Chapter-4: More on Direct Proof and Proof by Contrapositivenszakir
Proofs Involving Divisibility of Integers, Proofs Involving Congruence of Integers, Proofs Involving Real Numbers, Proofs Involving sets, Fundamental Properties of Set Operations, Proofs Involving Cartesian Products of Sets
Chapter-4: More on Direct Proof and Proof by Contrapositivenszakir
Proofs Involving Divisibility of Integers, Proofs Involving Congruence of Integers, Proofs Involving Real Numbers, Proofs Involving sets, Fundamental Properties of Set Operations, Proofs Involving Cartesian Products of Sets
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Problem Solving in Mathematics EducationJeff Suzuki
A major focus on current mathematics education is "problem solving." But "problem solving" means something very different from "Doing the exercises at the end of the chapter." An explanation of what problem solving is, and how it can be implemented.
* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Problem Solving in Mathematics EducationJeff Suzuki
A major focus on current mathematics education is "problem solving." But "problem solving" means something very different from "Doing the exercises at the end of the chapter." An explanation of what problem solving is, and how it can be implemented.
* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Solve equations in one variable algebraically.
* Solve a rational equation.
* Find a linear equation.
* Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
* Write the equation of a line parallel or perpendicular to a given line.
* Use like bases to solve exponential equations.
* Use logarithms to solve exponential equations.
* Use the definition of a logarithm to solve logarithmic equations.
* Use the one-to-one property of logarithms to solve logarithmic equations.
* Solve applied problems involving exponential and logarithmic equations.
* Solve quadratic equations by factoring.
* Solve quadratic equations by the square root property.
* Solve quadratic equations by completing the square.
* Solve quadratic equations by using the quadratic formula.
* Factor the greatest common factor of a polynomial.
* Factor a trinomial.
* Factor by grouping.
* Factor a perfect square trinomial.
* Factor a difference of squares.
* Factor the sum and difference of cubes.
* Factor expressions using fractional or negative exponents.
* Evaluate square roots.
* Use the product rule to simplify square roots.
* Use the quotient rule to simplify square roots.
* Add and subtract square roots.
* Rationalize denominators.
* Use rational roots.
* Solve a system of nonlinear equations using substitution.
* Solve a system of nonlinear equations using elimination.
* Graph a nonlinear inequality.
* Graph a system of nonlinear inequalities.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
1. 1.6 Rational and Radical
Equations
Chapter 1 Equations and Inequalities
2. Concepts and Objectives
⚫ Solve equations consisting of rational expressions
⚫ Solve equations with radicals and check the solutions
⚫ Solve equations that are quadratic in form
⚫ Solve work-rate problems
3. Rational Equations
⚫ A rational equation is an equation that has a rational
expression for one or more terms.
⚫ To solve a rational equation, multiply both sides by the
lowest common denominator of the terms of the
equation. Be sure to check your solution against the
undefined values!
Because a rational expression is not defined when its
denominator is 0, any value of the variable which makes
the denominator’s value 0 cannot be a solution.
5. Rational Equations (cont.)
⚫ Example: Solve
The lowest common denominator is , which is
equal to 0 if x = ‒1. Write this as .
−
+ =
+
2 3 5
2 1
x x
x
x
( )+2 1x
−1x
( ) ( ) ( )( )+ +
−
+ = +
+
2 1 2 1
2 3
2
5
1
2 1
x x
x
x x x x
6. Rational Equations (cont.)
⚫ Example: Solve
The lowest common denominator is , which is
equal to 0 if x = ‒1. Write this as .
−
+ =
+
2 3 5
2 1
x x
x
x
( )+2 1x
−1x
( ) ( ) ( )( )+ +
−
+ = +
+
2 1 2 1
2 3
2
5
1
2 1
x x
x
x x x x
( )( ) ( ) ( )( )+ − + = +1 2 3 2 5 2 1x x x x x
− − + = +2 2
2 3 10 2 2x x x x x
=7 3x
=
3
7
x
Since this is not ‒1, this is a
valid solution.
3
7
8. Rational Equations (cont.)
⚫ Example: Solve
The LCD is which is equal to . If x is
either 3 or ‒3, the denominator will be 0, so .
The only value of x which will satisfy the equation is 3,
but that is a restricted value, so the solution is .
− −
+ =
− + −2
2 3 12
3 3 9x x x
( )( )+ −3 3x x −2
9x
3x
( ) ( )− + + − = −2 3 3 3 12x x
− − + − = −2 6 3 9 12x x
− = −15 12x
=3x
10. Rational Equations (cont.)
⚫ Example: Solve
The LCD is xx ‒ 2, which means x 0, 2.
2
3 2 1 2
2 2
x
x x x x
+ −
+ =
− −
( ) ( ) ( )
( )2
2 2
3 2 1 2
2
2
x
x x x x
xx x x x x
+ −
+ = −
− − −
−
11. Rational Equations (cont.)
⚫ Example: Solve
The LCD is xx ‒ 2, which means x 0, 2.
2
3 2 1 2
2 2
x
x x x x
+ −
+ =
− −
( ) ( ) ( )
( )
( ) ( )
( )
2
2
3 2 1 2
2 2
3 2 2 2
3 2 2 2
2
3 3 0
3 1 0
2 2
0, 1
x
x
x x x x
x x x
x
xx x x x
x x
x x
x x
x
+ −
+ = − −
+ + − = −
+ + − = −
+ =
+ =
=
− −
−
−
1−
12. Power Property
⚫ Note: This does not mean that every solution of Pn = Qn
is a solution of P = Q.
⚫ We use the power property to transform an equation
that is difficult to solve into one that can be solved more
easily. Whenever we change an equation, however, it is
essential to check all possible solutions in the original
equation.
If P and Q are algebraic expressions, then every
solution of the equation P = Q is also a solution of
the equation Pn = Qn, for any positive integer n.
13. Solving Radical Equations
⚫ Step 1 Isolate the radical on one side of the equation.
⚫ Step 2 Raise each side of the equation to a power that is
the same as the index of the radical to eliminate the
radical.
⚫ If the equation still contains a radical, repeat steps 1
and 2.
⚫ Step 3 Solve the resulting equation.
⚫ Step 4 Check each proposed solution in the original
equation.
18. Solving Radical Equations (cont.)
⚫ Example: Solve + − + =3 1 4 1x x
( )
2
4 1x + +
( )
2
2
3 1 4 1
3 1 4 2 4 1
2 4 2 4
2 4
4 4 4
5 0
5 0
0, 5
x x
x x x
x x
x x
x x x
x x
x x
x
+ = + +
+ = + + + +
− = +
− = +
− + = +
− =
− =
=
19. Solving Radical Equations (cont.)
⚫ Example: Solve
Check:
Solution: {5}
+ − + =3 1 4 1x x
( )
2
2
3 1 4 1
3 1 4 2 4 1
2 4 2 4
2 4
4 4 4
5 0
5 0
0, 5
x x
x x x
x x
x x
x x x
x x
x x
x
+ = + +
+ = + + + +
− = +
− = +
− + = +
− =
− =
=
( )+ − + =3 0 1 0 4 1
− =1 4 1
− =1 2 1
− 1 1
( )+ − + =3 5 1 5 4 1
− =16 9 1
− =4 3 1
=1 1
20. Quadratic in Form
⚫ An equation is said to be quadratic in form if it can be
written as
where a 0 and u is some algebraic expression.
⚫ To solve this type of equation, substitute u for the
algebraic expression, solve the quadratic expression for
u, and then set it equal to the algebraic expression and
solve for x. Because we are transforming the equation,
you will still need to check any proposed solutions against
the original equation.
+ + =2
0au bu c
21. Quadratic in Form (cont.)
⚫ Example: Solve ( ) ( )− + − − =
2 3 1 3
1 1 12 0x x
22. Quadratic in Form (cont.)
⚫ Example: Solve
Let This makes our equation:
( ) ( )− + − − =
2 3 1 3
1 1 12 0x x
( )= −
1 3
1u x
+ − =2
12 0u u
( )( )+ − =4 3 0u u
= −4, 3u
23. Quadratic in Form (cont.)
⚫ Example: Solve
Let . This makes our equation:
So, and
( ) ( )− + − − =
2 3 1 3
1 1 12 0x x
( )= −
1 3
1u x
+ − =2
12 0u u
( )( )+ − =4 3 0u u
= −4, 3u
( )
( ) ( )
1 3
31/3 3
1 4
1 4
1 64
63
x
x
x
x
− = −
− = −
− = −
= −
( )
( ) ( )
1 3
31/3 3
1 3
1 3
1 27
28
x
x
x
x
− =
− =
− =
=
26. Work Rate Problems
⚫ If a job can be done in t units of time, then the rate of
work is of the job per time unit. Therefore,
⚫ If the letters r, t, and A represent the rate at which work
is done, the time, and the amount of work accomplished,
respectively, then
1
t
portion of the job completed = rate time
A rt=
27. Work Rate Problems (cont.)
⚫ Amounts of work are often measured in terms of the
number of jobs accomplished. For instance, if one job is
accomplished in t time units, then A = 1 and
1
r
t
=
28. Work Rate Problems (cont.)
⚫ Example: Lisa and Keith are raking the leaves in their
backyard. Working alone, Lisa can rake the leaves in 5
hr, while Keith can rake them in 4 hr. How long would it
take them to rake the leaves working together?
29. Work Rate Problems (cont.)
⚫ Example: Lisa and Keith are raking the leaves in their
backyard. Working alone, Lisa can rake the leaves in 5
hr, while Keith can rake them in 4 hr. How long would it
take them to rake the leaves working together?
Lisa’s rate: Keith’s rate:
x is the time it takes for both of them to rake the leaves
1 yard
5 hr
1 yard
4 hr
1 1
1
5 4
x x+ =
30. Work Rate Problems
⚫ Example, cont.
1 1
1
5 4
x x+ =
Lisa’s
rate
Keith’s
rate
( )
1 1
20 20 20 1
5 4
x x
+ =
4 5 20x x+ =
9 20x =
20
hr
9
x =
common
denominator