8.2: F1.02: Examples 4-5

8.2: F1.02: Examples 4-5

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Example 4. Negative numbers and the operation of subtraction.

Solution: When we do subtraction problems by hand or work with negative numbers, we usually write the negative sign and the subtraction symbol in exactly the same way, so we think of them as the same. But calculators treat them differently. Use your calculator to find .

Now try to find . Using the subtraction symbol won’t work here. The key you need is probably labeled . Find that key and experiment with it until you see how to use your calculator to find .

Most students go through entire algebra courses and never need the negative number key because they handle all the sign parts of the problem mentally. For example, we’d just say and then use the calculator for this resulting problem. But when we learn about trigonometry later in the course, we will need to be able to fully handle negative numbers, so we’ll need to use this key.

Discussion: Order of operations.

There are three types of mathematical expressions which we write, by hand, without parentheses, but which need parentheses when entering them into a calculator or spreadsheet.

Evaluate each expressionEnter thisWhy?

ParseError: EOF expected (click for details)

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, where

3 ^ (2*3) = or

In the original expression, the placement of the symbols indicates that the exponent is 2 times x. But when we have to just enter symbols one after the other – on the same line – we have to use parentheses to clarify what is in the exponent.

, where

(3^2-6*3)/(4*3+2) = or

(32-6*3)/(4*3+2) =

In the original expression, the placement of the symbols indicates that the entire numerator is divided by the entire denominator. But when we have to just enter symbols one after the other – on the same line – we have to use parentheses to clarify what is to be divided by what.

, where

(4*3+13) =

OR 4*3+13 = =

In the original expression, the fact that the expression was completely under the square root symbol made it clear. In the calculator, we have to use parentheses to say that.

Example 5. For each of the expressions above, evaluate it by hand and then evaluate it with your calculator. By hand

Plug these into your calculator using the expressions above, with parentheses

Evaluate each expressionEnter this

ParseError: EOF expected (click for details)

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, where

3 ^ (2*3) = 729

OR 729

, where

(3^2-6*3)/(4*3+2) = –0.642857

OR (32-6*3)/(4*3+2) = –0.642857

, where

(4*3+13) = 5

OR 4*3+13 = = 5

CC licensed content, Shared previously

  • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution

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A research study measured the pulse rates of 57 college men and found a mean pulse rate of 70.4211 beats per minute with a standard deviation of 9.9480 beats per minute. Researchers want to know if the mean pulse rate for all college men is different from the current standard of 72 beats per minute.

Pulse rates are quantitative. The sampling distribution will be approximately normally distributed because (n ge 30).

This is a two-tailed test because we want to know if the mean pulse rate is different from 72.

(overline) = sample mean
(mu_<0>) = hypothesized population mean
(s) = sample standard deviation
(n) = sample size

Our (t) test statistic is -1.198

Given that the null hypothesis is true and (mu=72), the probability of taking a random sample of (n=57) and finding a sample mean this or more extremely different is 0.235962. This is our p-value.

(p>.05), therefore we fail to reject the null hypothesis.

There is not sufficient evidence to state that the mean pulse of college men is different from 72.


Under the Whyte notation for the classification of steam locomotives, a 2-8-8-8-4 has two leading wheels, three sets of eight driving wheels, and four trailing wheels.

The equivalent UIC classification is to be refined to (1'D)D(D2') for these engines.

Only one 2-8-8-8-4 was ever built, a Mallet-type for the Virginian Railway in 1916. [1] Built by Baldwin Locomotive Works, it became the only example of their class XA, so named due to the experimental nature of the locomotive. Like the same railroad's large articulated electrics and the Erie Railroad 2-8-8-8-2s, it was nicknamed "Triplex".

An overview of Triplex engineering is given at Triplex (locomotive).

