Differential calculus
It is a branch of mathematics that studies how variables or continuous quantities change at a specific
instant or interval. Its main tool is the derivative, which allows you to calculate the instantaneous rate of
change of a function and is essential for optimizing processes.
Differential calculus allows us to model and predict everyday and scientific situations:
Physics and Engineering: To analyze the instantaneous velocity and acceleration of an object, as well
as the trajectory of projectiles or spacecraft.
Economics: To calculate marginal cost or revenue (how costs or profits change when producing an
additional unit).
Technology: In artificial intelligence and machine learning, it is used to adjust algorithms and optimize
models, improving their accuracy
The main components of differential calculus are:
1. Functions
They relate one variable to another.
Example: (f(x) = x^2 + 3x).
2. Variables
Amounts that can change.
The independent variable is usually represented by (x), and the dependent variable by (y).
3. Limits
They allow you to study the behavior of a function when a variable approaches a certain value.
They measure the instantaneous rate of change of a function.
f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
, 4. Referral Rules
Methods to calculate derivatives more easily.
They include power, product, ratio and chain rule.
5. Applications of the derivative
Slope calculation.
Speed and acceleration.
Process optimization (maximum and minimum).
Analysis of the growth and decrease of functions.
The fundamental elements of differential calculus are:
Functions → Boundaries → Continuity → Derivatives → Derivation Rules → Applications
These concepts work together to study and describe how quantities change in math and in real-world
situations.
,Properties of Real Numbers
Table of Contents:
Exercises
Conclusion
, Exercises
Activity 1. Axioms
I. Axiom to which it corresponds:
Axioms of addition
x+2=9
Propositions Axiom
a) x+2=9 Fact
b) x+2+(-2)=9+(-2) Additive Axiom (=)
c) x+(2+(-2))=9+(-2) Associative Axiom (+)
d) x+0=9+(-2) Investing Axiom (+)
e) x=9+(-2) Modulative Axiom (+)
f) x=7 Closing Axiom (+)
x + 1/4 = 10
Propositions Axiom
a) x + 1/4 = 10 Fact
b) x + 1/4 + (-1/4) = 10 + (-1/4) Additive Axiom (=)
c) x + ( 1/4 + (-1/4) ) = 10 + (-1/4) Axiom Associativity (+)
d) x + 0 = 10 + (-1/4) Investing Axiom (+)
e) x = 10 + (-1/4) Modulative Axiom (+)
f) x = 39/4 Closing Axiom (+)
and + 4 = -3
Propositions Axiom
a) and + 4 = -3 Fact
b) and + 4 + (-4) = -3 + (-4) Additive Axiom (=)
c) and + ( 4 + (-4) ) = -3 + (-4) Associative Axiom (+)
d) and + 0 = -3 + (-4) Investing Axiom (+)
e) y = -3 + (-4) Modulative Axiom (+)
f) y = -7 Closing Axiom (+)
x+5=7
Propositions Axiom
a) x+5=7 Fact
b) x + 5 + (-5) = 7 + (-5) Additive Axiom (=)
c) x + ( 5 + (-5) ) = 7 + (-5) Associative Axiom (+)
d) x + 0 = 7 + (-5) Investing Axiom (+)
e) x = 7 + (-5) Modulative Axiom (+)
f) x=2 Closing Axiom (+)
and + 0.3 = -2.4
It is a branch of mathematics that studies how variables or continuous quantities change at a specific
instant or interval. Its main tool is the derivative, which allows you to calculate the instantaneous rate of
change of a function and is essential for optimizing processes.
Differential calculus allows us to model and predict everyday and scientific situations:
Physics and Engineering: To analyze the instantaneous velocity and acceleration of an object, as well
as the trajectory of projectiles or spacecraft.
Economics: To calculate marginal cost or revenue (how costs or profits change when producing an
additional unit).
Technology: In artificial intelligence and machine learning, it is used to adjust algorithms and optimize
models, improving their accuracy
The main components of differential calculus are:
1. Functions
They relate one variable to another.
Example: (f(x) = x^2 + 3x).
2. Variables
Amounts that can change.
The independent variable is usually represented by (x), and the dependent variable by (y).
3. Limits
They allow you to study the behavior of a function when a variable approaches a certain value.
They measure the instantaneous rate of change of a function.
f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
, 4. Referral Rules
Methods to calculate derivatives more easily.
They include power, product, ratio and chain rule.
5. Applications of the derivative
Slope calculation.
Speed and acceleration.
Process optimization (maximum and minimum).
Analysis of the growth and decrease of functions.
The fundamental elements of differential calculus are:
Functions → Boundaries → Continuity → Derivatives → Derivation Rules → Applications
These concepts work together to study and describe how quantities change in math and in real-world
situations.
,Properties of Real Numbers
Table of Contents:
Exercises
Conclusion
, Exercises
Activity 1. Axioms
I. Axiom to which it corresponds:
Axioms of addition
x+2=9
Propositions Axiom
a) x+2=9 Fact
b) x+2+(-2)=9+(-2) Additive Axiom (=)
c) x+(2+(-2))=9+(-2) Associative Axiom (+)
d) x+0=9+(-2) Investing Axiom (+)
e) x=9+(-2) Modulative Axiom (+)
f) x=7 Closing Axiom (+)
x + 1/4 = 10
Propositions Axiom
a) x + 1/4 = 10 Fact
b) x + 1/4 + (-1/4) = 10 + (-1/4) Additive Axiom (=)
c) x + ( 1/4 + (-1/4) ) = 10 + (-1/4) Axiom Associativity (+)
d) x + 0 = 10 + (-1/4) Investing Axiom (+)
e) x = 10 + (-1/4) Modulative Axiom (+)
f) x = 39/4 Closing Axiom (+)
and + 4 = -3
Propositions Axiom
a) and + 4 = -3 Fact
b) and + 4 + (-4) = -3 + (-4) Additive Axiom (=)
c) and + ( 4 + (-4) ) = -3 + (-4) Associative Axiom (+)
d) and + 0 = -3 + (-4) Investing Axiom (+)
e) y = -3 + (-4) Modulative Axiom (+)
f) y = -7 Closing Axiom (+)
x+5=7
Propositions Axiom
a) x+5=7 Fact
b) x + 5 + (-5) = 7 + (-5) Additive Axiom (=)
c) x + ( 5 + (-5) ) = 7 + (-5) Associative Axiom (+)
d) x + 0 = 7 + (-5) Investing Axiom (+)
e) x = 7 + (-5) Modulative Axiom (+)
f) x=2 Closing Axiom (+)
and + 0.3 = -2.4
