Soal limit
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The document contains solutions to problems determining limits of functions. Solution 14 provides the graph of a function and determines the limits as x approaches various values. Solution 15 determines the values of constants a and b that make the limit of a function equal to 3 as x approaches 2 by setting up and solving a system of equations for the one-sided limits.
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The document discusses linear programming problems. It defines linear programming as finding non-negative values of variables to satisfy linear constraints and optimize a linear objective function. It provides examples of transportation problems and diet problems formulated as linear programs. It describes graphical and algebraic methods for solving linear programs, including introducing slack and surplus variables to transform inequalities to equations.
Calculus Homework Help
I am Piers L. I am a Calculus Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From, the University of Adelaide. I have been helping students with their homework for the past 7 years. I solve homework related to Calculus.
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I am Leonard K. I am a Differential Equations Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From California, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Differential Equations.
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Operation research bba15
This document presents an overview of linear programming and the graphical method for solving linear programming problems. It defines linear programming as a technique for optimal allocation of resources to obtain a particular objective. It discusses some common applications of linear programming like product mix problems and transportation problems. It then explains the graphical method for solving linear programming problems with two variables graphically. Using an example problem of maximizing an objective function subject to three constraints, it demonstrates how to plot the constraints on a graph and identify the optimal solution in the feasible region. The optimal solution is identified as the point that gives the maximum value of the objective function.
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Module code 124ms Tajvinder Virdee Sets and Log.docx
Module code: 124ms
Tajvinder Virdee
Sets and Logic Coursework 1
1. Find the truth value of P (Q R) if P is false, Q is true, and R is false. What is the truth
value of this expression if the brackets are removed?
Substituting in the truth values, we obtain
f (t f)
As there is a set of brackets, the brackets have to be done first as they have a higher order of
precedence. So (Q R), becomes (t f) which we can say is false, as if Q is true and R is
false the truth value of this implication is false. This then becomes
f f
which is equivalent to
f
If the brackets are removed, then the order of the procedure changes.
The expression now becomes
P Q R, where P is false, Q is true, and R is false.
Inserting the truth values of P, Q and R, we get
f t f
Since has higher priority than , we have to work out the f t first. This gives us the
value false as not both parts are true.
f f
This gives t
As false implies false, the truth value is true, meaning that the expression has truth value
true.
Both answers are different as once the brackets were removed the order of the procedure
changes, resulting in the steps taking to complete the expression to be different.
2. Consider the proposition
((C ( C S)) (S P)) S P.
(a) Interpreting these symbols as the atomic sentences
C : I am a careful programmer.
S : My program will meet its specification.
P : My program will pass the validation test.
Interpret the above proposition as an argument in natural English.
I started by drawing up a table, in which I explained the possible connectives and translated
them into natural English.
I then started to convert the proposition in English.
This is as can only be true if both parts are true,
so as both parts are false, the expression has the
truth value f (false).
Connectives (symbols) Natural English
“AND”
“OR”
“IMPLIES”, so if this ... then....
“IF AND ONLY IF”
“NOT”, so the negative of a sentence.
Module code: 124ms
Tajvinder Virdee
I am a careful programmer and if I am not a careful programmer, then my program will not
meet its specification, and if my program will meet its specification then my program will
pass the validation test. This implies that my program will meet its specification and then my
program will pass the validation test.
(b) By means of a truth table, find out whether the argument is valid.
I started by calculating how many rows my truth table would need. As there are only two
truth values, true and false, I would have to have the value 2 to the power n, where n would
be the amount of sentence letters, in the proposition.
In this case as there are 3 types of sentence letters, C, S and P, n would be 3.
So ...
Introduction to the Theory of Computation, Winter 2003 A. Hevia and J. Mao S...
Introduction to the Theory of Computation, Winter 2003 A. Hevia and J. Mao Solution to Problem Set 1
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Act_Fin_IAMT
The document discusses several topics related to calculus including:
1. How to graph functions using GeoGebra by inputting the function and seeing the outputted graph and table of values.
2. How to find the derivative of composite functions using formulas for derivatives of sums, differences, products, and quotients of functions.
3. The definitions of even and odd functions and how to determine if a given function is even or odd based on its behavior when the input is negated.
4. How to evaluate limit problems by directly substituting the limit value or using trigonometric limits such as lim x->0 sin(x)/x = 1.
OI.ppt
This document provides an introduction to optimization problems. It discusses key concepts like convex and non-convex functions, global vs local optima, and solving convex programs graphically and analytically. Specifically, it shows how to represent an optimization problem with an objective function and constraints, defines convexity, and explains that convex programs can be solved to find the global optimum while non-convex programs may only find a local optimum. It also demonstrates solving a simple convex program graphically by finding the point where the objective function contour is tangent to the constraint, and analytically by setting the gradients of the objective and constraints equal using Lagrange multipliers.
Calculus- Basics
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K to 12 - Grade 8 Math Learners Module Quarter 2
Here are the completed statements based on the conclusions:
1. n(A × B) = n(B × A).
2. A × B ≠ B × A.
The key conclusions are:
1. The cardinalities of the Cartesian products A × B and B × A are equal, since n(A × B) = n(B × A).
2. However, the sets A × B and B × A are not equal, since the ordered pairs will be arranged differently, so A × B ≠ B × A.
MATH MODULE GRADE 8
The document introduces the topic of relations and functions and outlines 3 lessons that will be covered - rectangular coordinate system, representations of relations and functions, and linear functions and applications. It provides learning objectives for each lesson and examples of how concepts like slope, intercepts, and graphs will be explored. A pre-assessment with 20 multiple choice questions is also included to gauge students' prior knowledge.
