Use of Pythagorean Theorem

One of the most important formulas in math is Pythagorean theorem which is named after Greek philosopher Pythagoras (even though its formulation is historically unknown). The importance of Pythagorean theorem lies in the fact that it can be used to find the length of the side of a right triangle, provided the length of the two other sides is known. It is given by:

Where c is the longest side  of the triangle (also known as hypotenuse), and a and b are the two other sides in no particular order.  The right triangle with its corresponding sides is shown in the figure below.

Now we can take a look at a simple example of how we can use Pythagorean theorem to find the length of a side.

Example:

Find the length of the hypotenuse of a right triangle in which the two other side have lengths of 3 and 4 units, accordingly.

Solution:

We can sketch the right triangle described in the problem statement. The triangle below is not drawn to scale.

By using Pythagorean theorem, we can calculate the length of the hypotenuse as shown below.

The length of the hypotenuse of this right triangle is 5 units.

Closing Remarks:

Right angles and triangles are very common. They appear all around us, thus the applications of Pythagorean theorem are endless. This makes it a very powerful formula which must be dominated by any math student. We will see it over and over again!

Basic Rules of Differentiation (Part 3)

In Part 1 and Part 2, we learned the rules of differentiation to handle common functions we might encounter. Now, we will learn about chain rule which will close this chapter of the basic rules of differentiation.

Chain Rule:

Chain rule is perhaps the most powerful of all rules of differentiation. This is due to the fact that many other rules of differentiation are based off it. In order to understand this rule, consider a function f(x) that is a composite function of the functions g(x) and h(x). In this case, the function h(x) is inside the function g(x).

It can be observed that g(x) is the outside function, and h(x) is the inside function. Then, the derivative of f(x) is given by:

This is known as chain rule. It basically says the derivative of f(x) is "the derivative of the outside (keep the inside function the same) times the derivative of the inside."

Examples:

Let's observe some examples of how we can apply chain rule. It is important to understand that the other rules of differentiation still apply.

Once we know how to apply chain rule, we can say that we know how to differentiate. Chain rule is the "pillar" of differentiation. If we understand it, we will be able to learn more complex rules of differentiation and even integration. 

Basic Rules of Differentiation (Part 2)

In Part 1, we learned two basic rules that allow us to differentiate polynomial functions. Now we will learn to deal with functions that involve the multiplication or division of functions. We will learn about product rule and quotient rule.

Product Rule:

Suppose we have a function f(x) that is the result of the multiplication (known as a product) between two functions g(x) and h(x). Then, the derivative of f(x) is given by:

Many teachers and students play with words to memorize this rule. Anyone can make up his/her own way to remember it, but it basically says the derivative of f(x) is "the first times derivative of the last, plus the last times derivative of the first."

Quotient Rule:

Suppose we have a function f(x) that is the result of the division (known as a quotient) of the functions g(x) by h(x). Then, the derivative of f(x) is given by:

In this case, we can say the derivative of f(x) is "the bottom times derivative of the top, minus the top times derivative of the bottom. All divided by the bottom squared."

Examples:

These two rules are quite popular, so let's look at some examples. Notice that we will apply the rules learned in Part 1 when dealing with the terms of the functions.

Now, as seen in the examples, we can differentiate more complicated functions!!!

Basic Rules of Differentiation (Part 1)

There are two basic rules that we can consider the "baby steps" to begin learning calculus. These are the derivative of a constant and the derivative of a variable raised to a constant exponent. 

Derivative of Constant:

The derivative of any constant is zero. Consider the following function f(x), where c is any constant.

Derivative of a Variable Raised to a Constant Exponent:

This is the second rule of differentiation that any calculus student must learn. Consider the following function f(x) where a is any real valued exponent.

In this case, we basically bring down the exponent to the front, and then, subtract 1 to the exponent.

Examples:

Now we will have a few examples that show how important these two rules are. Basically, the derivative of any polynomial function can be obtained from just these two rules.

Many real life situations can be described by polynomial functions, so it is safe to say that these two rules of differentiation will be used frequently.

Graphing a Simple Linear Function

In this post, we present an example of how to graph a linear function in the xy-plane from scratch.

Example

Graph the line described by the function y = -5x + 1.

Solution:

The function y = -5x + 1 is of the form y = mx + b. This indicates the graph is a straight line. For this reason, we can graph the line with only two points that we can connect to form the line.

Select any values of x and plug them into the given function to obtain the corresponding values of y. For this example, we use the values x = -1 and x = 1. 

The table below summarizes the results above.

Trace a line connecting these two points to obtain the graph for the function y = -5x + 1.

The Equation of the Line in Form y = mx + b

Definition

The equation of the line in form y = mx +b represents the basic relationship between two variables (x and y). m is the slope which is a measure of the inclination of the line. b is the y-intercept which is the value of y when x = 0.

The simplest of these functions is the equation y = x. This is a line with slope m = 1, and the y-intercept is b = 0. The graph for the function y = x is shown below.

As we can observe, the graph of y = x is an inclined line in the xy-plane.

The Slope

Mathematically, the slope can be described as the change in the dependent variable (y) over the change in the independent variable (x). This is given as:

This is true for any two points on the line. In other words, in order to calculate the slope, we need to know at least two points on the line.

The y-Intercept

As stated above, the y-intercept is the value of y when x = 0. This can be observed directly if the graph is known, or obtained mathematically from a point on the line (provided the slope was previously found).

Example

Write the equation in form y = mx + b for the line passing through the points (-2,-10) and (1,2).

Solution:

ALWAYS start with the slope. In this case, we can use the given points and plug them into the formula for the slope. Consider Point 1 as (-2,-10) and Point 2 as (1,2).

The slope is found to be 4. Therefore, the equation so far is y = 4x + b. We still need to find b.

In order to find b, we can use any of the two given points into what we have for the equation so far.

For simplicity, we can use (1,2).

The equation of the line passing through the points (-2,-10) and (1,2) is:

The graph of the line is:

From the line in the graph, it can be observed that y increases by 4 units as x increases by 1 unit. In addition, we can see that the y-intercept is -2.