Limits made easy and slope

Hi,
today I am going to explain to you very basic concept called the slope. This is going to be important for the basic understanding of derivation.


Slope describes how much is line steep and if it is decreasing, horizontal or vertical. There is simple equation for it.

m=\frac{y_2-y_1}{x_2-x_1}.

m is the slope of a line

m = \tan (\theta)\!

This equation equals the first one because trigonometry

So to get the slope of line at given point you simply pick one point on it which has two coordinates: x1 y1. Then you pick another point to the right side and it is going to have coordinates x2 y2.

Here you can see why tan(theta) is equal to m

The subtraction creates this kind of triangle from which you can either calculate the angle or the m value which is the steepness.

Lets make an example (from wikipedia), lets say we haw line that runs through two points: A=(1,2) B=(13,8)

When we have such a information we can calculate the slope because:

x1=1
y1=2
x2=13
y2=8

All parts of equation are set:m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}

To show you in practice that such slope is not very big, watch the gif below.

Slope is used often in everyday life for the steepness of road. Often you will see sign as this one in mountains.

This type means that for 100 meters in horizontal direction you will go up by 10 meters.

Next time I will talk about steepness but this time not only on line where the steepness is always the same but also on curve where it changes every moment though derivatives are capable of solving such problems.

Dragallur

PS: all pictures were from wikipedia page so I will not link them one by one.

 

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