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This document discusses various topics relating to functions and their graphs, including increasing and decreasing functions, relative maxima and minima, even and odd functions, piecewise functions, and difference quotients. Examples are provided for each topic to illustrate key concepts such as identifying where a graph is increasing, decreasing, or constant, finding relative extrema, determining if a function is even or odd, interpreting piecewise functions, and simplifying difference quotient expressions.
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This document provides a review of concepts covered in Module 4 including:
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This document discusses key concepts related to derivatives and their applications:
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- It introduces the use of derivatives to find maximum and minimum values, extreme points, and critical points of functions.
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- Functions and function notation. Key points are that a function assigns a single output to each input, and function notation (e.g. f(x)) represents the output of a function given a specific input.
- Finding roots of functions by setting the function equal to zero and solving.
- Composition of functions, where the output of one function becomes the input of another. The order of functions in composition matters.
- Other topics like inverse functions, trigonometric functions, exponentials and logarithms are also reviewed at a high level.
The document
Precalc 8.19.13
The document contains information about functions including definitions, examples of graphs, the vertical line test to determine if a relation is a function, examples of finding the domain of functions from equations, and practice problems determining the domain of various functions. Vocabulary terms defined include function, domain, and range. Functions are described as rules that assign each input to exactly one output, usually through an equation.
Calc i complete
This document provides notes for a Calculus I course taught by Paul Dawkins at Lamar University. The notes cover topics such as functions, inverse functions, trigonometric functions, exponential and logarithmic functions. The notes are intended to be a complete resource for learning Calculus I, though the author warns students that not everything covered in the notes is discussed in class. Students are advised to supplement the notes with attendance and their own class notes. The document contains examples and explanations to accompany the course material.
Uml - An Overview
The document discusses the Unified Modeling Language (UML) which is a general-purpose modeling language used to visualize, specify, construct, and document software systems. UML uses graphical notation to represent the design of software projects including concepts like use case diagrams, class diagrams, sequence diagrams, and more. It provides a standard way to visualize a system from different perspectives including structural and behavioral views.
Maxima & Minima for IIT JEE | askIITians
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Module 2 topic 1 notes
The document reviews key concepts about functions including domain, range, and evaluating functions. It provides examples of determining if a relation is a function using mapping diagrams and the vertical line test. It also gives examples of finding the domain and range of functions from graphs and equations. Practice problems are included for students to determine domains, ranges, and evaluate functions.
Lesson 2 - Functions and their Graphs - NOTES.ppt
The document provides lesson material on functions and their graphs. It includes sections on defining functions, determining if a relation is a function, functional notation, domain and range, graphing functions, and identifying intervals of increase/decrease. Additional topics covered are relative min/max values, step functions, even and odd functions, and piecewise-defined functions. Examples and exercises are provided to illustrate key concepts.
2.3 Functions
This document defines and discusses functions. It begins by defining a relation and function, noting that a function is a special type of relation where each input is mapped to exactly one output. It introduces function notation and discusses the domain, codomain, and range of a function. Examples are provided to illustrate determining if a relation defines a function. The document also covers identifying functions from equations or graphs, and the vertical line test. It concludes with a discussion of function notation and classwork assignments.
Order of Operations
The document discusses order of operations and evaluating algebraic expressions. It explains that parentheses and brackets should be evaluated first, followed by exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. Examples are provided to demonstrate simplifying expressions using order of operations and evaluating expressions by substituting values for variables.
Introduction to Java
This document provides an introduction to the Java programming language. It discusses Java's evolution and history from 1991 to present. It also covers Java fundamentals including data types, operators, decision making and looping constructs, classes and objects, arrays and strings. The document is intended as an overview of the major topics and features in Java.
125 2.2 and 2.3
1) The derivative of a constant is 0. The power rule states that when taking the derivative of a term with an x to a power, the power is subtracted by 1 and brought in front of the term.
2) When evaluating a derivative, you take the derivative of the function and plug in a value for x. For example, the derivative of f(x)=x^2 evaluated at x=3 is 2x=6.
3) In business and economics, the derivative is used to find marginal costs, revenues, and profits which provide the change in each per additional unit.
Edsc 304 lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The rate of change of a function is found by taking the difference quotient, which is similar to the slope formula. The domain of a function is the set of all possible x-values, while the range is the resulting y-values. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.
Special Topics on Functions, Sequences, and Series MAT117 .docx
Special Topics on Functions, Sequences, and Series
MAT/117 Version 9
1
Copyright © 2013 by University of Phoenix. All rights reserved.
University of Phoenix Material
Special Topics on Functions, Sequences, and Series
The goal of this week is to introduce more concepts and skills about functions that you started learning in
the previous week. In addition, you will learn to identify when a list of numbers have an arithmetic or
geometric pattern.
Composition of Functions
The composition of functions is denoted with the notation .
This operation consists on the idea that the input of one function is another function, or algebraically, this
will be written as . This means “The input of is .”
In Cognitive Tutor, you will learn composition of linear functions. However, it is possible to do composition
of functions with any type of functions. Here is an example between linear and quadratic functions:
Example
Let and – .
So,
( )
Careful: is not the same as . Order matters! See the example below using the same
functions above.
( )
One-to-One Functions
Previously, you learned that the basic idea of a function is that “every input has exactly one output,” or
“every x value has exactly one y value.”
For example, in the graph of the function below, you can see every x value corresponds with (matches)
exactly one y value (look closely how each vertical gridline intersects the graph only once; this is called
the Vertical Line Test).
Special Topics on Functions, Sequences, and Series
MAT/117 Version 9
2
Copyright © 2013 by University of Phoenix. All rights reserved.
If you look closely the graph above, there are different inputs that have the same output (or there are
different x values that correspond to the same y value).
Now, when each input of a function has a different output (or every x value has a different y value), then
we say the function has a one-to-one correspondence. Here is an example of a one-to-one function:
In the graph above, every x value has a different y value (look closely how each horizontal gridline
intersects the graph only once; this is called the Horizontal Line Test).
Understanding one-to-one functions is essential to understand inverse functions, which you will learn in
Cognitive Tutor. A real-life example of one-to-one functions is the following: Every person has a unique
social security number and each social security number corresponds to a person.
Exponential Functions
When a constant (a fixed number) is raised to a variable (say, x), we are working with exponential
functions. Algebraically, exponential functions have the form , where a > 0 but cannot be 1.
For example, . When the value of is greater than 1, the graph of these functions have this
form:
From the graph above, as ...
S1 0 derivadas
The document discusses the definition of the derivative and the four-step rule for calculating derivatives. It defines the derivative as the slope of the tangent line to a function's graph at a given point and as the limit of the difference quotient. The four-step rule involves adding an increment to x and y in a function, isolating the change in y, dividing by the change in x, and taking the limit as the change in x approaches zero. The objective is to understand the concept of the derivative from its geometric interpretation and apply formulas to calculate derivatives of algebraic and transcendental functions.
Lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The domain is the set of all possible x-values, and the range is the resulting y-values. To find the rate of change, use the difference quotient which is similar to the slope formula. The vertical line test determines if a graph represents a function - if any vertical line intersects the graph more than once, it is not a function. Domain restrictions specify allowed input values.
Math125 sections 1.5 and 1.6
This document provides an overview of key concepts related to limits and continuity in calculus, including:
1. Defining what a limit is graphically as the value a function approaches as the input gets closer to a number without reaching it.
2. Explaining how to find limits algebraically by direct substitution or simplifying expressions.
3. Distinguishing between one-sided and two-sided limits.
4. Identifying infinite limits and discontinuities that can occur with limits involving vertical asymptotes, holes, gaps or jumps in a function's graph.
5. Defining continuity as a function being continuous at a point if it is hole-free, gap-free, and
Polynomial Functions Review
The document covers three topics:
1) Composite functions and finding f[g(0)] for a given function.
2) The derivative and its applications, including proving the derivative of a function and finding the equation of a tangent line.
3) Polynomial functions, including finding the y-intercept, x-intercepts, critical points and intervals of increase/decrease to sketch the graph of a polynomial function.
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2 2 Bzca5e
- 1. Section 2.2 More on Functions and Their Graphs
- 2. Increasing and Decreasing Functions
- 4. The open intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates.
- 5. Example Find where the graph is increasing? Where is it decreasing? Where is it constant?
- 6. Example Find where the graph is increasing? Where is it decreasing? Where is it constant?
- 7. Example Find where the graph is increasing? Where is it decreasing? Where is it constant?
- 8. Relative Maxima And Relative Minima
- 10. Example Where are the relative minimums? Where are the relative maximums? Why are the maximums and minimums called relative or local?
- 11. Even and Odd Functions and Symmetry
- 14. Example Is this an even or odd function?
- 15. Example Is this an even or odd function?
- 16. Example Is this an even or odd function?
- 19. Example Find and interpret each of the following.
- 20. Example Graph the following piecewise function.
- 21. Functions and Difference Quotients
- 22. See next slide.
- 23. Continued on the next slide.
- 25. Example Find and simplify the expressions if
- 26. Example Find and simplify the expressions if
- 27. Example Find and simplify the expressions if
- 29. Example
- 30. (a) (b) (c) (d)
- 31. (a) (b) (c) (d)
- 32. (a) (b) (c) (d)