Math125 sections 1.5 and 1.6
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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
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The document discusses determining limits at infinity for functions. It explains that limits at infinity occur as the variable increases or decreases without bound. The graph of a function may approach horizontal asymptotes or cross them as the variable approaches positive or negative infinity. It also discusses using trigonometric squeeze theorem to determine limits at infinity and how rational functions have the same horizontal asymptotes at positive and negative infinity, while other functions may have different behaviors.
Calc 3.5
The document discusses determining limits at infinity for functions. It explains that limits at infinity occur as the variable increases or decreases without bound. The graph of a function may approach horizontal asymptotes or cross them as the variable approaches positive or negative infinity. It also discusses using trigonometric squeeze theorem to determine limits at infinity and how rational functions have the same horizontal asymptotes at positive and negative infinity, while other functions may have different behaviors.
1.5 all notes
This document discusses limits and how to calculate them. It defines a limit as a number a function approaches as the input value approaches a certain number. It provides examples of using graphs and tables on a calculator to find limits, and discusses direct substitution and the replacement theorem for evaluating limits. Special cases like one-sided limits, limits of polynomials, and limits involving radicals are also covered.
Graphing rational functions
The document discusses graphs of rational functions and how to analyze them. It covers identifying vertical and horizontal asymptotes analytically by finding values where the denominator is zero or analyzing behavior as x approaches positive/negative infinity. Examples show graphing points around asymptotes and confirming trends to sketch the graph. Key steps are finding intercepts and asymptotes, making a T-chart of values, and graphing with curves between points and asymptotes. Practice problems are provided to apply these skills.
Graphing rational functions
The document discusses graphs of rational functions and how to analyze them. It covers identifying vertical and horizontal asymptotes analytically by finding values where the denominator is zero or analyzing behavior as x approaches positive/negative infinity. Examples show graphing rational functions by plotting points near asymptotes and connecting them, confirming asymptotes numerically. Key steps are finding intercepts and asymptotes, making a table near asymptotes, and graphing points and asymptotes. The reader is assigned practice problems applying these skills.
Advanced functions part ii
This document provides an overview of key concepts related to graphing polynomials, including:
1. Definitions of terms like intervals of increase/decrease, odd/even functions, zeros, and multiplicities.
2. Steps for graphing polynomials which include determining behavior, finding intercepts and zeros, and joining points based on multiplicities.
3. Examples are provided to demonstrate finding zeros and their multiplicities, and graphing a polynomial based on the identified features.
4. Information that can be determined from a polynomial graph, such as degree, leading coefficient, end behavior, intercepts, and intervals of increase/decrease.
Rational Expressions
This document discusses rational functions and their properties. It defines rational functions as functions defined by a rational expression, which is a polynomial divided by a nonzero polynomial. It explains that the domain of a rational function excludes values where the denominator is zero, as the function is undefined at those points. These values can create holes or vertical asymptotes in the graph. Vertical asymptotes are vertical lines that the graph approaches but does not touch. The document provides steps for simplifying rational expressions and determining holes and vertical asymptotes from the denominator.
College Algebra 3.2
This document discusses key concepts for graphing polynomial functions of higher degree, including:
1. The maximum number of turning points is equal to the degree of the polynomial minus one.
2. The sign of the leading coefficient determines the end behavior of the graph at high values of x - positive leads to rising, negative leads to falling.
3. Zeros of the polynomial correspond to x-intercepts, and repeated linear factors indicate multiplicity of roots, which determines if the graph touches or crosses the x-axis at those points.
Graphing polynomials
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M17 t1 notes
The document provides step-by-step instructions for graphing rational functions. It explains that rational functions are fractions of polynomials, and defines key features like domains, zeros, vertical and horizontal asymptotes. Examples are given to demonstrate identifying these features and graphing the rational function based on them. Steps include finding intercepts, asymptotes, determining where the graph lies relative to the x-axis, and combining these elements to graph the full function.
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Indefinite Integral
Calculus is the study of change and is divided into differential and integral calculus. Differential calculus studies rates of change using derivatives, while integral calculus uses integration to find accumulated change. These concepts build on limits and algebra/geometry. Leibniz developed the notation and principles of calculus in the 1670s. Differential calculus uses derivatives to determine how quantities change, and integral calculus uses integrals and antiderivatives to determine quantities from rates of change. Differential equations relate functions to their derivatives and have general solutions representing families of curves.
CALCULUS chapter number one presentation
This document provides an overview of key concepts in calculus, including limits, derivatives, and continuity. It discusses how limits describe the behavior of a function as its input approaches a certain value. Specific topics covered include tangent lines and limits, areas and limits, decimals and limits, one-sided limits, infinite limits, vertical asymptotes, computing limits using limit laws, and continuity of functions. The document also introduces derivatives as a new function whose value at each x is the derivative of the original function at that point.
5.3 Graphs of Polynomial Functions
* Recognize characteristics of graphs of polynomial functions.
* Use factoring to find zeros of polynomial functions.
* Identify zeros and their multiplicities.
* Determine end behavior.
* Understand the relationship between degree and turning points.
* Graph polynomial functions.
* Use the Intermediate Value Theorem.
1553 linear & quadratic functions
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Section 1-5
The document discusses the concept of limits in calculus, including the formal definition of a limit as the value a function approaches as the input gets arbitrarily close to a given number. It provides examples of evaluating limits graphically and algebraically, such as determining the limit of a function as x approaches 3 is 6, even though the function is undefined at x=3 due to a removable discontinuity. The document also discusses using graphs to identify different types of discontinuities and limits.
Rs solving graphingquadraticequation
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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.
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- 4. Let’s see if I can explain this in words. This is not the same as just finding f (3). As you can see, there is no point there. To find the limit of f ( x ) as x approaches 3, put your finger over the picture at x = 3. With that open circle hidden, ask yourself, “what does it look like f(3) might be if you had to guess at it?” You would let your eyes follow the line right on up there really close to the hidden part. What y value is it getting closer and closer to? If you had the equation to work with, you could plug in values for x like 2.5, 2.8, 2.9, 2.99, and see what the y-values are “approaching.”
- 5. Put your finger over the graph at x = -2, and pretend like you don’t know what it is. Look at BOTH sides of your finger. The graph on either side of your finger need to be pointing at the same y value. In this case, the two sides of the finger seem to agree.
- 6. Compare: f(x) and g(x)
- 7. What if the two sides don’t agree? After you give it the finger: ANSWER:_________________________________.
- 9. Make sure you can do all the suggested problems. You might need to be reminded about more algebra stuff [order of operations, negative exponents, rational exponents, radicals, etc.]
- 10. D. How to find a limit by simplifying There is a problem with “direct substitution” here!
- 14. A limit only exists if the limit from the left and the limit from the right are equal to each other.
- 16. You try this one:
- 27. Jumps are discontinuities too: This one has a discontinuity at x = 3.
- 28. Vertical asymptotes are discontinuities. This one has a discontinuity at x = 2.