Materie

Matematica

Relazioni e funzioni

Integrazione indefinita e definita

Interpretazione dell0integrale definito di una funzione come area con segno dell0insieme di punti del piano compreso tra il suo grafico e le ascisse

Appunti di Matematica: Limiti e Derivate per la Scuola Superiore

Name: Knowunity
Brand: Knowunity

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Appunti di Matematica: Limiti e Derivate per la Scuola Superiore

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@saudade

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The document provides a comprehensive overview of calcolo differenziale (differential calculus), focusing on limits and derivatives. It covers key concepts such as the definition of derivatives, geometric interpretation, and rules for calculating derivatives of various functions. The material is suitable for high school students studying advanced mathematics.

• The text explains the concept of derivatives as the limit of the difference quotient, introducing the notion of tangent lines to curves.
• It presents derivation rules for elementary functions, including polynomials, trigonometric, exponential, and logarithmic functions.
• The document covers advanced topics such as higher-order derivatives, inverse function derivatives, and the chain rule for composite functions.

9/4/2023

10515

lim r
P-P
Si vuole capire quanto vale la retta tangente in un punto della funzione
lim
X-> Xo
Il rapporto
P
Xo
Es.
tangente in P
Xo
Y-Yo
M =

Vedi

Higher-Order Derivatives

This final page introduces the concept of higher-order derivatives, which are crucial for advanced applications in calcolo differenziale (differential calculus).

Higher-order derivatives are defined as follows:

Definition: The nth derivative of a function f(x) is the derivative of the (n-1)th derivative of f(x).

The notation for higher-order derivatives is presented:

f''(x) or y'' for the second derivative
f'''(x) or y''' for the third derivative
f⁽ⁿ⁾(x) or y⁽ⁿ⁾ for the nth derivative

Highlight: Higher-order derivatives are important in analyzing the behavior of functions, including concavity and inflection points.

The page provides examples of calculating higher-order derivatives:

Example: For y = x³, the derivatives are: y' = 3x² y'' = 6x y''' = 6 y⁽⁴⁾ = 0

This example illustrates how the order of the derivative can reduce a polynomial to a constant and eventually to zero.

The page also mentions the applications of higher-order derivatives in physics and engineering:

Vocabulary: In physics, the second derivative of position with respect to time represents acceleration, while the third derivative represents jerk.

This section concludes the document by showing how the concepts of derivatives can be extended to more complex analyses, setting the stage for further study in advanced calculus and differential equations.

Vedi

Trigonometric and Exponential Function Derivatives

This page expands on derivate fondamentali (fundamental derivatives) by covering trigonometric and exponential functions. It provides a comprehensive list of derivatives for these important function types.

The derivatives covered include:

Cosine function: y = cos x, y' = -sin x
Tangent function: y = tan x, y' = sec² x = 1 + tan² x
Cotangent function: y = cot x, y' = -csc² x = -(1 + cot² x)
Exponential function: y = e^x, y' = e^x
General exponential function: y = a^x, y' = a^x · ln a

Highlight: The exponential function e^x is unique in that it is its own derivative, which makes it particularly important in calculus and differential equations.

The page also introduces the derivative of the natural logarithm:

Natural logarithm: y = ln x, y' = 1/x

Vocabulary: The natural logarithm, denoted as ln x, is the logarithm with base e (Euler's number).

A summary table is provided at the end of the page, listing all the fundamental derivatives covered so far. This table is an excellent reference for students studying calcolo differenziale (differential calculus).

Example: The derivative of y = tan x is derived using the quotient rule and the derivatives of sin x and cos x: y' = (cos x · cos x + sin x · sin x) / cos² x = 1 / cos² x = sec² x

This page is crucial for students to memorize and understand these basic derivatives, as they form the foundation for more advanced topics in calculus.

Vedi

Inverse Function Theorem and Chain Rule

This page introduces two advanced concepts in calcolo differenziale (differential calculus): the inverse function theorem and the chain rule for composite functions.

The inverse function theorem is presented as follows:

Theorem: If y = f(x) is an invertible and differentiable function at x₀, then its inverse function x = f⁻¹(y) is differentiable at y₀ = f(x₀), and the derivative of the inverse is given by:

[f⁻¹(y)]' = 1 / f'(x)

The page provides examples of applying this theorem to find derivatives of inverse trigonometric functions:

Example: For y = arccos x, the derivative is y' = -1 / √(1 - x²)

Example: For y = arctan x, the derivative is y' = 1 / (1 + x²)

The chain rule for composite functions is then introduced:

Theorem: If y = g(f(x)) is a composite function where f is differentiable at x₀ and g is differentiable at f(x₀), then the composite function is differentiable at x₀ and its derivative is given by:

(g ∘ f)'(x) = g'(f(x)) · f'(x)

Highlight: The chain rule is crucial for differentiating complex functions that involve composition of simpler functions.

The page also includes a diagram illustrating the relationship between the domains and ranges of the functions involved in the chain rule.

Vocabulary: A composite function is a function that applies one function to the results of another function.

This section provides students with powerful tools for handling more sophisticated functions and prepares them for advanced topics in calculus.

Vedi

Introduction to Derivatives and Limits

This page introduces the fundamental concepts of derivatives and limits in calcolo differenziale. It explains how derivatives are used to find the slope of tangent lines to curves and their relationship to the rate of change of functions.

The page begins by presenting the limit definition of a derivative:

Definition: The derivative of a function f(x) at a point x₀ is defined as the limit of the difference quotient as x approaches x₀:

f'(x₀) = lim[x→x₀] (f(x) - f(x₀)) / (x - x₀)

This definition is crucial for understanding the significato geometrico della derivata (geometric meaning of the derivative).

Highlight: The derivative represents the slope of the tangent line to the function at a given point, which is also interpreted as the instantaneous rate of change.

The page also introduces the notation for derivatives:

y' = f'(x) = dy/dx

An example is provided to illustrate the concept:

Example: For the function y = 3x² - 7, the derivative at x₀ = 2 is calculated using the limit definition.

The page concludes by mentioning the term "rapporto incrementale" (difference quotient) and introduces the idea of one-sided derivatives.

Vocabulary: Rapporto incrementale refers to the expression (f(x) - f(x₀)) / (x - x₀) before taking the limit.

Vedi

Derivatives of Elementary Functions

This page focuses on the derivate fondamentali (fundamental derivatives) of common functions and introduces the concept of one-sided derivatives.

The page starts with an example of calculating left and right-hand derivatives for a piecewise function:

Example: For the function f(x) = {x - 3 if x ≤ 3, x - 1 if x > 3}, the left and right-hand derivatives at x = 3 are calculated.

This example illustrates how derivatives can be used to detect points where a function is not differentiable.

The page then provides a list of derivatives for elementary functions:

Constant function: y = k, y' = 0
Linear function: y = x, y' = 1
Quadratic function: y = x², y' = 2x

Highlight: The derivative of a constant function is always zero, while the derivative of x^n is nx^(n-1).

The page also introduces the concept of simplifying expressions before taking limits to avoid indeterminate forms.

Vocabulary: An indeterminate form occurs when a limit results in an expression like 0/0 or ∞/∞, requiring further analysis.

The derivations are shown step-by-step, emphasizing the importance of understanding the limit process in calculating derivatives.

Vedi

Derivative Rules and Operations

This page focuses on the rules for performing operations with derivatives, which are essential for solving more complex problems in calcolo differenziale (differential calculus).

The following rules are covered:

Constant multiple rule: If g(x) = k · f(x), then g'(x) = k · f'(x)
Sum rule: If h(x) = f(x) + g(x), then h'(x) = f'(x) + g'(x)
Difference rule: If y = f(x) - g(x), then y' = f'(x) - g'(x)
Product rule: If h(x) = f(x) · g(x), then h'(x) = f'(x) · g(x) + f(x) · g'(x)
Quotient rule: If y = f(x) / g(x), then y' = [f'(x) · g(x) - f(x) · g'(x)] / [g(x)]²

Highlight: These rules allow for the differentiation of complex functions by breaking them down into simpler parts.

