or your solution with a big values or weird numbers, such as many decimal numbers it will be difficult to see the intersect point is solution of the system. Substitution: Pro: you can get an exact solution Con: some time if they give you “overlapped” equations, or no real solutions, but you did not recognize them at first seconds, then you solve it and get zero or back the same only equation For example: x+2y = 3 and 2x+4y=6 2(32y) +4y = 6 – 4y +4y = 6 = right hand side =6. it is always true statement then you conclude is ALL REAL NUMBER SOLUTIONS. Elimination Pro: you can see the order of system, for example, they give you system of 3 variables, 3 equations, then you make down to 2 equations sand 2 variables, then you make down 1 equation and 1 variable, from here you can get 1 solution for example for solution x, then you go back system 2 by 2 to get another solution, for example for y, and so on so for... Con: it way requires you to work more and take care step by step because it is like a chain reaction, if you do wrong previous steps, you will get wrong answers/ solutions. The best one depends on your needs, strengths and on how much complicated system given, and what they want to require you to do, indeed. If just simple system, you can apply substitution or elimination. If you want to know how direction of lines, parabolas, circles, ellipses…etc and where is solution you can apply graph. If system with 3 by 3 and up, I recommend using elimination if you feel a complicated system, and use graph if you feel a simple system. Week 7 Discussion Question 2 1. How does the author determine what the first equation should be? The first equation is the equation formed out of the first information in the problem. The first equation is p + q = 63, where 63 is the sum of p and q as given in the question. 2. What about the second equation? The second equation is the equation formed out of the second information in the problem, where the required mixture of the two components p and q is formed using the data given in the problem. This step is called the “Mixture” step and the resulting equation is called “Mixture” equation. 3. How are these examples similar? These examples are similar in the sense that there are two unknowns in each question. They contain two sets of information that are translated into two linear equations. Thus, they give us a system of linear simultaneous equations, which is then solved to find the values of both the unknowns. 4. How are they different? The examples are different in the type of data they possess. The problems in this category could be on two investments, two chemicals mixture or on coins (pennies, nickels, dimes and quarters). How we form the two equations for a question depends upon how is the information stated in the problem. 5. Find a problem in the text that is similar to examples 2, 3, and 4. Post the problem for your classmates to solve. Example 6: A CoastGuard patrol boat travels 4 hr on a trip downstream with a 6mph current. The return trip against the same current takes 5 hr. Find the speed of the boat in still water. Week 9 Discussion Question There were some difficult obstacles for me in this class regarding the curriculum, and there were also some obstacles regarding time commitment and self discipline. I found the sections on graphing quadratics, especially the inequalities as difficult topics, but it was a direct catalyst for the quadratics functions and solving these types of equations. The topics that I learned about here can be utilized and applied to our daily lives but not in the direct approach that I worked or understood the week's topic. Each week provided me with useful concepts that will allow me to problem solve and use topics to solve various daily applications.
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