You can download the document that will teach you how to derive the formula for finding areas underneath curves below:

Alternatively, you can watch this video:

You can download the document that will teach you how to derive the formula for finding areas underneath curves below:

Alternatively, you can watch this video:

*[Please note: In order to derive the Taylor Series, you will need to understand how to differentiate. If you know how to differentiate, finding the Taylor Series won’t be much of a problem. You also need to know that 0!=1, 1!=1, 2!=2, 3!=6, x^0=1, x^1=x.]*

In this post I will be demonstrating how one can produce the **Taylor Series** from absolute scratch.

First of all, let’s look at the diagram above. Now, let’s suppose that the equation of the function above is:

Ok, so we have the equation for the function, however, it isn’t complete. C_0, C_1, C_2, C_3 etc are hidden constants. This means that our second task will be to discover these constants. We need to discover these constants to find the complete equation of the function so that we can arrive at the Taylor Series. Fortunately, this task won’t be too difficult. Let me show you how C_0, C_1, C_2, C_3 etc can be found fairly easily…

When x=0:

Now:

When x=0:

Also:

When x=0:

And, finally:

When x=0:

Alright, so now that we have discovered the hidden constants C_0, C_1, C_2 and C_3, our third task is to write down the complete equation of the function f(x+a). Thanks to the information we have above, the fact that x^0=1 and x^1=x, plus our ability to spot patterns, we will be able to do this quite quickly…

[*Image can be seen hereÂ if it appears to be too small on this page.]

And it turns out that the equation we have just above is the **Taylor Series** function. It can be simplified to look like this…

What is also interesting is that if we transform a=0, we get the **Maclaurin Series** function which can be used to discover formulas for things such as e^x.

If you have any questions regarding this post, please leave your comments below. Once again, thanks for stopping by! đź™‚

**Related:**

An **even functionÂ **exists when for all values of x.

The graph of anÂ **even functionÂ **must be symmetrical about the y-axis.

Examples ofÂ **even functionsÂ **have been listed below:

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AnÂ **odd function** exists when for all values of x.

Graphs ofÂ **odd functionsÂ **should have 180 degrees rotational symmetry about the origin (0,0).

Examples ofÂ **odd functionsÂ **are listed below:

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Most functions are neither even or odd.

An example of a function that is neither even or odd would be: