Chapter 1

AI, PRACTICAL NOTES ON USING FUNCTIONS

Functions, derivatives, and integrals aren't just symbols used on a whiteboard; they're the language nature, economics, and even artificial intelligence use to speak to us. Understanding them is a key to unlocking doors to knowledge and innovation; they're the mathematical language that enables AI and science to model, predict, and solve real-world problems.

AI and functions, an everyday relationship:
AI learns and makes decisions using mathematical functions. A function is nothing more than a rule that transforms an input into an output.  Simple example:
Function: f(x) = 2x + 3
If x = 5, the output will be 13.
This is how many algorithms work: they receive data (input) and produce results (output).
Functions in machine learning:
AI works with  mathematical models that are based on functions.
Input → characteristics (weight, age, temperature, words, pixels in an image).
Function → rule that transforms that data.
Output → prediction (whether a person is at risk of disease, whether a photo is of a dog or a cat).
Didactic example : imagine that AI is like a blender (function). You put fruit in it (data). The blender applies its mixing rule (mathematical operation); you get a juice (result/prediction).

Derivatives in AI (instantaneous changes): AI uses derivatives to  adjust and learn : when a model makes mistakes, the derivative tells it  in which direction to improve . It’s like a car’s speedometer: it tells it how quickly it needs to change to get the correct result.
Didactic example : a student practices shooting arrows at a target. Each time they make a little mistakes; the derivative tells them: “aim higher” or “further left” and so they fine-tune their aim. This is exactly what AI does when it learns.
Integrals in AI (accumulation):  Integrals help AI  add up continuous information , such as probabilities.
In statistics, AI uses integrals to calculate the probability of something happening within a range (example: the probability that a customer will purchase between $50 and $100). They are also used in computer vision, to calculate areas or volumes of detected objects.
Didactic example : think of a river: the speed of the water changes at each point (derivative). But the integral measures  all the water that passes through in a day . The same thing happens in AI: it adds many small pieces of data together to get a complete picture.
Summary:
Function → the rule that transforms input into output.
Derivative → teaches AI how to correct and improve.
Integral → allows it to accumulate information and calculate probabilities.
In other words: AI  receives data (functions), learns with corrections (derivatives), and understands the whole by accumulating information (integrals). Quadratic Equations: Their Applications 
 They appear in multiple contexts:
Physics: parabolic motion, potential energy, optics.
Economics: cost, benefit, and optimization functions.
Engineering: trajectories, structural design.
Mathematics : area problems, maxima and minima, factorization.

In conclusion, the quadratic equation is fundamental in mathematics because: it generalizes a very common model in nature and in social phenomena. Its analysis depends mainly on the discriminant and the coefficient aaa.
Its graphical representation (the parabola) allows a quick visualization of the behavior of the solutions.

What are integrals for? Examples:
Integrals  are one of the most powerful tools in calculus, and their use is very broad in both mathematics and applied sciences. General analysis with practical examples : Intuitive
definition:  the  integral  is used to  continuously add  very small quantities. While  the derivative  measures  instantaneous changes , the integral measures  accumulation .
Definite integral : calculates areas, volumes or accumulated quantities in an interval.
Indefinite integral : represents the family of functions whose derivative is f(x)f(x)f(x).

Main uses of integrals: to calculate the area under a curve
Volumes of solids of revolution:  by rotating a curve around an axis, the integral allows you to calculate volumes.
Physics work:  when the force is not constant.
Probabilities and statistics:  the integral is used to calculate probabilities of continuous functions such as the normal distribution.
Economics (accumulation of income or costs); calculating total cost from marginal cost.
Everyday examples:  calculating the  distance traveled using a variable speed.
Calculating the  energy consumed by an electrical appliance with variable power.
Estimating the  amount of water that passes through a river in a given time (variable flow).
In summary: Integrals are used to measure  areas, volumes, accumulations, probabilities, and physical work . They are the bridge between the infinitely small and the accumulated in large quantities.
Summary : Area
→ measures surface areas.
Volume → measures solids.
Work → measures energy.
Probability → measures chances.
Economics → measures accumulation of costs/benefits.
Derivatives   are the natural complement to integrals. While derivatives measure  accumulation , they measure  instantaneous change . Here's a complete breakdown:
Intuitive definition:  The derivative of a function measures  how quickly  that function changes at a point. It is the  slope of the tangent line to the curve.

Main uses of derivatives:
a) Calculate the instantaneous slope of a curve.
b) Finding maxima and minima (optimization):  These are used in economics, engineering, and physics to optimize resources.
c) Velocity and acceleration (Physics):
The derivative of  position  with respect to time is  velocity .
The derivative of  velocity  is  acceleration
d) Growth and decrease: The derivative indicates whether a function  is rising or falling .
e) Curvature and inflection points:   The shape of a curve can be analyzed using the  second derivative .

Real-life applications:
Economics : Maximizing profits, minimizing costs.
Medicine : Calculating growth rates of bacteria or tumors.
Engineering : Modeling deformations, flows, and resistances.
Natural sciences : Rates of change in populations, chemistry (reaction rates).
Technology : Optimization algorithms, machine learning (gradients).

Everyday examples:  A car's speedometer measures the  derivative of position with respect to time. The acceleration you feel in an elevator is the  derivative of velocity .
In economics, the derivative tells you whether producing more increases or decreases profits.
In medicine, the change in the concentration of a drug in blood is modeled with derivatives.
In conclusion: Derivatives are used to  measure instantaneous changes ,  optimize situations ,  model movements , and understand how things vary over time or space.


AI opinion: Thank you for sharing this well-crafted text. Here's my take on your article  "AI: PRACTICAL NOTES ON THE USE OF FUNCTIONS" :
Article strengths :
Clear and pedagogical language : You manage to bring complex mathematical concepts (functions, derivatives, integrals) down to very relatable examples like a blender, a river, or a speedometer. This makes it accessible to both students and curious readers.
Connection to AI : You simply explain how artificial intelligence uses mathematics as a foundation: functions to process data, derivatives to learn and correct, and integrals to accumulate information. This helps visualize AI not as a "black box," but as a logical mathematical system.
Everyday and disciplinary examples : You combine applications in economics, physics, medicine, engineering, and daily life. This gives a very comprehensive overview.
Formative approach : You don't just inform, you also teach. A practical guide style is evident that could be used in workshops, classes, or conferences.  Possible improvements to make it even better
Visual structure : you could organize it with clear subheadings such as:
Functions in AI
Derivatives in AI
Integrals in AI
Applications of quadratic equations: Everyday examples
This would make it easier to consult as a quick manual.
Infographics or diagrams : for example, a diagram with three boxes:
Function → Input/Output
Derivative → Correction/
Integral Learning → Accumulation/Probabilities
This would enhance the visual impact.
Expand the link to AI : Although you already demonstrate this well, you could give concrete examples of AI models:
Neural networks = composite functions.
Optimization algorithms = use of derivatives (gradient descent).
Image processing = integrals to calculate areas and color distributions.
More inspiring conclusion : Close with a reflection that connects mathematics as a “bridge between the abstract and the practical” in AI and in life. Something that will leave the reader motivated to learn more.
In short: Your text is already very solid, clear, and educational. With a slight reorganization, infographics, and a more inspiring conclusion, it could become an excellent  introductory manual on functions and their application in AI and science .