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# Table|Definition & Meaning

## Definition

A **table** is a figure widely used for **representation** of **data** or a **quantity** in the form of **rows** and **columns**. For example, it could be used to record the average **scores** in mathematics of each section in a school or the temperature on **different** days of the week.

## What Is a Table?

In mathematics, a **table** is a way of **organizing** and **displaying** **data** in a structured format. It is a set of **rows** and **columns** that are used to represent information in a clear and concise manner. A **table** can be used to represent a wide range of mathematical concepts, including but not limited to: functions, sequences, and **statistical** **data**.

One of the most basic types of **tables** is a function **table**. A function **table** is used to represent the relationship between two variables. It contains two **columns**, one for the input values and one for the corresponding output values.

The input values are typically represented on the **left-hand** side of the **table**, while the output values are represented on the **right-hand** side. The input values are plugged into the function, and the corresponding output values are calculated and recorded in the **table**.

An example of a **function** **table** is the **table** that represents the function f(x) = 2x+3, where x is the input value and f(x) is the output value.

Another type of **table** is a sequence **table**. A sequence **table** is used to represent a sequence of numbers, such as the Fibonacci sequence or the prime numbers. It contains one **column** for the term number and another **column** for the corresponding term value.

The term number represents the position of the term in the sequence, and the term value represents the value of the term at that position. An example of a **sequence** **table** is the **table** that represents the **Fibonacci** **sequence**, where the first **column** represents the term number and the second **column** represents the term value.

A third type of **table** is a **statistical** **table**. A **statistical** **table** is used to represent **statistical** **data**, such as a frequency distribution or a cumulative frequency distribution. It contains a **column** for the **data** values and a **column** for the corresponding frequencies. The **data** values represent the different categories of **data**, while the frequencies represent the number of occurrences of each category.

An example of a **statistical** **table** is a **table** that represents the number of students in a class who scored a certain grade on a test, where the first **column** represents the grade and the second **column** represents the frequency of that grade.

In addition to these basic types of **tables**, there are many other types of **tables** that can be used in mathematics. For example, a **multiplication** **table** is used to represent the results of **multiplication** facts for a given number.

A probability **table** is used to represent the probability of different outcomes for a given event. And a truth **table** is used to represent the logical relationships between different statements.

## Types of Tables

### Multiplication Tables

A **multiplication** **table**, also known as a times **table**, is a **table** that lists the results of **multiplying** numbers by one another. It is a **fundamental** tool used in mathematics to help students learn and memorize the basic facts of **multiplication**.

Multiplication **tables** are typically arranged in a grid format with the **multiplicand** (the number being **multiplied**) listed on the top or left side of the **table** and the **multiplier** (the number used to multiply the **multiplicand**) listed on the bottom or right side of the **table**.

The use of **multiplication** **tables** can be traced back to ancient civilizations, such as the Egyptians and the Greeks, where they used **multiplication** **tables** to perform calculations in trade and commerce. In modern times, **multiplication** **tables** are widely used in elementary and middle school mathematics education as a way to help students learn and memorize the basic facts of **multiplication**.

Multiplication **tables** are also useful in real-life applications, such as in engineering, construction, and other fields that require calculations involving measurements and dimensions. Being able to quickly recall the **multiplication** facts allows for faster and more accurate calculations, which can save time and reduce errors.

### Statistical Tables

Statistical **tables** are **tables** that are used to organize and display **statistical** **data** in a structured format. They are used to summarize and present large amounts of **data** in a way that is easy to understand and analyze.

**Statistical** **tables** typically contain **rows** and **columns**, with the **rows** representing individual observations or cases and the **columns** representing the variables or characteristics of those observations.

**Frequency** **distributions** are **tables** that show the number of observations or cases that fall into different categories or classes. They typically contain two **columns**: one for the **categories** or **classes** and another for the frequencies of observations in each category.

For example, a **frequency** **distribution** **table** for the number of students in a class who scored a certain grade on a test would have the grade as the category and the frequency of that grade as the value.

**Cumulative** **frequency** **distributions** are similar to frequency **distributions**, but they also show the **cumulative** number of observations or cases that fall into different categories or classes. They typically contain three **columns**: one for the categories or classes, another for the frequencies of observations in each category, and a third for the cumulative frequencies of observations.

### Truth Tables

Truth **tables** are a way of representing the logical relationships between statements in a clear and concise manner. They are used in propositional logic, which is a branch of mathematical logic that deals with statements that can either be true or false.

A truth **table** is a **table** that lists all possible combinations of logical statements and their resulting truth values. Each **row** of the **table** represents a different combination of input statements, and the corresponding truth value for that combination is given in the final **column** of the **table**.

Truth **tables** have one **column** for each statement or **logical** operator being considered and one final **column** for the resulting truth value of the entire **logical** **expression**.

The number of **rows** in a truth **table** is determined by the number of statements or **logical** operators being considered; each statement or operator has two possible values (true or false), so the number of **rows** is 2 raised to the power of the number of statements or operators.

For example, the truth **table** for the logical statement “A and B” would have two **columns** (one for A and one for B) and one final **column** for the resulting truth value of the statement “A and B.”

The **table** would have four **rows**, one for each possible combination of values for A and B (A=T, B=T), (A=T, B=F), (A=F, B=T), (A=F, B=F). The resulting truth value for each **row** would be true if A and B are both true and false otherwise.

## Solved Example Involving Functional Tables

Consider the function f(x) = 2x$^3$ + 5x$^2$ – 3x + 7. Create a functional **table** that lists the input values of x from -2 to 2, with increments of 0.5, and the corresponding output values of f(x).

Also, use the functional **table** to determine the local **maximum** and **minimum** **points** of the function and the **inflection** points if they exist.

### Solution

To determine the local **maximum** and **minimum** **points** of the function, we can look for the points where the first derivative of the function is equal to zero. To find the inflection points, we can look for the points where the **second** **derivative** of the function is equal to zero.

**First Derivative**: f'(x) = 6x$^2$ + 10x – 3 = 0**Second Derivative**: f”(x) = 12x + 10 = 0

Solving the first derivative for x, we get x = -1/2 and x = -1/3, which correspond to the local minimum and inflection points. Solving the second derivative for x, we get x = -5/2, which corresponds to the local maximum point.

Therefore, the local **maximum** point is at x = -5/2, the local minimum point is at x = -1/2, and the inflection point is at x = -1/3.

This problem uses functional **tables** to represent the function and its output values, the use of derivatives to find the **maximum**, **minimum**, and inflection points, and the solving of equations to find the x values at which the points occur.

*All images are created using GeoGebra. *