The XA was unable to sustain a speed greater than five miles an hour, since the six cylinders could easily consume more steam than the boiler could produce. When operating in compound the high pressure steam was divided between the cylinders of the center engine. The exhaust from one cylinder was piped to the front articulated engine. The exhaust from the other center engine cylinder was piped to the tender engine. The exhaust from the front engine was piped to the exhaust nozzle inside the firebox to generate draft through the firebox, through the fire tubes and out the exhaust stack. The exhaust from the tender engine went out of a stack at the rear of the tender water tank. Unfortunately it did not contribute to draft, being wasted. The tender had a four-wheel truck at the rear to help guide the locomotive into curves when drifting back downhill after pushing a train over the hill.

The XA was sent back to Baldwin in 1920 and was rebuilt as two locomotives, a 2-8-8-0,and a 2-8-2. Unlike their predecessor which lasted only a few years in service, these two locomotives remained in service until 1953.But sadly neither of the two locomotives were preserved.

8.2: F1.02: Examples 4-5

The PTHOUR file is the input point-source inventory file for hour-specific data and profiles. The file must be a list file (see Section, “Inventory list files” and the files listed in the PTHOUR file must be in Continuous Emissions Monitoring (CEM) format when processing hourly CEM inventory data files with an format identifier (#CEM) or the format can be specified in the PTHOUR list file (#LIST CEM). In addition, these files can have a #COUNTRY entry to set the country code (see Section, “Header records”. The default country code is 0, which corresponds to the United States in the default COSTCY file. Date range setting

The PTHOUR file may optionally contain a packet to control the range of dates for which Smkinven will read data. When this feature is needed, the following entry should appear on the first line of the PTHOUR file.


where MMDD is the two-digit month and day of the month at the start and end of the period of interest. For example, to request July 10th through 12th, the first line of the file should read:

The year is implied by the data themselves note that the files can only contain data for a single year. As an alternative to using the DATERANGE packet, you can manually break the year up into smaller periods in separate files, and only list the files of interest in the PTHOUR file. Note that reading in the whole year of data and extracting just a few days will take much more time than manually editing the file to contain just the days of interest. Also note that if you choose the manual editing option, you must select days that fully cover the modeling episode after accounting for time zone differences between the facility’s time zone and the modeling time zone (set by the OUTZONE option). CEM hour-specific format

SMOKE uses the CEM ORIS ID and Boiler ID fields to match sources in the CEM hour-specific inventory to those in the annual inventory. Note that the CEM data hours are in standard local time (no daylight saving time adjustment). The format of the CEM hour-specific data is shown in Table 8.22, “CEM Format for individual hour-specific data files”.

Table 8.22. CEM Format for individual hour-specific data files

Position Name Type Description
1 ORISID Char (6) DOE Plant Identification Code (required) (should match the same field in the PTINV file in IDA format)
2 BLRID Char (6) Boiler Identification Code (required) (should match the same field in the PTINV file in IDA format)
3 YYMMDD Int Date of data in YYMMDD format (required)
4 HOUR Integer Hour value from 0 to 23
5 NOXMASS Real Nitrogen oxide emissions (lb/hr) (required)
6 SO2MASS Real Sulfur dioxide emissions (lb/hr) (required)
7 NOXRATE Real Nitrogen oxide emissions rate (lb/MMBtu) (not used by SMOKE)
8 OPTIME Real Fraction of hour unit was operating (optional)
9 GLOAD Real Gross load (MW) (optional)
10 SLOAD Real Steam load (1000 lbs/hr) (optional)
11 HTINPUT Real Heat input (mmBtu) (required)
12 HTINPUTMEASURE Character(2) Code number indicating measured or substituted, not used by SMOKE.
13 SO2MEASURE Character(2) Code number indicating measured or substituted, not used by SMOKE.
14 NOXMMEASURE Character(2) Code number indicating measured or substituted, not used by SMOKE.
15 NOXRMEASURE Character(2) Code number indicating measured or substituted, not used by SMOKE.
16 UNITFLOW Real Flow rate (ft3/sec) for the Boiler Unit (optional must be present for all records or not any records – not yet used by SMOKE)

The code numbers used in columns 12 through 15 have the following meanings:

  • 01 = 'Measured'
  • 02 = 'Calculated'
  • 03 = 'Substitute'
  • 04 = 'Measured and Substitute'
  • 97 = 'Not Applicable'
  • 98 = 'Undetermined'
  • 99 = 'Unknown Code' FF10 hour-specific format

This FF10 format uses the header described in Section, “Header records”.

The user specifies the data provided using this command in the header: FF10_HOURLY_POINT.

Sample header records for hourly point sources are shown below:

Table 8.23. FF10 Format for individual point hour-specific data files

What Do the Numbers on Fertilizer Mean?

The three numbers on fertilizer represents the value of the three macro-nutrients used by plants. These macro-nutrients are nitrogen (N), phosphorus (P) and potassium (K) or NPK for short.

The higher the number, the more concentrated the nutrient is in the fertilizer. For example, numbers on fertilizer listed as 20-5-5 has four times more nitrogen in it than phosphorus and potassium. A 20-20-20 fertilizer has twice as much concentration of all three nutrients than 10-10-10.

The fertilizer numbers can be used to calculate how much of a fertilizer needs to be applied to equal 1 pound (453.5 gr.) of the nutrient you are trying to add to the soil. So if the numbers on the fertilizer are 10-10-10, you can divide 100 by 10 and this will tell you that you need 10 pounds (4.5 k.) of the fertilizer to add 1 pound (453.5 gr.) of the nutrient to the soil. If the fertilizer numbers were 20-20-20, you divide 100 by 20 and you know that it will take 5 pounds (2 k.) of the fertilizer to add 1 pound (453.5 gr.) of the nutrient to the soil.

A fertilizer that contains only one macro-nutrient will have 𔄘” in the other values. For example, if a fertilizer is 10-0-0, then it only contains nitrogen.

These fertilizer numbers, also called NPK values, should appear on any fertilizer you purchase, whether it is an organic fertilizer or a chemical fertilizer.

8.2: F1.02: Examples 4-5

The Tulalip Tribal Codes are current through legislation passed December 11, 2020.

Disclaimer: The Office of the Reservation Attorney has the official version of the Tulalip Tribal Codes. Users should contact the Tribes' Office for ordinances and resolutions passed subsequent to the date cited above.

To be notified when additions, amendments, or revisions are made to any of the Tulalip Tribal Codes, send your e-mail address to [email protected]

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Advanced Search Features

Stemming extends a search to cover grammatical variations on a word. For example, a search for fish would also find fishing. A search for applied would also find applying, applies, and apply.

Fuzzy searching will find a word even if it is misspelled. For example, a fuzzy search for apple will find appple.

Select a fuzziness level from the drop-down menu to set how many characters off a word can be from your search term and still count as a hit.

Synonym searching looks for words that have the same meaning as the word you entered. For example, a synonym search for eleventh will also find 11th.

Boolean Searches

A Boolean search request consists of words or phrases linked by connectors such as and and or that indicate the relationship between them. Examples:

Both words must be present

Either word can be present

Apple must occur within 5 words of pear

Apple must not occur within 5 words of pear

Only apple must be present

If you use more than one connector, you should use parentheses to indicate precisely what you want to search for. For example, apple and pear or orange juice could mean (apple and pear) or orange , or it could mean apple and (pear or orange) .

Noise words, such as if and the, are ignored in searches.

Search terms may include the following special characters:

Matches any single character. Example: appl? matches apply or apple.

Matches any number of characters. Example: appl* matches application

Stemming. Example: apply

matches apply, applies, applied.

Fuzzy search. Example: ba%nana matches banana, bananna.

Phonic search. Example: #smith matches smith, smythe.

9.14. Array Functions and Operators

Table 9-35 shows the operators available for array types.

