College Algebra 3.2
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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.
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This document discusses rules for taking derivatives of products and quotients. It introduces the product rule for derivatives of terms multiplied together, and the quotient rule for derivatives of terms divided by each other. It notes that while the power rule is easiest, these other rules provide shortcuts when the power rule cannot be used. An example problem uses both the product and quotient rules together to take the derivative.
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.
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This document provides instructions on solving quadratic equations using different methods: factoring, extracting square roots, the quadratic formula, and determining the discriminant. It explains that if the product of two terms is zero, one of the terms must be zero. This is used to solve equations by setting each factor equal to zero. Examples are provided to demonstrate solving quadratic equations by factoring, taking the square root of both sides, and using the discriminant to determine the number of solutions.
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This module discusses polynomial functions of degree greater than two. The key points are:
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2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
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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.
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Module 2 Lesson 2 Notes
This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.
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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.
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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.
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This document provides information about several advanced algebra topics:
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The document outlines the daily lesson log for a Grade 10 mathematics class. It includes 4 sessions on the topic of polynomial functions. The objectives are to illustrate, find intercepts of, and graph polynomial functions. Examples are provided for identifying key features of polynomials like degree, leading coefficient, constant term, and finding x- and y-intercepts by setting the polynomial equal to 0 or using the factored form. The behavior of graphs is also discussed based on whether the degree is odd or even, and the sign of the leading coefficient.
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A relation is a set of ordered pairs. A function is a relation where each domain value has only one range value. To determine if a relation is a function, use the vertical line test or check if each x-value only has one y-value. An equation defines a function if each x only corresponds to one y when solving the equation for y. Piecewise functions are defined by two or more equations over different parts of the domain. The slope of a line is rise over run and can be found by calculating change in y over change in x between any two points. You can write the equation of a line from its point-slope form or by finding the slope between two points and plugging into point-slope form
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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. The rate of change of a function can be found using the difference quotient or slope formula. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point. Domain restrictions limit the possible input values of a function. Horizontal and vertical asymptotes provide information about the behavior of a rational function as x approaches positive or negative infinity.
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1) The document discusses calculus concepts of derivatives and integrals. Derivatives measure the rate of change of a function, while integrals calculate the area under a function.
2) It provides examples of how to calculate derivatives and integrals, including the rules for derivatives of constants, powers, sums, products, and compositions. Integrals are calculated by dividing the area into rectangles.
3) As an application, it shows how to calculate the economic order quantity by setting the derivative of the total cost function equal to zero to find the minimum.
Calculus Application Problem #3 Name _________________________.docx
Calculus Application Problem #3 Name __________________________________________
The Deriving Dead! Due at the beginning of class ______________________
Introduction: Imagine that you are one of many people at a “party” and that, unknown to everyone else, one
person was bitten by a zombie on the way to the party! How quickly will the “zombiepocalypse” spread, and
what are the chances that you will leave the party as a zombie? The objective of this activity is to create a
mathematical model that describes the spread of a disease (such as a zombie virus) in a closed environment, and
then apply calculus concepts to this mathematical model.
Collecting the Data:
Let’s collect some data from an activity that will simulate the spread of a communicable disease over a period of
time, divided into “stages”.
The number of people in our “closed environment” is ________________
1
1
131211109876543210
Number
of Total
Infected
Number
of Newly
Infected
Stage
Number
Applying Calculus to the Data:
1. Using the data from the chart, make a scatterplot of the "Stage Number" (in L1) vs. the "Number of Total
Infected" (in L2). Sketch the scatterplot below. Connect the data points to create a continuous function for Y(t).
2. Using the data that was collected in the activity, answer the following questions about the derivative
function Y’(t), which represents the instantaneous rate of change of the number of infected at any stage.
Consider the domain to be [ 0 , 13 ].
a. When, if ever, is Y’(t) positive? ____________________________________
b. When, if ever, is Y’(t) negative? ___________________________________
c. When, if ever, is Y’(t) increasing? ____________________________________
d. When, if ever, is Y’(t) decreasing? ____________________________________
e. From your answers above, sketch a graph of Y’(t) below.
f. The t-value where Y’(t) changes from increasing to decreasing is the inflection point on Y(t).
According to the data in the chart, this occurs when t = _________, and the corresonding “y-value”
is ________.
(Note: We will check this later in the problem!)
Finding a Logistic Function that Models the Data
3. Since the data (should) appear to be a model for a logistic function, we need to find a function in the
form:
Y(t ) =
c
1 + a ⋅ e− b⋅t
,
where t represents the stage number and Y(t) represents the total number of infected people in stage t.
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activity,
c = _________
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solve for a. Show your work below.
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To find b, the last constant in the model, we need another ordered pair. Let’s use an ordered pair near the middle
of the data, say during Stage #7.
Record this ordered pair: ( 7 , _________)
Substitut ...
Advanced functions ppt (Chapter 1) part ii
This document provides definitions and examples to explain how to graph polynomial functions. It discusses determining the degree of the polynomial, the sign of the leading coefficient, end behavior, x- and y-intercepts, and intervals of increase and decrease. Examples are provided to demonstrate how to find zeros, or x-intercepts, and their multiplicities in order to properly graph the polynomial based on these features.
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This document discusses key concepts for graphing polynomial functions including:
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2. The sign of the leading coefficient indicates whether the graph faces up or down on both ends.
3. X-intercepts and y-intercepts are found by setting the polynomial equal to 0 or the variable equal to 0.
4. The multiplicity of intercepts determines whether the graph crosses or touches the x-axis at that point.
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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.
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The document provides step-by-step instructions for graphing different types of rational functions and polynomials. It includes two examples of graphing rational equations, showing how to find the greatest common factor, factor the numerator and denominator, identify holes and asymptotes, and graph. It also gives an example of solving a word problem involving perimeter and area of a rectangular field enclosed by fencing. Finally, it discusses how to graph polynomials based on their degree and signs.
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The document discusses various methods for computing limits, including:
1) Plugging the value directly into the function f(a) if it is defined
2) Factoring the function if f(a) is undefined to eliminate problems
3) Using the sandwich theorem to evaluate a limit if the function is between two other functions with known limits
It then provides an example using the sandwich theorem to show that limx→0 sin(x)/x = 1 by considering the areas of triangles on a unit circle as x approaches 0.
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4) Metasyntactic variables like (x:xs) follow conventions to bind pattern names and make code more readable.
5) Functions can match multiple patterns at once to handle different input structures.
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ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
College Algebra 3.2
- 8. END BEHAVIOR! (What is the graph doing at extreme left and extreme right?) The l.c. gives us info about the right side, and then we can figure out the left side with a little thinking.
- 12. Describe right and left handed behavior: