In the series: Note 8.

Version: 0.2

By: Albert van der Sel

Doc. Number: Note 8.

For who: for beginners.

Remark: Please refresh the page to see any updates.

Status: Ready.

Maybe you need to pick up "some" basic "mathematics" rather

So really..., my emphasis is on "rather

So, I am really not sure of it, but I hope that this note can be of use.

Ofcourse, I hope you like my "style" and try the note anyway.

Preceding notes:

Note 1: Basic Arithmetic.

Note 2: Linear Equations.

Note 3: Quadratic Equations and polynomials.

Note 4: The sine/cosine functions.

Note 5: How to differentiate and obtain the derivative function .

Note 6: Analyzing functions.

Note 7: The e

This note: Note 8: Primitive functions or Integral.

Each note in this series, is build "on top" of the preceding ones.

Please be sure that you are on a "level" at least equivalent to the contents up to, and including, note 7.

This note will be a very short note on Primitive fuctions, or sometimes called, the "antiderivatives", of a function.

Although some specific excercises can be spicy, the "core" idea is real easy, as you will see in a minute.

Remember the "derivative" function, we saw in note 5? Here we found for example, that suppose we have f(x)=x

Then the

In note 5, we also discussed some methods to calculate the "derivative", for several types of functions.

The

and therefore it determines the "slope", or gradient, of the tangent line to f(x), at "x".

This time, it's a bit the otherway around. Suppose we have f(x). Then what is F(x), in such a way that F

Suppose f(x)=4x

Then, what is F(x)?

Answer:

F(x)=x

Since the derivative of F(x) is F '(x)= f(x) =4x

Here, you knew how to obtain the derivative of x

actually, "F(x)=x

We introduce a new symbol, used in the equation of finding the primitive function, namely: ∫

Suppose we have funcion f(x). Then what is F(x), in such a way that F

Suppose f(x)=x

Then:

Thus F(x) = 1/(k+1) x

Again: Just think of for example the derivative of x

Thus, the primitive of 3x

You see how to interpret the "1/(k+1) x

Thus the derivative of 1/(k+1) x

Suppose h(x)=x

Then what is the primitive function H(x)?

You can view h(x) as the sum of x

It's true since the derivative of "1/6 x

You see? The "core" idea is not hard to understand. But I agree that specific exercises can be quite spicy.

Remember the "chain" rule from note 5?

It said that the derivative of the (compound) function f(x)=u(v(x)),

is f '(x) = u '(v(x)) . v '(x)

So..., it means that whenever you "reckognize" a function that "looks" like (or might be viewed as) u '(v(x)) . v '(x)

then it's primitive is u(v(x))

Doing many excercises will make you experienced in calculating primitives. For my purpose, understanding

the 'core' idea will suffice for the moment.

calculating the "area" below the function f(x) in two dimensions, like the XY plane.

Would we do a similar action in three dimensions, then we would calculate the "volume" bounded by some surface.

Later we will see more on the latter (not this note).

My statement now is, that in many cases:

Suppse we have the line f(x)=2x.

This line goes through the Origin of the XY plane (0,0), and also goes through the point (2,4).

You can probably see that a right-angled triangle exist, defined by the three points (0,0), (2,0) and (2.4).

This triangle is just exactly half of the rectangle with sidelengths 2 and 4.

So, the total surface area of that rectangle is 8, and thus for the right-angled triangle the Surface area is 4.

Now, let calculate the integral. It should return the value 4 as well:

Therefore, I hope that I made it "plausible" to be true in general as well.

If you believe that the example above is a representative example for a general case, then

we arrive as what is often seen as a "Fundamental Theorem of Integral Calculus":

Please remember that the indefinite integral (without the boundaries "a" and "b"),

And, the definite integral (over an interval)

Determine the definite integral of f(x)=2x

Answer:

We need to calculate:

Thus:

The next note (note 9) is a super quick intro into "complex numbers",

since I need those too for note 10 (which is about differential equations).