Table 9-35. array Operators

Operator Description Example Result
= equal ARRAY[1.1,2.1,3.1]::int[] = ARRAY[1,2,3] t
<> not equal ARRAY[1,2,3] <> ARRAY[1,2,4] t
< less than ARRAY[1,2,3] < ARRAY[1,2,4] t
> greater than ARRAY[1,4,3] > ARRAY[1,2,4] t
<= less than or equal ARRAY[1,2,3] <= ARRAY[1,2,3] t
>= greater than or equal ARRAY[1,4,3] >= ARRAY[1,4,3] t
@> contains ARRAY[1,4,3] @> ARRAY[3,1] t
[email protected] is contained by ARRAY[2,7] [email protected] ARRAY[1,7,4,2,6] t
&& overlap (have elements in common) ARRAY[1,4,3] && ARRAY[2,1] t
|| array-to-array concatenation ARRAY[1,2,3] || ARRAY[4,5,6]
|| array-to-array concatenation ARRAY[1,2,3] || ARRAY[[4,5,6],[7,8,9]] <<1,2,3>,<4,5,6>,<7,8,9>>
|| element-to-array concatenation 3 || ARRAY[4,5,6]
|| array-to-element concatenation ARRAY[4,5,6] || 7

Array comparisons compare the array contents element-by-element, using the default B-Tree comparison function for the element data type. In multidimensional arrays the elements are visited in row-major order (last subscript varies most rapidly). If the contents of two arrays are equal but the dimensionality is different, the first difference in the dimensionality information determines the sort order. (This is a change from versions of PostgreSQL prior to 8.2: older versions would claim that two arrays with the same contents were equal, even if the number of dimensions or subscript ranges were different.)

See Section 8.10 for more details about array operator behavior.

Table 9-36 shows the functions available for use with array types. See Section 8.10 for more discussion and examples of the use of these functions.

Deleting a Row or a Column in a Matrix

You can delete an entire row or column of a matrix by assigning an empty set of square braces [] to that row or column. Basically, [] denotes an empty array.

For example, let us delete the fourth row of a &minus

MATLAB will execute the above statement and return the following result &minus

Next, let us delete the fifth column of a &minus

MATLAB will execute the above statement and return the following result &minus


In this example, let us create a 3-by-3 matrix m, then we will copy the second and third rows of this matrix twice to create a 4-by-3 matrix.

Create a script file with the following code &minus

When you run the file, it displays the following result &minus

8.2: F1.02: Examples 4-5

So far, we have discussed estimating the mean and variance of a distribution. Our methods have been somewhat ad hoc. More specifically, it is not clear how we can estimate other parameters. We now would like to talk about a systematic way of parameter estimation. Specifically, we would like to introduce an estimation method, called maximum likelihood estimation (MLE). To give you the idea behind MLE let us look at an example.

I have a bag that contains $3$ balls. Each ball is either red or blue, but I have no information in addition to this. Thus, the number of blue balls, call it $ heta$, might be $, $1$, $2$, or $3$. I am allowed to choose $4$ balls at random from the bag with replacement. We define the random variables $X_1$, $X_2$, $X_3$, and $X_4$ as follows

  1. For each possible value of $ heta$, find the probability of the observed sample, $(x_1, x_2, x_3, x_4)=(1,0,1,1)$.
  2. For which value of $ heta$ is the probability of the observed sample is the largest?
    • Since $X_i sim Bernoulli(frac< heta><3>)$, we have egin onumber P_(x)= left< eginfrac< heta><3>& qquad extrm< for >x=1 & qquad 1-frac< heta><3>& qquad extrm< for >x=0 end ight. end Since $X_i

      Java mod examples

      Both remainder and modulo are two similar operations they act the same when the numbers are positive but much differently when the numbers are negative. In Java, we can use Math.floorMod() to describe a modulo (or modulus) operation and % operator for the remainder operation.

      • `Remainder (rem)“ = The result has the same sign (+ or -) as the dividend (first operand).
      • `Modulo (mod)“ = the result has the same sign (+ or -) as the divisor (second operand).


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