MAT 117 All DQs

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MAT 117 All DQs

Based on the readings and content for this course x
Based on the readings and content for this course, which topic did you find most useful or interesting? How will you use it later in life? What makes it valuable?

Ans:

Well, there are many small but significant things in real life, that I understand, are related to maths.
For example the screen size of a TV or a laptop. When we say 14 inch laptop we mean the diagonal of the screen is approximately 14 inches.
This is a direct application of Pythagorean Theorem.
And depending on the shape of the room determines the formula for area that I use. For example, if our room was a perfect square (which none of them are) I would utilize the formula a = s^2, since our rooms our rectangle, the formula we more commonly use is a = lw.

Again I understand the amount of money we spend on gas can be modeled by a mathematical function.
When we throw a ball up, the time taken for it to come down can be modeled by a quadratic equation.
There are so many things.
I won’t pick up any particular thing.
But after this course, I am able to look at many things in a more analytical way.
I can understand the mathematical logic behind them.

So all these are valuable to me

Do you always use the property of distribution when multiplying monomials and polynomials x
Do you always use the property of distribution when multiplying monomials and polynomials? Explain why or why not. In what situations would distribution become important? Provide an example for the class to practice with.

Ans:

The property of distribution is one important tool in solving the polynomial multiplications.

For example:

3x*(x+5) = 3x*x + 3x*5 = 3x^2 +15x

But this property is used only when one of the bracketed terms contains two or more terms of different order.

For example in the above case, the bracket consists of two different order terms, x and 5.

Let’s take another case:

3x*(x+2x)

This can be solved in two ways.

3x(x+2x) = 3x*x + 3x*2x = 3x^2 +6x^2 = 9x^2

Or we can say:

3x(x+2x) = 3x*3x = 9x^2

So in the 1st method we used the distribute property.

But in the 2nd case we did not.

So it depends on the particular problem and looking at the different terms we can decide whether or not to use distributive property.

Explain how to factor the following trinomials forms x
Explain how to factor the following trinomials forms:
x2 +
bx +
c and
ax2 +
bx +
c. Is there more than one way to factor this? Show your answer using both words and mathematical notation

Ans:

Actually I am not very sure how to answer this.
For me
x2 +
bx +
c and
ax2 +
bx +
c are not different from each other.
The 1st one is a special case of the second expression where a=1.
To factor the expression ax^2 + bx + c we first need to factor the middle term bx cleverly.
Now it’s to be understood that not all trinomials can be factored. But some of them can be.
Basically, we have to write b in the form b= p+q so that p*q= a*c. This is the only trick.
Then we can just take out the common factors.
Let’s see through an example.
Example 1:
x^2 + 2x +1
Here a=1 , b=2 and c=2
So the product a*c = 1*1 =1
We write b= 2= 1+1 so their product is also 1*1=1
So x^2 + 2x +1
= x^2 + x+ x +1
= x(x+1) + 1(x+1)
= (x+1)(+1)
Example -2:
4x^2 + 12x + 9
= 4x^2 + 6x + 6x +9
= 2x (2x+3) + 3 (2x+3)
=(2x+3)(2x+3)
Example -3 :
x^2 + 15x + 54
= x^2 + 9x + 6x + 54
= x (x+9) + 6 (x+9)
=(x+6)(x+9)

Explain the five steps for solving rational equations x
Explain the five steps for solving rational equations. Can any of these steps be eliminated? Can the order of these steps be changed? Would you add any steps to make it easier, or to make it easier to understand?

The 5 steps for solving a rational equation are as follows,
1) First of all we determine the least common denominator (LCD) of the terms associated with the equation
2) Then multiply each side of the equation by the LCD
3) Then simplify each term
4) Now solve the final equation
5) Then put the answers in the original equation to make a check.
I am not very sure which part I can omit or skip.
I will go through what others have written
Then I can have a better understanding I hope.

Explain the four steps for solving quadratic equations x
Explain the four steps for solving quadratic equations. Can any of these steps be eliminated? Can the order of these steps be changed? Would you add any steps to make it easier, or to make it easier to understand?

There may be different ways to solve quadratic equations.

For example one can use factoring or one may use quadratic formula.

For me quadratic formula is the best way to solve any quadratic equation.

The steps involved may be as follows:

Let’s consider the quadratic equation of the form : ax^2 + bx + c= 0

The first step will be to identify the coefficients a,b and c.

The second step will be to calculate the discriminant D= b^2 – 4ac.

The 3rd step may be to check the value of D.

If D > 0 then we have two real solutions.

