Rápido e Fácil | Função do 1º grau | Função afim

Dicasdemat Sandro Curió

13 May 201916:49

Summary

TLDRThis script delves into the concept of functions and their applications, using a factory's cost structure as an example. It explains how a fixed cost plus a variable cost per unit forms a first-degree function, illustrating the linear relationship between cost and production quantity. The script guides through the process of identifying the function's domain and range, and how to derive the function's formula from real-world data. Practical examples and a step-by-step approach make the abstract concept of first-degree functions tangible and applicable.

Takeaways

😀 A function is a mathematical relationship between two sets where one set of values (domain) generates another set of values (image).
🔍 The domain of a function consists of the input values that generate the output, while the image represents the resulting values.
📚 The concept of function is crucial in mathematics, especially in understanding the relationship between input and output sets.
📈 The script provides an example of a linear function, which is a first-degree function, with a cost function in a production scenario.
💰 The cost function includes a fixed cost plus a variable cost per unit produced, which is a common model in economics and business.
📉 The fixed cost is independent of the number of units produced, while the variable cost changes linearly with the number of units.
📝 The script explains how to derive the cost function by identifying the fixed and variable costs and their relationship to the number of units produced.
📊 The process of graphing a linear function is demonstrated, showing how to plot points and draw the line that represents the function.
🔢 The script also covers how to find the slope (angular coefficient) and y-intercept (linear coefficient) of a linear function from a graph.
📌 Understanding the slope and y-intercept is essential for determining the equation of a linear function, which can then be used to predict costs or other outcomes.
🎓 The explanation is aimed at helping learners grasp the concept of functions, particularly first-degree functions, and their practical applications.

Q & A

What is the basic concept of a function in the context of sets?
-A function represents a relationship between two sets where each element from the first set (domain) is associated with exactly one element in the second set (image).
What is the domain of a function?
-The domain of a function is the set of all possible input values (elements from set A) that are used to generate the output values (elements from set B).
What is the image of a function?
-The image of a function is the set of all output values that result from applying the function to each element of the domain.
What is the difference between the domain and the codomain of a function?
-The domain is the set of inputs for the function, while the codomain is the set of all possible outputs, which includes the actual image of the function.
How is the cost function of a production process described in the script?
-The cost function is described as having a fixed cost plus a variable cost per unit produced, where the total cost is calculated by adding the fixed cost to the variable cost multiplied by the number of units produced.
What is the fixed cost in the production cost function mentioned in the script?
-The fixed cost in the production cost function is 40 reais, which is the cost that does not depend on the number of units produced.
What is the variable cost per unit in the production cost function?
-The variable cost per unit is 10 reais, which is added to the total cost for each unit produced.
How can you determine the total cost of producing a certain number of units using the given cost function?
-To determine the total cost, multiply the number of units produced by the variable cost per unit and then add the fixed cost.
What is the practical example given in the script to illustrate the application of a first-degree function?
-The practical example is a factory with a fixed cost of 40 reais and a variable cost of 10 reais per unit produced, where the function is used to calculate the total cost based on the number of units produced.
How does the script explain the process of graphing a first-degree function?
-The script explains graphing by identifying points on the graph, such as the fixed cost point when no units are produced, and then using two points to draw the line that represents the first-degree function.
What is the significance of the slope in the context of a first-degree function?
-The slope in a first-degree function represents the rate of change of the dependent variable (cost) with respect to the independent variable (number of units produced), indicating how much the cost increases for each additional unit.
How can you find the equation of a first-degree function from a graph?
-You can find the equation by identifying two points on the line, calculating the slope (change in y over change in x), and then using one of the points to solve for the y-intercept (b) in the equation y = mx + b.
What is the cost of producing 30 units according to the script's example?
-The cost of producing 30 units is 340 reais, calculated as 10 times 30 (for the variable cost) plus the fixed cost of 40 reais.

Outlines

00:00

📚 Introduction to Functions and Their Basic Concepts

This paragraph introduces the concept of a function as a relationship between two sets. It explains that a function is a rule that assigns each element from the domain set (A) to exactly one element in the image set (B). The paragraph uses an example to illustrate how a value from set A generates a corresponding value in set B. It also discusses the domain and image of a function, clarifying that the domain consists of the values that generate the image, while the image consists of the values being generated. Additionally, it touches on the idea of a function's formula, which is used to determine the output (y) based on the input (x), and introduces the concept of a linear function as an example of a first-degree function.

05:00

🔢 Application of Linear Functions in Cost Calculation

The second paragraph applies the concept of a linear function to a practical scenario involving cost calculation in a manufacturing context. It describes a situation where there is a fixed cost plus a variable cost per unit produced. The paragraph formulates a linear function that represents the total cost as a function of the number of units produced, with the fixed cost being R$40 and the variable cost being R$10 per unit. It explains how to determine the cost of producing a certain number of pieces by substituting the number of units into the function. The paragraph also demonstrates how to calculate the cost for producing 30 pieces, resulting in a total cost of R$340, and briefly mentions the importance of understanding linear functions in various competitive exams and everyday life.

10:01

📈 Graphical Representation and Calculation of Linear Functions

This paragraph delves into the graphical representation of a linear function and the process of determining its equation from a graph. It explains how to find the slope (angular coefficient) of the line by using the change in y (delta y) over the change in x (delta x) between two points on the line. The paragraph provides a step-by-step method for calculating the slope using two given points on the graph and then uses this slope to formulate the equation of the line. It also discusses how to find the y-intercept (b) by substituting a point from the graph into the line's equation, solving for b. The paragraph concludes with an example of how to use the derived linear function to calculate the cost of producing a different number of pieces, emphasizing the linear growth of costs.

15:03

📝 Practical Example and Calculation of Production Costs

The final paragraph presents a practical example of calculating production costs using a linear function. It provides a scenario where the cost of producing nine pieces is to be determined. The paragraph shows the calculation process by substituting the number of pieces (x = 9) into the linear function equation derived earlier. It emphasizes the fixed cost component of the production, which remains constant regardless of the number of pieces produced, and calculates the total cost for producing nine pieces, which is R$85. The paragraph concludes with a summary of the steps involved in solving linear function problems, including finding the angular coefficient, linear coefficient, and using the function to make practical calculations.

Mindmap

Keywords

💡Function

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In the context of the video, the function represents a mathematical concept that describes the relationship between two sets, where one set of values (the domain) is mapped to another set (the image). The video script uses the term to explain how a value from one set generates a corresponding value in another set, illustrating this with an example of a function from set A to set B.

💡Domain

The domain of a function is the set of all possible inputs for the function. In the video, the domain is the set of values that are being used to generate the output values (the image). The script clarifies that the domain consists of the values that are generating the image, such as the numbers 1, 2, and 3 in the example provided.

💡Image

The image of a function is the set of all possible outputs. In the script, the image is described as the set of values generated from the domain. For instance, if the domain values are increased by 5, the resulting set of numbers forms the image of the function.

💡First-degree function

A first-degree function, also known as a linear function, is a function of the form f(x) = ax + b, where 'a' is the coefficient of the linear term and 'b' is a constant. The video script introduces the concept of a first-degree function by explaining its structure and providing an example of a cost function in a production scenario.

💡Coefficient

In the context of a first-degree function, the coefficient refers to the multiplier of the variable term. The script mentions the coefficient as the rate of change in the function, indicating how much the output (cost, in the example) increases for each unit increase in the input (number of pieces produced).

💡Fixed cost

Fixed cost is a cost that does not change with the level of output. In the video, the fixed cost is represented by the constant value in the first-degree function that represents the cost of production, regardless of the number of units produced.

💡Variable cost

Variable cost is a cost that changes with the level of output. The script explains variable cost as the cost per unit produced, which is added to the fixed cost to determine the total cost of production.

💡Graph

A graph is a visual representation of data, typically used to display the relationship between two quantities. The video script describes how to graph a first-degree function by plotting points and drawing a line that represents the cost as a function of the number of pieces produced.

💡Slope

Slope is a measure of the steepness of a line, indicating the rate of change between two variables. In the script, the slope of the line is found by calculating the change in the output (delta y) over the change in the input (delta x), which is used to determine the coefficient of the first-degree function.

💡Linear relationship

A linear relationship is a direct proportionality between two quantities, which can be represented by a straight line on a graph. The video script uses the term to describe how the cost of production increases with the number of pieces produced, forming a first-degree function.

💡Production cost

Production cost is the total cost incurred by a company to produce a good or service. In the script, the concept is used to illustrate how the cost is calculated using a first-degree function, which includes both fixed and variable costs.

Highlights

Introduction to the concept of a function as a relationship between two sets.

Explanation of domain and image in the context of functions.

Clarification of common confusion between domain and image in functions.

Illustration of how a function maps values from one set to another.

Example of a function with a simple addition operation to demonstrate the concept.

Discussion on the practical application of linear functions in daily life.

Introduction to the cost function in a production scenario with fixed and variable costs.

Derivation of the cost function formula based on the number of items produced.

Graphical representation of the cost function and its interpretation.

Explanation of the slope in the context of a linear function representing cost increase per unit.

Calculation of production cost for a specific number of items using the derived function.

Demonstration of how to graph a linear function using two points.

Finding the slope of a line given two points on the graph.

Determination of the y-intercept to complete the linear function equation.

Practical example of calculating production costs for different quantities using the linear function.

Emphasis on the importance of understanding the fixed cost in production scenarios.

Summary of the process to derive a linear function from a graph, including finding the slope and y-intercept.

Encouragement to apply the knowledge of linear functions to solve practical problems.