This document defines functions and discusses the pigeonhole principle. It defines injections, surjections and bijections between sets. The pigeonhole principle states that if more pigeons are placed in fewer holes, then some hole must contain more than one pigeon. The document provides examples applying the principle, including that in a group of n people there must be two who shake hands with the same number of others. It also proves theorems about subsets summing to the same number and groups including clubs or strangers.
We will define what is a function “formally”, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.
The document provides examples of solving linear equations. It explains that a linear equation is one where the expressions on both sides are linear, such as 3x + 10 = 34. An example problem involves calculating the cost of x pizzas including delivery. By setting up and solving the equation 3x + 10 = 34, it is determined that x = 8 pizzas. The document discusses manipulating linear equations through steps like subtraction to solve for the variable.
The document discusses operations that can be performed on polynomial expressions. It defines terms and like-terms in polynomials, and explains that like-terms can be combined while unlike terms cannot. It provides examples of combining like-terms, expanding polynomials using the distributive property, multiplying terms and polynomials, and simplifying the results.
The document discusses systems of linear equations. It explains that to solve for two unknowns, two equations are needed, and to solve for three unknowns, three equations are required. A system of linear equations consists of two or more linear equations with two or more variables. A solution to the system is a set of numbers for each variable that satisfies all equations. Two examples are provided to demonstrate solving systems of two equations with two unknowns to find the price of items.
The document discusses solving systems of linear equations using the substitution method. It provides an example of a system involving coupons that can be exchanged for pizza slices and donuts. It then shows an example of using the substitution method to solve the system 2x + y = 7 and x + y = 5. The method involves solving one equation for one variable in terms of the other and substituting it into the other equation.
We will define what is a function “formally”, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.
The document provides examples of solving linear equations. It explains that a linear equation is one where the expressions on both sides are linear, such as 3x + 10 = 34. An example problem involves calculating the cost of x pizzas including delivery. By setting up and solving the equation 3x + 10 = 34, it is determined that x = 8 pizzas. The document discusses manipulating linear equations through steps like subtraction to solve for the variable.
The document discusses operations that can be performed on polynomial expressions. It defines terms and like-terms in polynomials, and explains that like-terms can be combined while unlike terms cannot. It provides examples of combining like-terms, expanding polynomials using the distributive property, multiplying terms and polynomials, and simplifying the results.
The document discusses systems of linear equations. It explains that to solve for two unknowns, two equations are needed, and to solve for three unknowns, three equations are required. A system of linear equations consists of two or more linear equations with two or more variables. A solution to the system is a set of numbers for each variable that satisfies all equations. Two examples are provided to demonstrate solving systems of two equations with two unknowns to find the price of items.
The document discusses solving systems of linear equations using the substitution method. It provides an example of a system involving coupons that can be exchanged for pizza slices and donuts. It then shows an example of using the substitution method to solve the system 2x + y = 7 and x + y = 5. The method involves solving one equation for one variable in terms of the other and substituting it into the other equation.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
The document discusses solving linear equations by factoring using an example of determining the number of pizzas ordered. It formulates the problem as the equation 3x + 10 = 34 where x is the number of pizzas. It solves the equation by subtracting 10 from both sides, dividing both sides by 3, and determining that x = 8 pizzas were ordered. The document then provides more details on linear equations, their structure, and their general solution method.
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
This document discusses recursive definitions, algorithms, and program correctness. It provides examples of recursively defining functions and sets using a basis step and recursive step. Examples of recursive algorithms are given for computing factorials and exponentials. The difference between recursion and iteration is explained. Program verification and correctness are defined, and examples are worked through to show initial and final assertions for program segments. The document concludes with an announcement about an upcoming exam.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses mathematical expressions and how to combine and manipulate them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms can be combined by adding or subtracting their coefficients, while unlike terms cannot be combined. Multiplying an expression distributes the number to each term using the distributive property.
The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents:
1) The multiply-add rule: ANAK = AN+K
2) The divide-subtract rule: AN/AK = AN-K
3) The power-multiply rule: (AN)K = ANK
4) Additional rules including that A0 = 1 and A-K = 1/AK
The document discusses linear equations and how to solve them. It begins by providing an example of solving a multi-step linear equation to find the number of pizzas ordered given the total cost. It then defines linear equations as those containing only first degree terms of the variable and no higher powers. The document states that linear equations are easy to solve by manipulating the equation to isolate the variable. It provides examples of single-step linear equations and explains the basic principle is to apply the opposite operation to both sides to isolate the variable.
6.1 system of linear equations and matricesmath260
The document discusses solving systems of linear equations using the elimination method. It provides an example of a system with two unknowns, x and y, representing the cost of a hamburger and fries. The system is translated into two equations, which are then subtracted to eliminate one variable. The resulting single equation is solved for the remaining variable, which is substituted back into the original system to solve for both unknowns. A second example system with three unknowns, x, y, and z, is also given and the elimination method is shown being applied step-by-step to reduce the system.
49 factoring trinomials the ac method and making listsalg1testreview
The document discusses factoring trinomials. It explains that trinomials are polynomials of the form ax2 + bx + c. There are two types of trinomials: those that are factorable, which can be written as the product of two binomials, and those that are not factorable. The ac-method uses tables to determine whether a trinomial is factorable or not by finding values for u and v that fit the table. If values are found, the trinomial can be factored using the grouping method. An example factors the trinomial x2 - x - 6 by grouping.
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
The document discusses systems of linear equations. It defines a system of linear equations as a collection of two or more linear equations with two or more variables. A solution to a system is a set of numbers, one for each variable, that satisfies all equations simultaneously. The document provides an example system involving the costs of hamburgers and fries to illustrate how to solve a system of two equations with two unknowns.
The document summarizes the development of number systems from ancient times to modern real numbers. It begins with early humans using tally marks to count sheep and cattle. Ancient Egyptians had a base-10 system using symbols for 10 and 100 that were repeated to represent larger numbers. Over time, more advanced societies developed more complex systems until the modern real number system was established. This system includes natural numbers, integers, rational numbers, irrational numbers, and real numbers. It discusses the properties of these different types of numbers.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and represent calculation procedures. Evaluation is the process of replacing variables in an expression with input values and calculating the output. This allows expressions to be used to solve real-world problems by modeling relationships between variables.
The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
Higher order derivatives are obtained by repeatedly taking the derivative of a function or its derivatives. The order of a derivative refers to how many times differentiation has been performed. To find a higher order derivative, one simply takes the derivative of the existing derivative. For example, to get the third derivative f'''(x) of a function f(x), one would take the derivative of the second derivative f''(x). Higher derivatives provide important information about the curvature and flexibility of a function at different points.
This document contains a presentation on polynomials. It defines what a polynomial is and discusses how the degree of a polynomial relates to the number of zeros. It provides an example of a cubic polynomial and discusses the relationship between the zeros and coefficients of a cubic polynomial. Specifically, it states that the sum of the zeros equals the negative of the coefficient of x^2, the sum of the products of the zeros equals the coefficient of x, and the product of the zeros equals the negative of the constant term. The document then provides two example questions - one asking to find the zeros of a quadratic polynomial and verify the relationship between the zeros and coefficients, and another asking to find a quadratic polynomial with given zeros.
This document discusses roots of polynomials. It defines polynomials and monomials, and explains that the roots of a polynomial are the values that make the polynomial equal to zero. It covers the fundamental theorem of algebra, which states that a polynomial of degree n has n roots. Descartes' rule for the number of positive roots is introduced. Properties of the roots and factors of polynomials are outlined. Examples of finding the roots of polynomials are provided at the end.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
The document discusses solving linear equations by factoring using an example of determining the number of pizzas ordered. It formulates the problem as the equation 3x + 10 = 34 where x is the number of pizzas. It solves the equation by subtracting 10 from both sides, dividing both sides by 3, and determining that x = 8 pizzas were ordered. The document then provides more details on linear equations, their structure, and their general solution method.
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
This document discusses recursive definitions, algorithms, and program correctness. It provides examples of recursively defining functions and sets using a basis step and recursive step. Examples of recursive algorithms are given for computing factorials and exponentials. The difference between recursion and iteration is explained. Program verification and correctness are defined, and examples are worked through to show initial and final assertions for program segments. The document concludes with an announcement about an upcoming exam.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses mathematical expressions and how to combine and manipulate them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms can be combined by adding or subtracting their coefficients, while unlike terms cannot be combined. Multiplying an expression distributes the number to each term using the distributive property.
The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents:
1) The multiply-add rule: ANAK = AN+K
2) The divide-subtract rule: AN/AK = AN-K
3) The power-multiply rule: (AN)K = ANK
4) Additional rules including that A0 = 1 and A-K = 1/AK
The document discusses linear equations and how to solve them. It begins by providing an example of solving a multi-step linear equation to find the number of pizzas ordered given the total cost. It then defines linear equations as those containing only first degree terms of the variable and no higher powers. The document states that linear equations are easy to solve by manipulating the equation to isolate the variable. It provides examples of single-step linear equations and explains the basic principle is to apply the opposite operation to both sides to isolate the variable.
6.1 system of linear equations and matricesmath260
The document discusses solving systems of linear equations using the elimination method. It provides an example of a system with two unknowns, x and y, representing the cost of a hamburger and fries. The system is translated into two equations, which are then subtracted to eliminate one variable. The resulting single equation is solved for the remaining variable, which is substituted back into the original system to solve for both unknowns. A second example system with three unknowns, x, y, and z, is also given and the elimination method is shown being applied step-by-step to reduce the system.
49 factoring trinomials the ac method and making listsalg1testreview
The document discusses factoring trinomials. It explains that trinomials are polynomials of the form ax2 + bx + c. There are two types of trinomials: those that are factorable, which can be written as the product of two binomials, and those that are not factorable. The ac-method uses tables to determine whether a trinomial is factorable or not by finding values for u and v that fit the table. If values are found, the trinomial can be factored using the grouping method. An example factors the trinomial x2 - x - 6 by grouping.
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
The document discusses systems of linear equations. It defines a system of linear equations as a collection of two or more linear equations with two or more variables. A solution to a system is a set of numbers, one for each variable, that satisfies all equations simultaneously. The document provides an example system involving the costs of hamburgers and fries to illustrate how to solve a system of two equations with two unknowns.
The document summarizes the development of number systems from ancient times to modern real numbers. It begins with early humans using tally marks to count sheep and cattle. Ancient Egyptians had a base-10 system using symbols for 10 and 100 that were repeated to represent larger numbers. Over time, more advanced societies developed more complex systems until the modern real number system was established. This system includes natural numbers, integers, rational numbers, irrational numbers, and real numbers. It discusses the properties of these different types of numbers.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and represent calculation procedures. Evaluation is the process of replacing variables in an expression with input values and calculating the output. This allows expressions to be used to solve real-world problems by modeling relationships between variables.
The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
Higher order derivatives are obtained by repeatedly taking the derivative of a function or its derivatives. The order of a derivative refers to how many times differentiation has been performed. To find a higher order derivative, one simply takes the derivative of the existing derivative. For example, to get the third derivative f'''(x) of a function f(x), one would take the derivative of the second derivative f''(x). Higher derivatives provide important information about the curvature and flexibility of a function at different points.
This document contains a presentation on polynomials. It defines what a polynomial is and discusses how the degree of a polynomial relates to the number of zeros. It provides an example of a cubic polynomial and discusses the relationship between the zeros and coefficients of a cubic polynomial. Specifically, it states that the sum of the zeros equals the negative of the coefficient of x^2, the sum of the products of the zeros equals the coefficient of x, and the product of the zeros equals the negative of the constant term. The document then provides two example questions - one asking to find the zeros of a quadratic polynomial and verify the relationship between the zeros and coefficients, and another asking to find a quadratic polynomial with given zeros.
This document discusses roots of polynomials. It defines polynomials and monomials, and explains that the roots of a polynomial are the values that make the polynomial equal to zero. It covers the fundamental theorem of algebra, which states that a polynomial of degree n has n roots. Descartes' rule for the number of positive roots is introduced. Properties of the roots and factors of polynomials are outlined. Examples of finding the roots of polynomials are provided at the end.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
The document defines and describes various types of number systems. It discusses natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. It also describes their properties and relationships. Different types of polynomials are defined based on their degree, number of terms, zeros, and factors. Methods for factorizing polynomials including taking common factors, grouping, and splitting the middle term are explained. Algebraic identities are also introduced.
5.7 Interactive Classroom Roots and Zeros.pptAARow1
The document provides examples and explanations of key concepts relating to polynomials, including:
- Determining the number and type of roots of a polynomial equation using factoring and the fundamental theorem of algebra.
- Using Descartes' Rule of Signs to determine the possible numbers of positive and negative real zeros of a polynomial function.