If D < 0 then we have two imaginary solutions. If D=0 then we have one real solution. The last step will be to put the value in the quadratic formula and simplify to get the values of x. X= [-b ± SQRT (b^2 – 4ac)]/ 2a I think we can skip the 2nd and 3rd step. Once we identify the values of a, b and c, we can simply use the quadratic formula to get the results. From the concepts you have learned in this course x From the concepts you have learned in this course, provide a real-world application of something that you think has been the most valuable to you? Why has it been valuable? Answer 1: As silly as it may seem, what I am finding that has been of the most benefit to me is one of the more basic properties that we learned. And that was back in our second or third week of the course when we started looking into the properties how to measure area of a square or rectangle. The reason that I have found this information to be so beneficial to me is because we are doing home renovations. And when we make trips to Lowe's for materials such as flooring, it is measured in squared footage. So for me to know how many boxes of flooring we need to buy, I first need to know the dimensions of the room that we are covering. And depending on the shape of the room determines the formula for area that I use. For example, if our room was a perfect square (which none of them are) I would utilize the formula a = s^2, since our rooms our rectangle, the formula we more commonly use is a = lw. Answer 2: Well, there are many small but significant things in real life, that I understand, are related to maths. For example the screen size of a TV or a laptop. When we say 14 inch laptop we mean the diagonal of the screen is approximately 14 inches. This is a direct application of Pythagorean Theorem. Similarly the area of square floor. Again I understand the amount of money we spend on gas can be modeled by a mathematical function. When we throw a ball up, the time taken for it to come down can be modeled by a quadratic equation. There are so many things. I won't pick up any particular thing. But after this course, I am able to look at many things in a more analytical way. I can understand the mathematical logic behind them. How are these concepts of direct x How are these concepts of direct, inverse, and joint variation used in everyday life? Provide examples for each. ans These are some of the concepts which I can visualize in daily life. The concepts of direct, inverse and joint variation are used a lot in daily life although we may not notice it. Let’s see them by some simple examples. Direct variation: Suppose 1 ice-cream costs $10. So 10 ice-creams will cost $100. This is a simple example of direct variations. We can find many more examples of this. Indirect variation: Suppose one person takes 40 minutes to do complete a work. Then how much will 4 persons take to complete the work together. The answer is 4 persons will take 10 minutes to complete it. This is an example of inverse variation. Joint Variation: Mathematically, we can say that if x varies directly with y and if x varies inversely with z then we can write: x= K y/z where K is a constant. So this is a joint variation. But right now I cannot remember a real life application. I will go through fellow students’ answers. I can get some idea from there. How do you factor the difference of two squares x How do you factor the difference of two squares? How do you factor the perfect square trinomial? How do you factor the sum and difference of two cubes? Which of these three makes the most sense to you? Explain why. How do you factor the difference of two squares? This is my personal favorite and I guess this one is the easiest of them all to factor. If we have somthing like a^2 - b^2, then we can factor it out like: a^2 - b^2 = (a+b)(a-b) For example : Factor: 16a^2 - 9 16a^2 - 9 = (4a)^2 - 3^2 = (4a+3)(4a-3) How do you factor the perfect square trinomial? The only trick in this kind of trinomials is to identify that this is a perfect square. It has to be in the form a^2 + 2*a*b + b^2 (or a^2 -2*a*b + b^2). Then it's easy to write it as (a+b)^2 or (a-b)^2 For example: 4x^2 + 12x + 9 =(2x)^2 + 2*2a*3 +3^2 =(2x+3)^2 How do you factor the sum and difference of two cubes? When we have a sum of two cubes we can factor it as : a^3 + b^3 = (a+b)(a^2 -ab +b^2) Similarly, if we have a difference of two cubes, then we can write it as: a^3 - b^3 = (a-b)(a^2 +ab+b^2) I hope I have done them all correctly. If your neighbor asked you to explain what you learned in this course x If your neighbor asked you to explain what you learned in this course, what would you tell her? Well. It will be a bit difficult for me to explain to someone what I learned through this class. There are so many things and I am not very sure how to explain all these things to a lay man. I can give example of a few things, like how to find the area of a wall or a flooring or how to calculate the speed. I can try to explain about the negative integers. I can tell them how the speed of a car is calculated. There are so many things which I can tell them. But then there are some things which are difficult to explain for me. It will be difficult to explain the quadratic equations or the radicals Other than those listed in the text how might the Pythagorean theorem be used in everyday life x Other than those listed in the text how might the Pythagorean theorem be used in everyday life? Provide examples of each. Well this is one more concept which I understand has a lot of applications in daily life although we may not know this exactly. The best example that comes to mind is the regarding the TV screens. When we talk about a 21" TV we actually mean the diagonal of the screen is 21" which comes from the Pythagorean theorem. Same is the case for laptops. I think carpenters also use these principles a lot in their profession. Also when we use a ladder to reach at a certain height of a wall, it’s the Pythagorean theorem which comes into play. For example suppose we need to reach a height of 8 meters of a wall. And we can place the ladder at 6 meters from the wall. So the minimum length of the ladder must be SQRT(62+82)=SQRT(36+64)=SQRT(100)=10 meters. Right now I cannot come up with anymore example. But I will search and come up with more Quadratic equations, which are expressed in the form x Quadratic equations, which are expressed in the form of ax2 + bx + c = 0, where a does not equal 0, may have how many solutions? Explain why. Ans: Any quadratic equation of the form ax^2 + bx +c =0 can have at most two solutions. I am not sure how to explain this. As I understand the maximum number of solution of any equation equals the highest power of x in that equation. In case of a quadratic equation, the discriminant D= b2 - 4ac, tells us how many real number solutions the equation ax2 + bx + c= 0 has. When D = negative, has two non real imaginary number solutions. When D = 0, it has only one solution, it is a real number b/c zero is the perfect square and can be factored as a square When D = positive it has two different real number solutions This is as much as I understand. I will look at other answers to know more. What are the two steps for simplifying radicals x What are the two steps for simplifying radicals? Can either step be deleted? If you could add a step that might make it easier or easier to understand, what step would you add? The following steps should be taken to simplify a radical expression: step 1 - First the largest perfect nth power factor of the radicand must be found. step 2 - Then factor out and simplify the perfect nth power. We look at this with a simple example: I am considering square root for simplicity. √24 + √54 = √(4*6) + √(9*6) (Write 24 = 4*6 = 2*2*6 and 54 = 9*6 = 3*3*6) = 2√6 + 3√6 (√4 = and √9 =3) = √6 (2+3) (Factoring out √6) = 5√6 What constitutes a rational expression x What constitutes a rational expression? How would you explain this concept to someoneunfamiliar with it? Ans: A rational expression is an expression which has variables in both the numerator and/or denominator. It’s a kind of fraction with polynomials In case of a rational expression we have to make sure that the denominator is not equal to 0 because division by zero is not allowed. Anything divided by zero is undefined. In 3x/(x^2 - 16), the value of x can't be -4 or 4, since it makes the denominator equal to 0 When we explain this to someone, first of all I should explain what is a fraction. For example : 2/3 is a fraction or x/y is a fration. In case of x/y, x is numerator and y is denominator. If we replace x and y by expressions like: (2x+5)/(3x-8), it becomes a rational expression. What four steps should be used in evaluating expressions x We need to apply the PEDMAS rule for this. Step's are: Step 1:Parenthesis or brackets solved first. There are three types of parenthesis [ ],{},( ). The order of them is ( ) first then { } and at the last [ ]. Step 2: Exponential, if there is any exponential terms as a^x in the brackets then we need to simplify that. Step 3:D-> Division and M–> Multiplication we need to perform them from left to right if they both are present in the same expression under same brackets.

Step 4:A–> Addition and S–> Subtraction. We need to perform them from left to right if they both are present in the same expression under same brakets.We first need to solve ( ) expression and then { } expression and then [ ] at the last.

We can skip some of these steps if there is no need of them. Like if there is no term of exponential including then no need to apply that step. In the same way we can skip some steps only if they are not present in the question

What is the greatest common factor x
What constitutes a rational expression? How would you explain this concept to someone unfamiliar with it?

A rational expression is an expression which has variables in both the numerator and/or denominator. It’s a kind of fraction with polynomials

We have to make sure that the denominator is not equal to 0 because division by zero is not allowed. Anything divided by zero is undefined.

For example 3x/(x^2 – 16) is a rational expression.
Here 3x is the numerator and x^2-16 is the denominator.
Again here, value of x can’t be -4 or 4, since it makes the denominator equal to 0.

What is the quadratic formula x
What is the quadratic formula? What is it used for? Provide a useful example, not found in the text.

The quadratic formula is used to solve quadratic equations.
It is one of the best tools to solve any quadratic equation.
Suppose we consider an equation of the form : ax^2 + bx + c =0
Then the quadratic formula says that we can give the solutions as :

X= [-b ± SQRT (b^2 – 4ac)]/ 2a
Let’s consider an example:
x^2 + 4x +4 =0
Solve for x.
Here a= 1, bn=4 and c=4.
So putting in the formula we get:
x= [-b ± SQRT (b^2 – 4ac)]/ 2a
=[-4 ± SQRT (4^2 – 4*1*4)]/ 2*1
=[-4 ± SQRT (16 – 16)]/ 2
= -4/2
=-2
So the solution is x=-2

What is the relationship between exponents and logarithms x
What is the relationship between exponents and logarithms? How would you distinguish between the two? Provide an example for the class to practice translating between the two.

Ans:

For me again this is a tricky question to answer.

I will try to explain it as I understand it.

When we mean exponents I understand something like ax = b.

This is an exponential notation.

Here a is the base and x is the exponent.

The above equation can be represented in a different way as:

x = loga b

This is called a logarithmic representation.

Example:

Consider the following exponential : 102 = 100

This can be written in logarithmic notation as: log10 100 =2

Problem for class:

Covert the following exponential to logarithim.

25=32

What one area from the readings in Week Three are you most comfortable with x
What one area from the readings in Week Three are you most comfortable with? Why do you think that is? Using what you know about this area, create a discussion question that would trigger a discussion—that is, so there is no single correct answer to the question

Well. I must say it’s a bit difficult to answer this question.
I am not very sure which part do I like most.
But I guess I like the concept of factoring expressions which are in the form of difference of two squares.
For example : a^2 – b^2 = (a+b)(a-b)
I also like to factor the expressions which are in the form of a^2 + 2ab + b^2 so that we can write them in the form
a^2 + 2ab + b^2 = (a+b)^2
But I like these two because they are easy to done. The best part as per me is to factor a trinomial where we have to split the middle term and then factor it out. I love that concept

What role do radical numbers play in your current or future profession x
What role do radical numbers play in your current or future profession? Provide a specific example and relate your discussion to your classroom learning this week.

Ans:

I do not really know how this concept of radicla number can fit into aby day to day proffesion.

As far as I understand, this will have more use in science and technology.

Banking proffesionals may have a use of it.

Also perhaps architects might be using it.

Other than this I do not know how this concepts come into play in other proffesions.

I will look into others’ responses to have a clue.

Which of the four operations on functions do you think is the easiest to perform x
Which of the four operations on functions do you think is the easiest to perform? What is the most difficult? Explain why.

If I am correct we are talking about the four operations which are addition, subtraction, multiplication and division.
When talking of functions, these operations becomes complicated as compared to normal variables.
But for me I thing addition is the easiest one and also subtraction.
The multiplication and division may be a bit more difficult.
I will look at what other fellow classmates are talking about and try to learn more on this.

Which of the special products are you most comfortable with and why x
Which of the special products are you most comfortable with and why? Using what you know about this special product, create an example for the class to practice factoring.

Ans:

For me the factoring of difference of two squares is the most comfortable one.
First of all it is easier to recognize a square as compared to a cube.
And the corresponding factor is also so simple.
a^2 – b^2 = (a+b)(a-b)
I don’t see any other formula so simple.
Let’s see an example:
16x^2 -9y^2 = (4x)^2 – (3y)^2 = (4x+3y)(4x-3y)
Let’s consider a difference of cubes.
8x^3 – 27 = (2x)^3 – 3^3 = (2x-3)(4x^2+6x+9)
For me the first one was definitely easier to identify and solve.

Why is scientific notation so important in today x
Why is scientific notation so important in today’s society? Find a practical use of scientific notation and share it with the class as well as your thoughts as to why it was vital in this situation.

Ans:

Scientifics notations give us a proper and better way to represent very large or very small numbers.

For example, 450000 can be written as : 4.5 * 105.

Since 450000 is a small number we don’t realize the importance of scientific notation.

Consider the number 4500000000000000000000.

It will be a cumbersome job to write or use this number.

Instead we can write this as: 4.5*1020.

This makes our life much easier if we want to use this number.

I think the scientific notations have a lot of use in science and engineering.

I searched on internet to find a good use of this and I found an interesting application.

The distance between the Sun and the Earth is almost d=15*107 km

Speed of light is c = 3 *105 km/s

So time taken for light to reach on Earth from Sun is: t= d/c = 15*107/3*105

= (15/3)*102

= 5*102 sec

=500 sec

= 8 min 20 sec

Now if we have to do this calculation without scientific notations, it will be very difficult.

Write a word problem involving a quadratic function x
Write a word problem involving a quadratic function. How would you explain the steps in finding the solution to someone not in this class?

I searched through internet and found the relation for a free falling body.
So I made this problem myself.
Hope this will work.

Word Problem:
The distance traveled by a free falling body under the action of gravity is given by the relation:
h=(1/2)gt^2
where h is the height
g is the acceleration due to gravity = 9.8 m/s^2
and t is the time taken.
Suppose a stone is dropped from a height of 10 meters.
Find the time taken to reach the ground.
Solution:
From the equation we can find the time as:
h=(1/2)gt^2
or t^2 = 2h/g
or t = sqrt (2h/g) [we do not consider the negative root as time cannot be negative]
or t = sqrt (2*10/9.8) = sqrt(20/9.8) = 1.42 sec

I hope I have done it correctly.

Please let me know if I made any mistakes.