Math 4 introduction - What is Mathematics for?
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The document contains several passages that discuss the purpose and value of mathematics education. Some key points made are: - Mathematics teaches logical thinking and problem solving skills that are useful for a variety of careers including law, military, and business. - Studying mathematics strengthens and exercises the mind in the same way that physical exercise strengthens the body. - The goal of mathematics education is to teach people to reason clearly and accurately, not just for jobs but to improve human reasoning abilities overall.
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Math 4 graphing rational functions
The document discusses rational functions and their key properties for graphing. It defines a rational function as a quotient of two polynomial functions, with the domain being all real numbers except where the denominator is zero. The zeros of the numerator are the zeros of the rational function, and the zeros of the denominator determine the vertical asymptotes. The horizontal asymptote is the line a rational function approaches as x approaches positive and negative infinity. An example demonstrates how to find the horizontal asymptote of a rational function by evaluating it at large positive and negative values of x.
More theorems on polynomial functions
The document discusses theorems related to polynomial functions including the remainder theorem, factor theorem, rational zeros theorem, and fundamental theorem of algebra. It then provides examples of applying these theorems to find zeros of polynomials, factor polynomials, and sketch graphs. Additional topics covered include conjugate, upper/lower bound, and Descartes' rule of signs theorems. The document concludes with exercises for students.
Theorems on polynomial functions
The document discusses several theorems related to polynomial functions:
1) The Remainder Theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a).
2) The Factor Theorem states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0.
3) The Rational Zero Theorem provides conditions for rational zeros of a polynomial with integral coefficients.
4) The Fundamental Theorem of Algebra states that every polynomial function with complex coefficients has at least one zero in the set of complex numbers.
Polynomial functions
The document discusses polynomial functions of the form f(x) = anxn + ... + a1x + a0. It provides information to identify properties of a polynomial function based on its graph, including whether the highest term's exponent n is odd or even, the sign of the leading coefficient an, the lowest possible value of n, and the number of real zeros. It also gives rules for the behavior of graphs based on the sign of an and whether n is odd or even. The document concludes by listing the information needed to graph polynomial functions and providing two examples of functions to graph.
Math 4 axioms on the set of real numbers
This document discusses the axioms that define a field and the set of real numbers. It outlines the closure, associativity, commutativity, distributive property, identity element, and inverse element axioms for fields. It also covers the equality axioms of reflexivity, symmetry, transitivity, and property extensions for addition and multiplication. The goal is to define the necessary axioms for the set of real numbers to form a field.
Math 4 axioms on the set of real numbers
The document discusses the axioms that define a field and the set of real numbers. It outlines the closure, associativity, commutativity, distributive property, identity element, and inverse element axioms for fields. It also covers equality axioms and proves theorems like cancellation for addition and the involution property using the field axioms.
Completeness axiom
The document defines upper and lower bounds of a set as numbers greater than or equal to and less than or equal to elements of the set, respectively. It also defines the least upper bound and greatest lower bound as the smallest upper bound and largest lower bound. Several examples of sets are given with their greatest lower and least upper bounds identified. The homework assigns finding the greatest lower and least upper bounds of additional sets.
Specific function examples
The document discusses different types of functions including linear, quadratic, absolute value, and square root functions. It provides the definitions and key properties of each function such as domain, range, intercepts, vertex, and transformations that modify the graph. Examples are worked through demonstrating how to find specific characteristics of each function and graph transformations.
Inverse of functions
The document defines one-to-one functions and their inverses. A one-to-one function maps each element in the domain to a unique element in the range. The inverse of a one-to-one function undoes the original mapping. To find the inverse function algebraically, interchange the x and y variables and isolate y. Graphically, flip the graph across the line y=x to find the inverse. Examples are provided of finding inverses both algebraically and graphically. Homework problems involve finding specific inverses and their domains and ranges.
Functions
The document defines functions, relations, domains, ranges, and different types of functions such as even, odd, and composite functions. It provides examples of evaluating functions at given values, performing operations on functions, and composing functions. Graphs of functions and their properties such as the vertical line test are also discussed. Homework problems involve identifying functions, finding their domains and ranges, evaluating, operating on, and composing various functions.
Permutations and combinations examples
This document contains 13 examples of permutations and combinations problems from a Mathematics 4 class. It provides example problems involving passcode combinations, palindromes, paths on a grid, lottery number selection, triangles in an octagon, quiz answers, multiple choice exams, hamburger toppings, NBA championship series outcomes, poker hands, and donut flavors. The document is intended to illustrate different applications of permutations and combinations in mathematics.
Permutations
1. The document contains examples and formulas for permutations and combinations. It discusses linear permutations, distinguishable permutations, circular permutations, and ring permutations.
2. One example asks how many five-letter words can be formed from a set of vowels and consonants if they must alternate with no repetition.
3. The combination formula is presented, and examples ask how to select teams and committees from groups of people.
Counting examples
The document provides 12 examples of counting problems using the fundamental counting principle. It explains that if one event can occur in n ways and a second in m ways, the total number of outcomes is n*m. The examples include counting combinations of digits, tournament placements, outfit combinations, license plates, dice rolls, classroom seating arrangements, true/false tests, multiple choice tests, and card shuffles.
Sequences and series
The document provides information about different types of infinite sequences, including explicitly defined sequences, recursively defined sequences, arithmetic sequences, harmonic sequences, and geometric sequences. It gives examples and formulas for finding terms, sums, and analyzing properties of each type of sequence. Key points covered include the definitions of explicit and recursive sequences, the formulas for the nth term and partial sum of arithmetic and geometric sequences, and examples of finding terms and analyzing sequences.
Powers and Roots of Complex numbers
The document discusses operations on complex numbers, including multiplication, raising to powers, and De Moivre's theorem. It provides examples of multiplying complex numbers algebraically and in polar form. It also gives examples of raising complex numbers to powers using the rules zn = rnθn and De Moivre's theorem, which states that (r cis θ)n = rn cis nθ.
Right triangle problems
This document discusses solving right triangle problems using trigonometric functions like sine, cosine, and tangent. It provides examples of right triangles where certain sides or angles are given and asks the reader to solve for unknown values. Applications discussed include finding the length of a pole or distance across a river using right triangles defined by angles of elevation or depression. Bearings and courses of ships are also defined and used in an example of finding distance and bearings between two moving ships after a period of time.
Inverse trigonometric functions
The document discusses inverse trigonometric functions. It defines the inverse sine function as sin^-1x = arcsin(x), with domain [-1,1] and range [-π/2, π/2]. It provides examples of evaluating inverse trig functions like sin^-1(1/2) = π/6. The inverse cosine function is similarly defined as cos^-1x = arccos(x), with domain [-1,1] and range [0,π]. The document concludes with a short quiz evaluating inverse trig expressions.
Graphing trigonometric functions
The document discusses how to graph trigonometric functions like sine and cosine. It explains that the sine function has a range from -1 to 1 and is decreasing in quadrants 1 and 2. The cosine function has the same range and properties as sine, except its zeros occur at odd integer multiples of π/2. Both functions have an amplitude of 1 and a period of 2π. The document also describes how changing the coefficients in trigonometric functions affects their amplitude, period, and horizontal or vertical shifts.
Circles and Tangent Lines
The document provides examples and explanations of finding equations of circles that are tangent to lines or pass through given points. It discusses using formulas for perpendicular distance from a line or point to determine a circle's radius and center. It also shows solving systems of equations algebraically or using geometric constructions to find centers and radii when given additional constraints like points or lines that a circle must be tangent to or pass through.
Circles - Degenerate and Null cases
This document discusses special cases of circle equations, including degenerate circles which represent a single point, and null sets which represent no points. It provides examples of converting circle equations to standard form, identifying degenerate and null sets, and finding the radius and equation of a circle given two points. Practice problems are also presented for students to identify circle equations as representing circles, points, or null sets, and to find the equation of a circle given its radius endpoints.
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Walmart Business+ and Spark Good for Nonprofits.pdf
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
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Event Link:- https://meetups.mulesoft.com/events/details/mulesoft-mysore-presents-mule-event-processing-models/
Agenda
● What is event processing in MuleSoft?
● Types of event processing models in Mule 4
● Distinction between the reactive, parallel, blocking & non-blocking processing
For Upcoming Meetups Join Mysore Meetup Group - https://meetups.mulesoft.com/mysore/YouTube:- youtube.com/@mulesoftmysore
Mysore WhatsApp group:- https://chat.whatsapp.com/EhqtHtCC75vCAX7gaO842N
Speaker:-
Shivani Yasaswi - https://www.linkedin.com/in/shivaniyasaswi/
Organizers:-
Shubham Chaurasia - https://www.linkedin.com/in/shubhamchaurasia1/
Giridhar Meka - https://www.linkedin.com/in/giridharmeka
Priya Shaw - https://www.linkedin.com/in/priya-shaw
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Math 4 introduction - What is Mathematics for?
- 1. What is Mathematics for?
- 2. Kevin is two years older than his wife Angela. In 15 years, the sum of their ages will be one more than thrice Kevin’s age now. What are their present ages? A bank teller has some P50 bills and some P200 bills. The teller has five more P200 bills than P50 bills. The total value of the money is P7250. Find the number of P50 bills. A gardener has 26 kg of a mixture of a fertilizer and weed killer. If one kilo of the mixture is replaced by weed killer the result is a mixture that is 5% weed killer. What percent of the original mixture was weed killer? A ladder is to be constructed with 16 rungs whose lengths decrease uniformly from 20 inches at the base to 16 inches at the top. Find the total length of material needed for the rungs. The angle of elevation from the ground to the top of a cliff is 320 at point A. Point B is exactly 100 feet nearer the cliff, and the angle of elevation from that point is 420. Find the height of the cliff.
- 6. Give 100 loaves to five men so that the shares are in arithmetic progression and the sum of the two smallest is 1/7 of the three greatest. - a problem found in the Rhind Papyrus, Egyptian textbook of mathematics c. 1650 BC
- 7. Simplify: 1 1 1 1 2 2 2 − 2 x y x y − 1 1 1 1 2 − 2 2 2 x y x y 8 2 2 x y x− y x y −2 x− y x y y 2 x 2