Modeling with quadratics
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The document discusses modeling quadratic functions to solve problems involving the product of consecutive integers. It provides an example of finding the smallest pair of consecutive even integers whose product is 48. It shows setting up the quadratic equation x(x+2)=48, factoring to get (x+8)(x-6)=0, and solving to find the integers are -6 and -8, which is the smallest pair.
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An extended Roman domination function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the conditions that (i) every vertex u for which f(u) is either 0 or 1 is adjacent to at least one vertex v for which f(v) =3. (ii) if u and v are two adjacent vertices and if f(u)=0 then f(v)≠0, similarly if f(u)=1 then f(v)≠1. The weight of an extended Roman domination function is the value f(V)=∑_(u∈V)▒〖f(u).〗 The minimum weight of an extended Roman domination function on graph G is called the extended Roman domination number of G, denoted by γ_(R_ϵ ) (G).The Hexagonal networks are popular mesh-derived parallel architectures. In this paper we present an upper bound for the extended Roman domination number of hexagonal networks.
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7. z = 1 − y2
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10. x2 + z2 = 16
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⟨
0, 1
2
,
√
3
2
⟩
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2,
√
3
2
, 1
2
⟩
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√
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4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of
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√
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√
3 , 1). Let θ be the
angle between L and the vector u⃗ = 1√
2
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This document presents research on the application of vector spaces and subspaces in information technology careers. It provides theoretical concepts and practical examples of vector spaces and subspaces to apply in solving specific problems in information technology and communication engineering. Examples include using the Wronskian determinant to determine if sets of functions are linearly independent or dependent. The research concludes that using multimedia applications can help illustrate geometric vector space concepts in 2D and 3D, aiding understanding of algebra linear topics. Vector spaces and subspaces are increasingly important across many fields of study and engineering.Factorising Quadratics
The document provides instructions on factorizing quadratic equations. It begins by explaining what quadratic equations are and provides examples. It then discusses factorizing quadratics where the coefficient of x^2 is 1 by finding two numbers whose product is the last term and sum is the middle term. The document continues explaining how to factorize when the coefficient of x^2 is not 1 and predicts the signs of the factors based on the signs of the terms in the quadratic equation. It provides examples of factorizing different quadratic equations.
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Dynamic programming can be used to solve optimization problems involving overlapping subproblems, such as finding the most valuable subset of items that fit in a knapsack. The knapsack problem is solved by considering all possible subsets incrementally, storing the optimal values in a table. Warshall's and Floyd's algorithms also use dynamic programming to find the transitive closure and shortest paths in graphs by iteratively building up the solution from smaller subsets. Optimal binary search trees can also be constructed using dynamic programming by considering optimal substructures.
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Calculus I
Directions: (10 pts. each) Answer each of the following questions below. In order to receive ANY credit for a question, you must SHOW YOUR WORK using proper notation and clear and concise logic. You're graded on both the accuracy of your answers AND your explanations that sufficiently support your answers. Unless otherwise stated, you're to give the EXAXCT VALUES of answers instead of decimal approximations. In order to receive ANY credit for any applied/word problem (i.e. Problems #29 - ), you MUST declare a variable (unless the variable(s) have already been declared in the problem) and set up and solve an appropriate mathematical expression that can be used to answer the question. Proper units must also be included in answers to applied problems. NO CREDIT WILL BE GIVEN FOR EITHER GUESSING OR CHECKING POSSIBLE ANSWERS WITHOUT SOLVING THE PROBLEM. YOU CANNOT USE CALCULUS TO SOLVE THESE PROBLEMS.
Finally, write ONLY FINAL ANSWERS ON THESE PAGES; you must show your work both according to homework guidelines and on YOUR OWN PAPER.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Multiply or divide as indicated. Write your answer in factored form.
1) x22 - 9x + 14 · xx22 -- 1618x x ++ 4877 1)
2)
x
-
12
x
+
32
Simplify the complex rational expression.
4
x
2
-
4
x
-
32
-
1
x
-
8
2)
1 + 1 x + 4
Find the difference quotient for the function and simplify it.
3) g(x) = 6x2 + 14x - 1 3)
Find the domain and range of the function. Write your answers using interval notation.
4)
g(z)
=
16
-
z
2
4)
Find a formula for the function graphed.
5) 5)
Determine if the function is even, odd, or neither. You must use algebra to justify your answer; otherwise, no full credit will be given. NO CREDIT is given for an answer without a mathematical explanation.
6) f(x) = x -+7 9 6)
State the domain of the composition.
7)
(
g
H
h)(x) with g(x)
=
x
+
5
and h(x)
=
8
x
+
7
7)
Compute
f(x
+
h)
-
f(x)
h
(h
J
0) for the given function
.
8) f(x) = 4x - 8 8)
9)
f(x)
=
5
x
2
+
6
x
9)
10)
f(x)
=
1
9
x
10)
Solve the equation by multiplying both sides by the LCD.
11) 32x - x 3+ 1 = 1 11)
12)
Solve the equation.
x
+
6
+
2
-
x
=
4
12)
13)
(
4
x
-
2
)
/
3
2
+
6
=
15
13)
14)
3
x
+
4
=
x
-
1
14)
Find the real solutions of the equation by factoring.
15) x3 + 8x2 - x - 8 = 0 15)
Solve the equation by making an appropriate substitution.
16) (x2 - 2x)2 - 11(x2 - 2x) + 24 = 0 16)
Solve the logarithmic equation.
17) log2(x + 7) + log2(x - 7) = 2 17)
Solve the exponential equation. Express the solution set in terms of natural logarithms.
18) 4x + 4 = 52x + 5 18)
Solve the inequality and express the solution in interval notation.
19) 7Ax - 1A L 2 19)
Solve the inequality. Write your answer using interval notation.
20) x 18- 5 > x 15+ 1 20)
Write the equation as f(x) = a(x - h)2 + k. Identify the vertex, range, and axis of symmetry of the function.
21) f(x) = x2 + 5x + 2 21)
23) log
F.
Laws of Exponent
1. The document discusses laws of exponents, including: an = a × a × ... × a (n times); (an)m = an×m; and an × am = a(n+m).
2. It provides examples of applying these laws, such as evaluating expressions like (23)2, (x4)3, and (a2)5.
3. The key conclusions are that for powers of powers, you multiply the exponents, for products of powers you add the exponents, and for quotients of powers you subtract the exponents.
Algebra Notes Factor and Simplify Quadratics
This document provides instructions and examples for solving "riddles" involving factoring quadratic expressions. It explains that the riddles involve two numbers whose product and sum are given. There are four possible combinations of signs for the product and sum. Examples are worked through step-by-step, showing how to determine the signs of the two numbers based on the information given, find the factors of the product, and write the factored quadratic expression that represents the solution. The goal is to help students recognize different patterns that arise when factoring quadratics and factor expressions into standard form.
02 Notes Divide and Conquer
These are the notes for the slides 02 Divide and Conquer of the class of analysis of algorithms. Take a look to the slides to complete all the ideas.
Vectorise all the things - long version.pptx
Does your code work beautifully on a few dozen examples, but leave you waiting for hours when looping through all your data? Do you wish there was a way to speed things up without subjecting yourself to learning C?
In this talk, you’ll learn some simple tricks, borrowed from linear algebra and implemented in NumPy, which deliver significant performance gains in Python code. Turbo boost your code without losing readability or grappling with complex coding concepts.
Q3- G4 WEEK 5 ACTIVITY power point presentation
Rafael is tasked with finding the missing numbers in several equations. He has been trying for almost an hour but cannot find the missing numbers. The document provides 10 equations with missing numbers for Rafael to solve. It also provides information about different types of heat transfer including conduction, convection and radiation.
10.2 using combinations and the binomial theorem
The document discusses combinations and the binomial theorem. It provides formulas for combinations and explains how to use Pascal's triangle to determine the coefficients in binomial expansions. Examples are worked out expanding (2x + 1)^4, (x - 2y)^3, and finding the coefficient of x^4 in (2x - 7)^9. The key points are that combinations involve choosing objects without regard to order, while permutations consider order; Pascal's triangle determines the coefficients in binomial expansions; and the powers of the terms follow a pattern of the first term decreasing and the second increasing.
Quadratic eq cth
This document discusses solving quadratic equations through various methods:
1) Factoring quadratic equations by using the zero-product principle and setting each factor equal to zero.
2) Using the square root property to solve equations of the form u^2 = d by taking the square root of both sides.
3) Completing the square to transform quadratic equations into perfect square trinomials that can be factorized.
4) Applying the quadratic formula to solve any quadratic equation in the form ax^2 + bx + c = 0.
Module 3 topic 2 notes
This document discusses solving multi-step linear equations in 3 sentences or less:
The document provides examples and steps for solving multi-step linear equations, which may require using the distributive property, combining like terms, or having variables on both sides of the equation. It also discusses setting up and solving equations related to consecutive integers and word problems involving linear equations with more than one step.
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- 1. Modeling with Quadratics The product of consecutive integers.
- 2. Quadratics have many appearances! 1. A rocket is fired into the air with an initial velocity of ____. The height of the rocket may be found with ____. 2. The length of a rectangle is two more than twice the width. If the area of the rectangle is ____, what is the length? 3. The product of two consecutive odd integers is ___. Find the larger of the integers.
- 3. Here is one of my favorite types of questions The product of two consecutive even integers is 48. Find the smallest integer pair.
- 4. First, write the function Consecutive even means one after the other, like 2 & 4. So let x = first integer and x + 2 = second integer... Then, x(x + 2) = 48
- 5. Simplify and solve by factoring and setting factors = 0 x(x + 2) = 48 x^2 + 2x - 48 = 0 (x + 8)(x - 6) = 0 X + 8 = 0 or x - 6 = 0 X = -8 0r x = 6 X + 2 = -6 or x + 2 = 8
- 6. Analyze answer and place in context The product of two consecutive even integers is 48. Find the smallest integer pair. Since -6 and -8 are smaller than 6 and 8, that is the solution to this problem!