Each rule is accompanied by an example to illustrate its application:

Example: For y = 3x^5, using the constant multiple rule: y' = 3 · 5x^4 = 15x^4

Example: For y = sin x + 8x^4, using the sum rule: y' = cos x + 32x^3

Example: For y = x² · e^x, using the product rule: y' = 2x · e^x + x² · e^x = e^x(2x + x²)

The page emphasizes the importance of these rules in simplifying the process of finding derivatives for complicated functions.

Vocabulary: The product rule and quotient rule are particularly important for differentiating functions that involve multiplication or division of two or more terms.

This section provides students with the tools to tackle a wide range of derivative problems, building upon the fundamental derivatives learned earlier.

Vedi

More Derivatives and General Rules

This page continues with derivate fondamentali (fundamental derivatives) for more complex functions and introduces general rules for derivatives.

The page covers the following derivatives:

General power function: y = x^n, y' = nx^(n-1)
Square root function: y = √x, y' = 1 / (2√x)
Sine function: y = sin x, y' = cos x

Highlight: The derivative of the sine function is the cosine function, which is a key result in trigonometry and calculus.

The page provides detailed derivations for these functions, emphasizing the use of limit definitions and algebraic manipulations.

Example: For y = x^n, the derivation uses the binomial theorem to expand (x+h)^n before taking the limit.

The page also introduces a general rule for derivatives of the form x^k:

y = x^k, y' = kx^(k-1)

This rule is applicable for any real number k, generalizing the power rule for derivatives.

Vocabulary: The power rule states that the derivative of x^n is nx^(n-1) for any real number n, except when n = 0.

The page concludes with an example of applying this rule:

Example: For y = x^17, the derivative is y' = 17x^16.

This section lays the groundwork for understanding more complex derivative rules and applications in calcolo differenziale (differential calculus).

Vedi

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

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Adoro questa app ❤️, la uso praticamente sempre quando studio.

Materie

Matematica

Relazioni e funzioni

Integrazione indefinita e definita

Interpretazione dell0integrale definito di una funzione come area con segno dell0insieme di punti del piano compreso tra il suo grafico e le ascisse

Appunti di Matematica: Limiti e Derivate per la Scuola Superiore

Sal

@saudade

2.046 Follower

Segui

9/4/2023

10515

4ªl/5ªl

Matematica

332

Appunti gratuiti dei migliori studenti - Sbloccali ora!

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Higher-Order Derivatives

This final page introduces the concept of higher-order derivatives, which are crucial for advanced applications in calcolo differenziale (differential calculus).

Higher-order derivatives are defined as follows:

Definition: The nth derivative of a function f(x) is the derivative of the (n-1)th derivative of f(x).

The notation for higher-order derivatives is presented:

f''(x) or y'' for the second derivative
f'''(x) or y''' for the third derivative
f⁽ⁿ⁾(x) or y⁽ⁿ⁾ for the nth derivative

Highlight: Higher-order derivatives are important in analyzing the behavior of functions, including concavity and inflection points.

The page provides examples of calculating higher-order derivatives:

Example: For y = x³, the derivatives are: y' = 3x² y'' = 6x y''' = 6 y⁽⁴⁾ = 0

This example illustrates how the order of the derivative can reduce a polynomial to a constant and eventually to zero.

The page also mentions the applications of higher-order derivatives in physics and engineering:

Vocabulary: In physics, the second derivative of position with respect to time represents acceleration, while the third derivative represents jerk.

