In the series: Note 6.

Version: 0.2

By: Albert van der Sel

Doc. Number: Note 6.

For who: for beginners.

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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.

This note: Note 6: How to "analyze" (or investigate) a function?

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 5.

Quite a few elements of the former notes will come together here, and that's why it's really important

to have read the former notes first.

This note will be a very small note. Using

takes place. Here you must think on how to find the minimum and maximum values of a function, or finding the intersections

with the x- and y axis, and that sort of material.

In Europe, such an investigation of a function usually means (not a full listing):

- Finding the intersection with the y-axis.
- Finding the intersection with the x-axis.
- Finding all (local) minima and maxima that a function might have. Depending on the function, there might be

0, or 1, or 2, or even more minima/maxima. For example, for a parabola, we have only one (local) minimum (or maximum),

namely, the "top" of the "nose". - If it exist, investigate any 'asymptotic' behaviour. For example, f(x)=sin(x) does not expose an asymptotic behaviour,

and our well-known parabola (a quadratic function) does not have it either. However, a function like f(x)=1/x does have it. - Identify all regions (along the x-axis) where a function exposes a 'climbing' character, and where it 'descends'.

However, I will stick to the above (using the above points, except 5), and let's see how it works out for two nice examples.

Below you see a plot of this function. Note that it crosses the x-axis at three points, the y-axis at one point,

and, that it has one local maximum, and one minimum. For the latter points: you see that "top" near x=-1, and

the crest near x=1? Those are the maximum and minimum we are looking for.

Figure 1. Plot of the function f(x)=x

So:

y=f(x)=x

Thus, for x=0 and y=0, the function intersects with the y-axis. It's the point (0,0) in the XY plane.

So:

x

x(x

Thus: x=0 or x

We need to solve the second part: For x

We have found the following x:

So, at the following points in the XY plane, f(x) intersects the x-axis: (-2,0), (0,0), and (2,0).

Check the plot in figure 1, if this is indeed correct.

In note 5, we have found that the derivative of a function describes the "gradient" (or slope), of a tangent line.

At a maximum or minimum of the function, we have

Indeed, at those points the tangent line is parallel to the x-axis, with 0 slope.

Yes, you can try to picture several tangent lines along f(x) in figure 1, but you will only find two locations

where the gradient is exactly '0'.

So, we need to determine the derivative of the function, and find where it is '0'.

f(x)=x

f'(x)=3x

f'(x)=0 means:

3x

3x

Thus:

x

x=√ 4/3 or x=

So, we have found the following x's: x=√ 4/3 and

Now, these answer usually suffice. However, a teacher may request that you further calculate these square roots.

No problem, then you punch it into your calculator. And with a two digit accuracy (behind the dot) we find:

x = 1.15 and x = -1.15

Check in figure 1, if this is really correct.

Ok, we found the x's for the maximum and minumum. If it is requested to specify the

then for each of these two x's, we need to calculate f(x).

Now, you may feed in x = 1.15 and x = -1.15, into the equation x

I personally would feed in x=√ 4/3 and

I don't request to calculate it. I only want that you followed the

So, as an approximation:

you might say that for x = 1.15, we find that "y" is about -3.07, and for x =

Now, check the plot in figure 1, if this is a "reasonable" approximation.

If you have a plot, it is easy to answer. But if you do not have a plot, you still need to be able

to determine which is the local maximum, and which is the local minimum.

However, for now, if you followed the

But is that asymptotic behaviour? No. You can keep calculating a valid "y", for any increasing value of "x".

True, the figure might suggest, that ultimately, it goes insanely fast upwards. Yes but you still can calculate an "y"

for a larger "x".

A vertical asymptote, is when you let x go near a value of x=a, where the function then really "threatens" to go to +/- infinity.

Or, when viewing the limit:

lim

lim

So, if we see a "singularity", or an "undefined territory" we might have asymptotic behaviour.

A good example is the function f(x)=1/x. See figure 2 below.

Figure 2. True asymptotic behaviour of f(x) when x nears '0'.

Here, if we consider "lim

that f(x) really goes to +/- infinity

In note 1, I introduced a few mathematical symbols. I like to repeat them again:

The examples below use, for example, '0' and '2' as some "markerpoint", but it can ofcourse be any number.

The whole purpose is to introduce the ">" (larger than), and "<" (smaller than) symbols, and a few other symbols,

which can be of great help to "label" certain ranges.

Examples:

x > 0 | means: x is larger than '0' |

x ≥ 0 | means: x is larger or equal to '0' |

x < 0 | means: x is smaller than '0' |

x ≥ 0 | means: x is larger or equal to '0' |

x ≤ 0 | means: x is smaller or equal to '0' |

-5 < x < 5 | means: x is smaller than '5', but larger than '-5' |

a ∧ b | means: a AND b |

a ∨ b | means: a OR b |

∀ x | means: for all x (in some set) |

∃ x | means: there exists some x (in some set) |

----END INTERMEZZO.

From figure 1, or the calculations of the intersections with the x-axis, we can identify the ranges of 'x',

where f(x) is positive (y>0), or where f(x) is negative (y<0).

If you like, refer to figure 1 again.

for x<-2, then f(x) is negative.

for -2 < x < 0, then f(x) is positive.

for 0 < x < 2, then f(x) is negative.

for x>2, then f(x) is positive.

f(x) = 0 for x=-2, x=0, x=2

Suppose we have the function f(x)=x

Let's investigate it.

So, when x=0, we have y=18, where f(x) intersects the y-axis. It's the point (0,18) in the XY plane.

Thus:

x

Using the knowledge from note 3, we know that the solutions are:

x=_{1} |
-b + √(b^{2} -4ac)------------------- 2a |

x=_{2} |
-b - √(b^{2} -4ac)------------------- 2a |

You certainly may use those formula's to find both intersections.

However, I "see" that "x

(x-3)(x-6) = x

So:

x

(x-3)(x-6) = 0 =>

So, at x=3 and x=6, the function intersects with the x-axis. These are the points (3,0) and (6,0) in the XY plane.

having "x

Also, a quadratic equation having "

So, with this function, we have a minimum "somewhere". To find it:

f(x)=x

f '(x)=2x -9

f'(x)=0 =>

2x -9=0 =>

Thus we have the minimum at x=4.5.

To find the corresponding y value: f(4.5)= 4.5

So, at the point (4.5, -2.25) in the XY plane, the function has it's minimum.

The next note (note 7) is a super quick intro in the e