- Using synthetic substitution to find the real zeros of a polynomial function.
- Writing a polynomial function of least degree that has given complex zeros, using the fact that complex conjugates are also zeros.
1. The document discusses universal quantification and quantifiers. Universal quantification refers to statements that are true for all variables, while quantifiers are words like "some" or "all" that refer to quantities.
2. It explains that a universally quantified statement is of the form "For all x, P(x) is true" and is defined to be true if P(x) is true for every x, and false if P(x) is false for at least one x.
3. When the universe of discourse can be listed as x1, x2, etc., a universal statement is the same as the conjunction P(x1) and P(x2) and etc., because this
Fundamentals of AlgebraChu v. NguyenIntegral ExponentsDustiBuckner14
Fundamentals of Algebra
Chu v. Nguyen
Integral Exponents
Exponents
If n is a positive integer (a whole number, i.e., a number without decimal part) and x is a number, then
The number x is called the base and n is called the exponent.
The most common ways of referring to are “ x to the nth power,”
“ x to the nth,” or “the nth power of x.”
Integral Exponents (cont.)
For any non-zero number x and a positive integer n
and
Note: is not defined
and
Rules Concerning Integral Exponents
Following are five rules in which m and n are positive integers:
Rule 1: ; for example,
Rule 2: ; for example
or
Rules Concerning Integral Exponents (Cont.)
Rule 3: ; for example
or
Rule 4: ; for example
or
Rule 5: ; for example
or
Basic Rules for Operating with Fractions
Since dividing by zero is not defined, we assume that the denominator
is not zero.
Following are the eight basic rules for operating with fractions.
Rule 1: ; for example
Rule 2: ; for example
Rule 3: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 4: ; for example
Rule 5: ; for example
Rule 6: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 7: ; for example
Rule 8: ; for example
Notes: a*b +a*x may be expressed as a(b + x)
a*b + 1 may be written as a(b + ), and
m*x – y may be expressed as m(x - )
Square Root
Generally, for a>0 , there is exactly one positive number x such that
, we say that x is the root of a, written as
for
When n = 2, we say that x is the square root of “a” and is denoted by
or or
For example:
or
Practices
Carrying out the following operations:
24 ; 2-2 ; 2322, ; 252-5 ; and (2x3)5
; ; ; and
3.
4.
5.
n
m
n
m
x
x
x
+
=
n
x
1
0
)
2
(
2
2
2
=
=
=
-
+
-
x
x
x
x
1
0
=
x
0
0
n
n
n
x
x
x
x
=
=
+
0
0
n
n
x
x
1
=
-
3
2 ...
The pigeonhole principle states that if n objects are put into m containers where n > m, then at least one container must contain more than one object. It was first described by Dirichlet in 1834 and has various applications in mathematics and computer science. Some examples include proving that among a group of people, some pair will have the same birthday, and that collisions are inevitable in a hash table where there are more possible keys than indices. The principle can be used to solve problems involving divisibility, counting, and allocation of objects into groups.
I am Jayson L. I am a Mathematical Statistics Homework Expert at statisticshomeworkhelper.com. I hold a Master's in Statistics, from Liverpool, UK. I have been helping students with their homework for the past 5 years. I solve homework related to Mathematical Statistics.
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The document discusses the pigeonhole principle, which states that if n items are put into m containers where n > m, then at least one container must contain more than one item. It provides various formulations and applications of the principle in areas like data compression, hash tables, and the Chinese Remainder Theorem. The principle is believed to have first been stated by Dirichlet in 1834 and is also known as the drawer or shelf principle. Examples are given to illustrate how it can be used to solve problems related to birthdays, friend relationships, and geometry.
The document discusses the pigeonhole principle, which states that if n objects are put into m containers where n > m, then at least one container must contain more than one object. It provides various formulations and applications of the principle in areas like data compression, hash tables, and the Chinese Remainder Theorem. The history of the principle is traced back to Dirichlet, who described it as the "drawer principle" or "shelf principle" in 1834. Examples are given for problems involving birthdays, friend relationships, and geometry that can be solved using the pigeonhole principle.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Mel Anthony Pepito
This document provides an overview of the key points covered in a calculus lecture on derivatives of logarithmic and exponential functions:
1) It discusses the derivatives of exponential functions with any base, as well as the derivatives of logarithmic functions with any base.
2) It covers using the technique of logarithmic differentiation to find derivatives of functions involving products, quotients, and/or exponentials.
3) The document provides examples of finding derivatives of various logarithmic and exponential functions.
I am Jayson L. I am a Mathematical Statistics Homework Expert at excelhomeworkhelp.com. I hold a Master's in Statistics, from Liverpool, UK. I have been helping students with their homework for the past 5 years. I solve homework related to Mathematical Statistics.
Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call +1 678 648 4277 for any assistance with the Mathematical Statistics Homework.
Lesson 25: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document discusses indeterminate forms and L'Hopital's rule. It introduces indeterminate forms as limits that can have different values depending on the approach, such as 0/0 or infinity/infinity forms. It then presents L'Hopital's rule, which states that if the limit of the numerator and denominator of a quotient both approach 0, infinity, or negative infinity, the limit can be evaluated by taking the derivative of the numerator and denominator and rearranging terms. Examples are provided to demonstrate how L'Hopital's rule can be used to evaluate indeterminate forms. The document also provides biographical information about Guillaume de l'Hopital, after whom the rule is named.
* Evaluate a polynomial using the Remainder Theorem.
* Use the Factor Theorem to solve a polynomial equation.
* Use the Rational Zero Theorem to find rational zeros.
* Find zeros of a polynomial function.
* Use the Linear Factorization Theorem to find polynomials with given zeros.
* Use Descartes’ Rule of Signs.
The document discusses the upper and lower bound theorem for real roots of polynomial functions. The theorem states that if the remainder of synthetic division of a polynomial f(x) by x-b has only non-negative numbers, b is an upper bound for the real roots of f(x)=0. If the remainder alternates in sign when dividing f(x) by x-a, a is a lower bound. An example demonstrates finding the bounds between -3 and 2 for a 4th degree polynomial. The intermediate value theorem and fundamental theorem of algebra are also summarized.
I am Racheal W. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Massachusetts Institute of Technology, USA.
I have been helping students with their homework for the past 7 years. I solve assignments related to probability.
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You can also call on +1 678 648 4277 for any assistance with probability assignments.
This document provides an introduction to number theory, covering divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines key concepts such as division, factors, multiples, and the division algorithm. It presents theorems about divisibility and the relationship between greatest common divisors and least common multiples. Examples are provided to illustrate divisibility, the division algorithm, greatest common divisors, least common multiples, and modular arithmetic. The document serves as an overview of fundamental topics in number theory.
This document provides an introduction to number theory, covering divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines key concepts such as division, factors, multiples, and the division algorithm. It presents theorems about divisibility and the relationship between greatest common divisors and least common multiples. Examples are provided to illustrate divisibility, the division algorithm, greatest common divisors, least common multiples, and modular arithmetic. The document serves as an overview of fundamental topics in number theory.
Cricket management system ptoject report.pdfKamal Acharya
The aim of this project is to provide the complete information of the National and
International statistics. The information is available country wise and player wise. By
entering the data of eachmatch, we can get all type of reports instantly, which will be
useful to call back history of each player. Also the team performance in each match can
be obtained. We can get a report on number of matches, wins and lost.
Covid Management System Project Report.pdfKamal Acharya
CoVID-19 sprang up in Wuhan China in November 2019 and was declared a pandemic by the in January 2020 World Health Organization (WHO). Like the Spanish flu of 1918 that claimed millions of lives, the COVID-19 has caused the demise of thousands with China, Italy, Spain, USA and India having the highest statistics on infection and mortality rates. Regardless of existing sophisticated technologies and medical science, the spread has continued to surge high. With this COVID-19 Management System, organizations can respond virtually to the COVID-19 pandemic and protect, educate and care for citizens in the community in a quick and effective manner. This comprehensive solution not only helps in containing the virus but also proactively empowers both citizens and care providers to minimize the spread of the virus through targeted strategies and education.
Sri Guru Hargobind Ji - Bandi Chor Guru.pdfBalvir Singh
Sri Guru Hargobind Ji (19 June 1595 - 3 March 1644) is revered as the Sixth Nanak.
• On 25 May 1606 Guru Arjan nominated his son Sri Hargobind Ji as his successor. Shortly
afterwards, Guru Arjan was arrested, tortured and killed by order of the Mogul Emperor
Jahangir.
• Guru Hargobind's succession ceremony took place on 24 June 1606. He was barely
eleven years old when he became 6th Guru.
• As ordered by Guru Arjan Dev Ji, he put on two swords, one indicated his spiritual
authority (PIRI) and the other, his temporal authority (MIRI). He thus for the first time
initiated military tradition in the Sikh faith to resist religious persecution, protect
people’s freedom and independence to practice religion by choice. He transformed
Sikhs to be Saints and Soldier.
• He had a long tenure as Guru, lasting 37 years, 9 months and 3 days
Online train ticket booking system project.pdfKamal Acharya
Rail transport is one of the important modes of transport in India. Now a days we
see that there are railways that are present for the long as well as short distance
travelling which makes the life of the people easier. When compared to other
means of transport, a railway is the cheapest means of transport. The maintenance
of the railway database also plays a major role in the smooth running of this
system. The Online Train Ticket Management System will help in reserving the
tickets of the railways to travel from a particular source to the destination.
2. This Lecture
We will define what is a function formally, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications.
• Examples and definitions (injection, surjection, bijection)
• Pigeonhole principle and applications
3. Functions
function, f, from set A to set B
associates an element , with an element
:f A B→
( )f a B∈ .a A∈
The domain of f is A.
The codomain of f is B.
Definition: For every input there is exactly one output.
4. Functions
domain = R
codomain = R+
-{0}
domain = R+
-{0}
codomain = R
domain = R
codomain = [0,1]
domain = R+
codomain = R+
5. Functions
f(S) = |S|
f(string) = length(string)
f(student-name) = student-ID
f(x) = is-prime(x)
domain = the set of all sets
codomain = non-negative integers
domain = the set of all strings
codomain = non-negative integers
domain = positive integers
codomain = {T,F}
not a function,
since one input could have
more than one output
[two students may have same name]
6. ≤ 1 arrow in
A B
f( ) =
Injections (One-to-One)
is an injection iff no two inputs have the same output.:f A B→
, .
( ( ) ( ')) ( ')
′∀ ∈
= → =
a a A
f a f a a a |A| ≤ |B|
7. Surjections (Onto)
A B
1 arrow in
is a surjection iff every output is possible.:f A B→
f( ) =
. ( )b B a A f a b∀ ∈ ∃ ∈ = |A| ≥ |B|
8. Bijections
A B
is a bijection iff it is surjection and injection.:f A B→
f( ) =
exactly one arrow in
|A| = |B|
9. Function Domain Codomain Injective? Surjective
?
Bijective?
f(x)=sin(x) Real Real
f(x)=2x
Real Positive
real
f(x)=x2
Real Non-
negative
real
Reverse
string
Bit strings
of length n
Bit strings
of length n
Exercises
Whether a function is injective, surjective, bijective
depends on its domain and the codomain.
10. Inverse Sets
A B
Given an element y in B, the inverse set of y := f-1
(y) = {x in A | f(x) = y}.
11. Inverse Function
A B
f( ) =
exactly one arrow in
Informally, an inverse function f-1
is to “undo” the operation of function f.
There is an inverse function f-1
for f if and only if f is a bijection.
12. Inverse Function
A B
f( ) =
exactly one arrow in
Informally, an inverse function f-1
is to “undo” the operation of function f.
There is an inverse function f-1
for f if and only if f is a bijection.
13. Composition of Functions
Two functions f:X->Y’, g:Y->Z so that Y’ is a subset of Y,
then the composition of f and g is the function g 。 f: X->Z, where
g 。 f(x) = g(f(x)).
X
Y
Z
Y’f g
14. Function f Function g g 。 f
injective?
g 。 f
surjective?
g 。 f
bijective?
f:X->Y
f surjective
g:Y->Z
g injective
f:X->Y
f surjective
g:Y->Z
g surjective
f:X->Y
f injective
g:Y->Z
g surjective
f:X->Y
f bijective
g:Y->Z
g bijective
f:X->Y f-1
:Y->X
Exercises
15. This Lecture
• Examples and definitions (injection, surjection, bijection)
• Pigeonhole principle and applications
17. Pigeonhole Principle
then some hole must have at least two pigeons!
Pigeonhole principle
A function from a larger set to a smaller set cannot be injective.
(There must be at least two elements in the domain that have
the same image in the codomain.)
18. Example 1
Question: Let A = {1,2,3,4,5,6,7,8}
If five integers are selected from A,
must a pair of integers have a sum of 9?
Consider the pairs {1,8}, {2,7}, {3,6}, {4,5}.
The sum of each pair is equal to 9.
If we choose 5 numbers from this set,
then by the pigeonhole principle,
both elements of some pair will be chosen,
and their sum is equal to 9.
19. Example 2
Question: In a party of n people, is it always true that there are
two people shaking hands with the same number of people?
Everyone can shake hand with 0 to n-1 people, and there are n people,
and so it does not seem that it must be the case, but think about it carefully:
Case 1: if there is a person who does not shake hand with others,
then any person can shake hands with at most n-2 people,
and so everyone shakes hand with 0 to n-2 people,
0 to n-2 => n-1 possible values (i.e., cardinality of codomain = n-1)
There are n people (i.e., cardinality of domain = n)
so
the answer is “yes” by the pigeonhole principle.
20. Example 2
Question: In a party of n people, is it always true that there are
two people shaking hands with the same number of people?
Everyone can shake hand with 0 to n-1 people, and there are n people,
and so it does not seem that it must be the case, but think about it carefully:
Case 2: if everyone shakes hand with at least one person, then
any person shakes hand with 1 to n-1 people,
1 to n-1 => n-1 possible values (i.e., cardinality of codomain = n-1)
There are n people (i.e., cardinality of domain = n)
so
the answer is “yes” by the pigeonhole principle.
21. Birthday Paradox
In a group of 367 people, there must be two people having the same birthday.
Suppose n <= 365, what is the probability that in a random set of n people,
some pair of them will have the same birthday?
We can think of it as picking n random numbers from 1 to 365 without repetition.
There are 365n
ways of picking n numbers from 1 to 365.
[You have 365 choices to pick the first one, same for the second and so on…]
There are 365·364·363·…·(365-n+1) ways of picking n numbers from 1 to 365
without repetition.
[You have 365 choices to pick the first one, 364 for the second and so on…]
So the probability that no pairs have the same birthday is
equal to 365·364·363·…·(365-n+1) / 365n
This is smaller than 50% for 23 people, smaller than 1% for 57 people.
22. Generalized Pigeonhole Principle
If n pigeons and h holes,
then some hole has at least
n
h
pigeons.
Generalized Pigeonhole Principle
Cannot have < 3 cards in every hole.
♠ ♥ ♣ ♦
24. Subset Sum
90 numbers, each with at most 25 digits.
So the total sum is at most 90x1025
Let A be the set of all subsets of the 90 numbers.
Let B be the set of integers from 0 to 90x1025
.
(pigeons)
(pigeonholes)
By pigeonhole principle, there are two different subsets with the same sum.
25. Club vs Strangers
Theorem: Every collection of 6 people includes a club of 3 people,
or a group of 3 strangers.
Let’s agree that given any two people, either they have met or not.
If every people in a group has met, then we’ll call the group a club.
If every people in a group has not met, then we’ll call a group of strangers.
Let x be one of the six people.
By the (generalized) pigeonhole principle, we have the following claim.
Claim: Among the remaining 5 people, either 3 of them have met x,
or 3 of them have not met x.
26. Club vs Strangers
Theorem: Every collection of 6 people includes a club of 3 people,
or a group of 3 strangers.
Claim: Among the remaining 5 people, either 3 of them have met x,
or 3 of them have not met x.
Case 1: “3 people have met x”
Case 1.1: No pair among those people met each other.
Then there is a group of 3 strangers.
OK!
Case 1.2: Some pair among those people have met each other.
Then that pair, together with x, form a club of 3 people.
OK!
27. Club vs Strangers
Theorem: Every collection of 6 people includes a club of 3 people,
or a group of 3 strangers.
Claim: Among the remaining 5 people, either 3 of them have met x,
or 3 of them have not met x.
Case 2: “3 people have not met x”
Case 2.1: Every pair among those people met each other.
Then there is a club of 3 people.
OK!
Case 2.2: Some pair among those people have not met each other.
Then that pair, together with x, form a group of 3 strangers.
OK!
28. Quick Summary
Make sure you understand basic definitions of functions.
These will be used in the next lecture for counting.
The pigeonhole principle is very simple,
but there are many clever uses of it to prove non-trivial results.
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