title: typewriter notation

001. typewriter notation

Note that there are six questions in this exercise. Be sure to continue scrolling down until you get to the end of the exercise.

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Question: `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). Then evaluate each expression for x = 2.

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Your solution:

= x - 2 / x + 4

= 2 - 2 / 2 + 4

= 2 - 1 + 4

= 1 + 4

= 5

In this equation, 2 is divided by x first and then is subtracted from the first x.

= (x - 2) / (x + 4)

= (2 - 2) / (2 + 4)

= 0 / 6

= 0

In this equation, we solve the two equations in the parentheses and then divide to find the solution.

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Self-critique rating #$&* 3

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Question: `q002. Explain the difference between 2 ^ x + 4 and 2 ^ (x + 4). Then evaluate each expression for x = 2.

Note that a ^ b means to raise a to the b power. This process is called exponentiation, and the ^ symbol is used on most calculators, and in most computer algebra systems, to represent exponentiation.

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Your solution:

= 2 ^ x + 4

= 2 ^ 2 + 4

= 4 + 4

= 8

In this equation we must raise 2 to the 2nd power first and then add 4.

= 2 ^ (x + 4)

= 2 ^ (2 + 4)

= 2 ^ (6)

= 64

In this equation we must solve 2 + 4 first since they are located inside parentheses. After that we raise 2 to the 6th power.

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Self-critique rating #$&* 3

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Question: `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2?

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Your solution:

x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x

The numerator in this equation is 3. Numerators are the values on the top of the fraction.

The denominator in this equation is [ (2x-5)^2 * 3x + 1 ]. Denominators are values on the bottom of the fraction.

solve for x =2

= x - 3 / [ (2x - 5)^2 * 3x + 1 ] - 2 + 7x

= 2 - 3 / [ (2(2) - 5)^2 * 3(2) + 1 ] - 2 + 7(2)

= 2 - 3 / [ (4 - 5)^2 * 6 +1 ] - 2 + 14

= 2 - 3 / [ -1^2 * 6 + 1 ] - 2 + 14

= 2 - 3 / [ 1 * 6 + 1 ] - 2 + 14

= 2 - 3 / [ 6 + 1 ] - 2 + 14

= 2 - 3/7 - 2 + 14

= 2 - (0.4285714286)- 2 + 14

= 2 - 0.4285714286 + 12

= 14 - 0.4285714286

= 13.57142857

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.

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Your solution:

Solve where x = 4

= (x - 5) ^ 2x-1 + 3 / x-2

= (4 - 5)^2(4) - 1 + 3/4 -2

= (-1)^2 * 4 - 1 + 3/4 -2

= 1 * 4 - 3 + 3/4

= 4 - 3 + 3/4

= 1 + 3/4

= 1 3/4

= 1.75

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q005. At the link

http://www.vhcc.edu/dsmith/genInfo/introductoryproblems/typewriter_notation_examples_with_links.htm

(copy this path into the Address box of your Internet browser; alternatively use the path

http://vhmthphy.vhcc.edu/ > General Information > Startup and Orientation (either scroll to bottom of page or click on Links to Supplemental Sites) > typewriter notation examples

and you will find a page containing a number of additional exercises and/or examples of typewriter notation.Locate this site, click on a few of the links, and describe what you see there.

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Your solution:

There is a problem with the first given link:

http://www.vhcc.edu/dsmith/genInfo/introductoryproblems/typewriter_notation_examples_with_links.htm

This is a broken link. Using the second link I was able to view the webpage which contained several examples of typewriter notation.

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Question: `q006 Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions.

However it's very important to understand order of operations, and students do get used to this way of doing it.

You should of course write everything out in standard notation when you work it on paper.

It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation.

Indicate your understanding of why it is important to understand this notation.

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Your solution:

It is important to understand both notations because we will be using both ways in this course.

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Self-critique: OK

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Self-critique rating #$&* 3"

&#Good responses. Let me know if you have questions. &#

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title: Describing Graphs

002. Describing Graphs

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Question: `q001. You will frequently need to describe the graphs you have constructed in this course. This exercise is designed to get you used to some of the terminology we use to describe graphs. Please complete this exercise and email your work to the instructor. Note that you should do these graphs on paper without using a calculator. None of the arithmetic involved here should require a calculator, and you should not require the graphing capabilities of your calculator to answer these questions.

Problem 1. We make a table for y = 2x + 7 as follows: We construct two columns, and label the first column 'x' and the second 'y'. Put the numbers -3, -2, -1, -, 1, 2, 3 in the 'x' column. We substitute -3 into the expression and get y = 2(-3) + 7 = 1. We substitute -2 and get y = 2(-2) + 7 = 3. Substituting the remaining numbers we get y values 5, 7, 9, 11 and 13. These numbers go into the second column, each next to the x value from which it was obtained. We then graph these points on a set of x-y coordinate axes. Noting that these points lie on a straight line, we then construct the line through the points.

Now make a table for and graph the function y = 3x - 4.

Identify the intercepts of the graph, i.e., the points where the graph goes through the x and the y axes.

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Your solution:

y = 2x + 7

x | Y

=======

-3 | 1

-2 | 3

-1 | 5

0 | 7

1 | 9

2 |11

3 |13

=======

points form a striaght line

===

y = 3x - 4

x | Y

=======

-3 |-13

-2 |-10

-1 | -7

4/3| 0

0 | -4

1 | -1

2 | 2

3 | 5

=======

y = 3x - 4

y = 3(-3) -4

y = -9 -4

y = -13

y = 3x - 4

y = 3(-2) - 4

y = -6 -4

y = -10

y = 3x - 4

y = 3(-1) - 4

y = -3 -4

y = -7

y = 3x - 4

y = 3(0) - 4

y = 0 - 4

y = -4

y = 3x - 4

y = 3(1) - 4

y = 3 - 4

y = -1

y = 3x - 4

y = 3(2) - 4

y = 6 - 4

y = 2

y = 3x - 4

y = 3(3) - 4

y = 9 - 4

y = 5

y = 3x - 4

0 = 3x - 4

4 = 3x

4/3 = x

This graph is another linear graph. The points are in a straight line. The points that intersect the y axis when x= 0 and y = -4. The points that intersect the x axis when x=4/3 and y = 0

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Question: `q002. Does the steepness of the graph in the preceding exercise (of the function y = 3x - 4) change? If so describe how it changes.

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Your solution:

Based on the points that we solved for, it seems that the graph forms a straight line. The slope remained constant at 3/1, which means that the steepness would remain the same.

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q003. What is the slope of the graph of the preceding two exercises (the function is y = 3x - 4;slope is rise / run between two points of the graph)?

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Your solution:

The slope is 3/1

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Self-critique rating #$&* 3

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Question: `q004. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = 0 and x = 3.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?

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Your solution:

y = x^2

Graph y = x^2 between x = 0 and x = 3.

x | Y

=======

0 | 0

1 | 1

2 | 4

3 | 9

=======

y = x^2

y = 0^2

y = 0

y = x^2

y = 1^2

y = 1

y = x^2

y = 2^2

y = 4

y = x^2

y = 3^2

t = 9

The graph is an exponential graph that is increasing. The slope increases on each given point starting at 0, 1, 1.5, 1.66667 which means that the graph is increasing at an increasing rate with every point set.

confidence rating #$&*

Actually quadratic rather than exponential, but we'll make that distinction soon.

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q005. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = -3 and x = 0.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

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Your solution:

y = x^2

Graph y = x^2 between x = -3 and x = 0.

x | Y

=======

-3|9

-2|4

-1|1

0|0

=======

y = x^2

y = -3^2

y = 9

y = x^2

y = -2^2

y = 4

y = x^2

y = -1^2

y = 1

y = x^2

y = 0^2

y = 0

The graph is an exponential graph that is decreasing. The slope decreases on each given point starting at 5, 3, 1 which means that the graph is decreasing at a decreasing rate with every point set.

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q006. Make a table of y vs. x for y = `sqrt(x). [note: `sqrt(x) means 'the square root of x']. Graph y = `sqrt(x) between x = 0 and x = 3.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

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Your solution:

y = `sqrt(x)

Graph y = `sqrt(x) between x = 0 and x = 3

x | Y

=======

0|0

1|1

2|1.41213562

3|1.732050808

=======

y = `sqrt(x)

y = `sqrt(0)

y = 0

y = `sqrt(x)

y = `sqrt(1)

y = 1

y = `sqrt(x)

y = `sqrt(2)

y = 1.41213562

y = `sqrt(x)

y = `sqrt(3)

y = 1.732050808

The graph is increasing at a decreasing rate. The slope decreases on each given point.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q007. Make a table of y vs. x for y = 5 * 2^(-x). Graph y = 5 * 2^(-x) between x = 0 and x = 3.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

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Your solution:

y = 5 * 2^(-x)

Graph y = 5 * 2^(-x) between x = 0 and x = 3

x | Y

=======

0|5

1|2.5

2|1.25

3|0.625

=======

y = 5 * 2^(-x)

y = 5 * 2^(-0)

y = 5 * 2^(0)

y = 5 * 1

y = 5

y = 5 * 2^(-x)

y = 5 * 2^(-1)

y = 5 * 0.5

y = 2.5

y = 5 * 2^(-x)

y = 5 * 2^(-2)

y = 5 * 0.25

y = 1.25

y = 5 * 2^(-x)

y = 5 * 2^(-3)

y = 5 * 0.125

y = 0.625

The graph is decreasing at a decreasing rate. The slope decreases on each given point.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q008. Suppose you stand still in front of a driveway. A car starts out next to you and moves away from you, traveling faster and faster.

If y represents the distance from you to the car and t represents the time in seconds since the car started out, would a graph of y vs. t be increasing or decreasing?

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Your solution:

The graph would be increasing at an increasing rate. The further away from me the car got the faster it would be traveling.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Given Solution:

&#Good work. See my notes and let me know if you have questions. &#

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