MATH MODULE GRADE 8
The document introduces the key concepts that will be covered in a module on relations and functions, including the rectangular coordinate system, representations of relations and functions, and linear functions and their applications. It outlines 3 lessons that will examine how to predict the value of a quantity given the rate of change, and provides sample problems to assess students' prior knowledge on these topics before beginning the lessons.
5HBC: How to Graph Implicit Relations Intro Packet!
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2. Using parametric mode and the quadratic formula to solve the implicit function for x as a parametric function of t.
3. Using function mode, solving for x as a function of y, and using DrawInv to graph the inverse relation.
4. Using function mode and the Solve() command to numerically solve the implicit equation for y as a function of x.
5. Using polar mode by rewriting the implicit equation in terms of r and θ and graphing r
B_W3 Quiz #1Graphing Exponential Growth & Decay (15 questio.docx
B_W3: Quiz #1
Graphing Exponential Growth & Decay (15 questions)
1.
Which of the following is an example of an exponential function?
A.
f(x) = 4
B.
f(x) = 4x
C.
D.
f(x) = 6x
2.
Which of the following is an exponential decay function?
A.
B.
C.
D.
3.
The graph of has _______.
A.
an asymptote at x = 0.
B.
no asymptotes.
C.
an asymptote at y = 0.
D.
an asymptote at y = x.
4.
The graph of shifts the graph of _______ .
A.
left 1 unit
B.
right 1 unit
C.
up 1 unit
D.
down 1 unit
5.
Which exponential function matches the graph?
A.
B.
C.
D.
6.
Which is the graph of ?
A.
B.
C.
D.
7.
Which is the graph of ?
A.
B.
C.
D.
8.
Which exponential function matches the graph?
A.
B.
C.
D.
9.
What is the equation of the following graph?
A.
f(x) = 2x + 2
B.
f(x) = 2x - 2
C.
f(x) = 2x - 2
D.
f(x) = 2x + 2
10.
What is the equation of the following graph?
A.
f(x) = 2x + 7
B.
f(x) = 2x - 7
C.
f(x) = 2x - 7
D.
f(x) = 2x + 7
11.
What is the transformation that occurs to the equation y = 2x if it changes to y = 2x + 5?
A.
The graph moves 5 units to the left.
B.
The graph moves 5 units to the right.
C.
The graph moves 5 units down.
D.
The graph moves 5 units up.
12.
What is the transformation that occurs to the equation y = 2x if it changes to y = 2x-3?
A.
There is a transformation of 3 units to the right.
B.
There is a transformation of 3 units to the left.
C.
There is a transformation of 3 units up.
D.
There is a transformation of 3 units down.
13.
What is the transformation that occurs to the parent function if the equation changes to ?
A.
The graph moves 1 unit up.
B.
The graph reflects across the y-axis.
C.
The graph reflects across the x-axis.
D.
The graph moves 1 unit down.
14.
What is the transformation that occurs to the equation y = 2x if the equation changes to y = 2x-3 + 4?
A.
The transformation moves the graph 3 units to the right and 4 units down.
B.
The transformation moves the graph 3 units to the left and 4 units up.
C.
The transformation moves the graph 3 units to the left and 4 units down.
D.
The transformation moves the graph 3 units to the right and 4 units up.
15.
What is the transformation that occurs to the equation y = 2x if it changes to y = -2x – 2?
A.
The graph is reflected across the y-axis and moves 2 units up.
B.
The graph is reflected across the x-axis and moves 2 units up.
C.
The graph is reflected across the x-axis and moves 2 units down.
D.
The graph is reflected across the y-axis and moves 2 units down.
B_W3 Quiz #2
Solving Exponential Equations (10 questions)
1. Determine the value for the "?" 4?=64
A.
16
B.
3
C.
4
D.
12
2. Solve 3x+1 = 37
A.
x = 3
B.
x = 7
C.
x = 6
D.
x = 1
3. Solve:
A.
x = -4
B.
x = -2
C.
x = -5
D.
x = -10
4. What is the next step in t ...
Linear programming.pdf
1. The document provides information about linear programming including definitions of key terms, steps to solve linear programming problems, examples worked out in detail, and exercises.
2. Linear programming involves optimizing (maximizing or minimizing) an objective function subject to certain constraints. It was first introduced by a Russian mathematician in the 1930s-1940s to optimize resources like manpower and materials during war time.
3. Examples worked out in the document show how to set up the constraints and objective function mathematically based on word problems, sketch the feasible region, find the corner points, and determine the optimal solution that maximizes or minimizes the objective function.
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Soal limit
- 1. SOLUTION 14 : Consider the function i.) The graph of f is given below. ii.) Determine the following limits. a.) . b.) . c.) We have that does not exist since does not equal . d.) . e.) . f.) We have that since .
- 2. g.) We have that (The numerator is always -1 and the denominator is always a positive number approaching 0.) , so the limit does not exist. h.) . i.) We have that does not exist since does not equal . j.) . k.) . l.) . Click HERE to return to the list of problems. SOLUTION 15 : Consider the function Determine the values of constants a and b so that exists. Begin by computing one-sided limits at x=2 and setting each equal to 3. Thus, and . Now solve the system of equations a+2b = 3 and b-4a = 3 . Thus, a = 3-2b so that b-4(3-2b) = 3 iff b-12+ 8b = 3 iff 9b = 15
- 3. iff . Then . Click HERE to return to the list of problems.