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

[Migliori voti con il supporto dell'intelligenza artificiale

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Trigonometric and Exponential Function Derivatives

The derivatives covered include:

Cosine function: y = cos x, y' = -sin x
Tangent function: y = tan x, y' = sec² x = 1 + tan² x
Cotangent function: y = cot x, y' = -csc² x = -(1 + cot² x)
Exponential function: y = e^x, y' = e^x
General exponential function: y = a^x, y' = a^x · ln a

Highlight: The exponential function e^x is unique in that it is its own derivative, which makes it particularly important in calculus and differential equations.

The page also introduces the derivative of the natural logarithm:

Natural logarithm: y = ln x, y' = 1/x

Vocabulary: The natural logarithm, denoted as ln x, is the logarithm with base e (Euler's number).

Example: The derivative of y = tan x is derived using the quotient rule and the derivatives of sin x and cos x: y' = (cos x · cos x + sin x · sin x) / cos² x = 1 / cos² x = sec² x

This page is crucial for students to memorize and understand these basic derivatives, as they form the foundation for more advanced topics in calculus.

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

[Migliori voti con il supporto dell'intelligenza artificiale

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Inverse Function Theorem and Chain Rule

This page introduces two advanced concepts in calcolo differenziale (differential calculus): the inverse function theorem and the chain rule for composite functions.

The inverse function theorem is presented as follows:

Theorem: If y = f(x) is an invertible and differentiable function at x₀, then its inverse function x = f⁻¹(y) is differentiable at y₀ = f(x₀), and the derivative of the inverse is given by:

[f⁻¹(y)]' = 1 / f'(x)

The page provides examples of applying this theorem to find derivatives of inverse trigonometric functions:

Example: For y = arccos x, the derivative is y' = -1 / √(1 - x²)

Example: For y = arctan x, the derivative is y' = 1 / (1 + x²)

The chain rule for composite functions is then introduced:

Theorem: If y = g(f(x)) is a composite function where f is differentiable at x₀ and g is differentiable at f(x₀), then the composite function is differentiable at x₀ and its derivative is given by:

(g ∘ f)'(x) = g'(f(x)) · f'(x)

Highlight: The chain rule is crucial for differentiating complex functions that involve composition of simpler functions.

The page also includes a diagram illustrating the relationship between the domains and ranges of the functions involved in the chain rule.

Vocabulary: A composite function is a function that applies one function to the results of another function.

This section provides students with powerful tools for handling more sophisticated functions and prepares them for advanced topics in calculus.

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

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Introduction to Derivatives and Limits

The page begins by presenting the limit definition of a derivative:

Definition: The derivative of a function f(x) at a point x₀ is defined as the limit of the difference quotient as x approaches x₀:

f'(x₀) = lim[x→x₀] (f(x) - f(x₀)) / (x - x₀)

This definition is crucial for understanding the significato geometrico della derivata (geometric meaning of the derivative).

Highlight: The derivative represents the slope of the tangent line to the function at a given point, which is also interpreted as the instantaneous rate of change.

The page also introduces the notation for derivatives:

y' = f'(x) = dy/dx

An example is provided to illustrate the concept:

Example: For the function y = 3x² - 7, the derivative at x₀ = 2 is calculated using the limit definition.

The page concludes by mentioning the term "rapporto incrementale" (difference quotient) and introduces the idea of one-sided derivatives.

Vocabulary: Rapporto incrementale refers to the expression (f(x) - f(x₀)) / (x - x₀) before taking the limit.

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

[Migliori voti con il supporto dell'intelligenza artificiale

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Derivatives of Elementary Functions

This page focuses on the derivate fondamentali (fundamental derivatives) of common functions and introduces the concept of one-sided derivatives.

The page starts with an example of calculating left and right-hand derivatives for a piecewise function:

Example: For the function f(x) = {x - 3 if x ≤ 3, x - 1 if x > 3}, the left and right-hand derivatives at x = 3 are calculated.

This example illustrates how derivatives can be used to detect points where a function is not differentiable.

The page then provides a list of derivatives for elementary functions:

Constant function: y = k, y' = 0
Linear function: y = x, y' = 1
Quadratic function: y = x², y' = 2x

Highlight: The derivative of a constant function is always zero, while the derivative of x^n is nx^(n-1).

The page also introduces the concept of simplifying expressions before taking limits to avoid indeterminate forms.

Vocabulary: An indeterminate form occurs when a limit results in an expression like 0/0 or ∞/∞, requiring further analysis.

The derivations are shown step-by-step, emphasizing the importance of understanding the limit process in calculating derivatives.

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

[Migliori voti con il supporto dell'intelligenza artificiale

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Iscrivendosi si accettano i Termini di servizio e la Informativa sulla privacy.

Derivative Rules and Operations

This page focuses on the rules for performing operations with derivatives, which are essential for solving more complex problems in calcolo differenziale (differential calculus).

The following rules are covered:

Constant multiple rule: If g(x) = k · f(x), then g'(x) = k · f'(x)
Sum rule: If h(x) = f(x) + g(x), then h'(x) = f'(x) + g'(x)
Difference rule: If y = f(x) - g(x), then y' = f'(x) - g'(x)
Product rule: If h(x) = f(x) · g(x), then h'(x) = f'(x) · g(x) + f(x) · g'(x)
Quotient rule: If y = f(x) / g(x), then y' = [f'(x) · g(x) - f(x) · g'(x)] / [g(x)]²

Highlight: These rules allow for the differentiation of complex functions by breaking them down into simpler parts.

Each rule is accompanied by an example to illustrate its application:

Example: For y = 3x^5, using the constant multiple rule: y' = 3 · 5x^4 = 15x^4

Example: For y = sin x + 8x^4, using the sum rule: y' = cos x + 32x^3

Example: For y = x² · e^x, using the product rule: y' = 2x · e^x + x² · e^x = e^x(2x + x²)

The page emphasizes the importance of these rules in simplifying the process of finding derivatives for complicated functions.

Vocabulary: The product rule and quotient rule are particularly important for differentiating functions that involve multiplication or division of two or more terms.

This section provides students with the tools to tackle a wide range of derivative problems, building upon the fundamental derivatives learned earlier.

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

[Migliori voti con il supporto dell'intelligenza artificiale

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Iscrivendosi si accettano i Termini di servizio e la Informativa sulla privacy.

More Derivatives and General Rules

This page continues with derivate fondamentali (fundamental derivatives) for more complex functions and introduces general rules for derivatives.

The page covers the following derivatives:

General power function: y = x^n, y' = nx^(n-1)
Square root function: y = √x, y' = 1 / (2√x)
Sine function: y = sin x, y' = cos x

Highlight: The derivative of the sine function is the cosine function, which is a key result in trigonometry and calculus.

The page provides detailed derivations for these functions, emphasizing the use of limit definitions and algebraic manipulations.

Example: For y = x^n, the derivation uses the binomial theorem to expand (x+h)^n before taking the limit.

The page also introduces a general rule for derivatives of the form x^k:

y = x^k, y' = kx^(k-1)

This rule is applicable for any real number k, generalizing the power rule for derivatives.

Vocabulary: The power rule states that the derivative of x^n is nx^(n-1) for any real number n, except when n = 0.

The page concludes with an example of applying this rule:

Example: For y = x^17, the derivative is y' = 17x^16.

This section lays the groundwork for understanding more complex derivative rules and applications in calcolo differenziale (differential calculus).

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

[Migliori voti con il supporto dell'intelligenza artificiale

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Iscriviti con l'e-mail

Iscrivendosi si accettano i Termini di servizio e la Informativa sulla privacy.

Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

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Appunti gratuiti dei migliori studenti - Sbloccali ora!

[Appunti gratuiti per ogni materia, realizzati dai migliori studenti

[Migliori voti con il supporto dell'intelligenza artificiale

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

Scarica

Google Play

Scarica

App Store

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 12 Paesi

950 K+

Studenti che hanno caricato appunti

Non siete ancora sicuri? Guarda cosa dicono gli altri studenti...

Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS