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Computing Values of Functions

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly out output.  An example of a commonly used function that related each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x). The input variables are often referred to as the arguments of the function.

Functions are very important in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions are defined by a formula or an algorithm that tells how to compute the output for a given input. Some functions are given by a graph of the function. A function can also be described implicitly or as a solution of a differential equation.

The input and output of a function can be expressed as an ordered pair. These pairs are ordered so that the first element is the input and the second is the output. If the input and outputs are real numbers, this ordered pair is viewed as the Cartesian coordinates of a point on a graph of the function.

A function is defined by its set of inputs. These inputs are called the domain. A set containing the outputs is called the codomain or range. The set of all paired inputs and outputs are called the functions graph.

A function f with domain X and codomain Y is commonly denoted by

F: X→Y

The elements of X are called arguments of f. For each argument X there is a corresponding unique Y in the codomain called the function value at x. 

Euler and Modified Euler's method

I need help using Euler an the improved Euler methods: (look at attachment for better formula display) 1. Consider the initial value problem y' = 2xy, y(1) = 1. Use the Euler's method and improved Euler's method with h = 0.1and h = 0.05 to obtain approximate values of the solution at x = 1.5. At each step compare the approxim

Functions, Limits & Continuity

1. Find the doubling time of an investment earning 8% interest if interest is compounded continuously. 2. Find the limit of f(x) = (7x+7)/(4x+4) as x approaching positive infinity and negative infinity. 3. Find the slope of the function's graph at the given point, then find an equation for the line tangent to the graph th

Stability and Lyapunov functions

Following is the problem that I solved the first part: Consider the system x' = f(x), where f: R^2 into R^2, is defined by: f(x) = [ (x1)^3 + (x1) (x2)^2 ] [ (x1)^2 (x2) + (x2)^3 ] a- Find all equilibrium points of the system. b- Use an appropriate Lyapunov function to determine the stability of the equilibr

Statistics Variable Identified Populations

A campaign conducts a poll of 1500 registered voters, and finds that 54% intend to vote for candidate A, 32% intend to vote for candidate B (their candidate), and 14% are undecided. Undecided voters, and those intending to support candidate A, are asked what single issue might cause them to support candidate B. The topic of ca

Z x Z Functions and Range

See the attached file. Consider the set of integers Z. The set Z x Z consists of all ordered pairs of integers. In symbols, Z x Z={(x,y):x,y∈Z} For example, (2,-5) , (0,0) , (-127,10) , and (-5,2) are all distinct elements of Z x Z. Notice that the order matters; (2,-5) and (-5,2) are different elements of Z x Z.

Time projections and tables

Three individual steps to be taken before the service could begin: (1) write instructions and procedures, (2) select techniques to operate the equipment, and (3) procure the equipment. It would be possible to save time on the project, by paying some premiums to complete certain activities faster than the normal schedule listed b

average price for a Big Mac

Do not send in PDF format. Sec. 2.1 # 38. The set of countries in which the average price for a Big Mac is between $2.00 and $2.99 Switzerland $5.46 Denmark $4.97 Sweden $4.46 United Kingdom $3.61 Germany $3.58 New Zealand $3.16 United States $3.00 Turkey $2.80 Peru $2.74 Canada $2.60 Chile $2.56 Japan $2.50

Cauchy sequences

5.2.1. Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are equivalent sequences of rationals, then (a_n)[n=1,infinity] is a Cauchy sequence if and only if (b_n)[n=1,infinity] is a Cauchy sequence. 5.2.2. Let epsilon > 0. Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are eventually epsilon-close, then (a_n)

Xbar and R charts, Is the process in control?

A manufacturing company produces rods. Each rod is required to be 20 millimeters in diameter. Each hour, random samples of size n = 4 rods are measured to check process control. Five hours of observations yielded the following: Diameter Time Rod 1 Rod 2 Rod 3 Rod 4 9 A.M. 19.8 20.4 19.9 20.3 10 A.M. 20.1 20.2 19.9 19.8


You are going to invest $20,000 in a portfolio consisting of assets X, Y, Z as follows: Asset Annual Return probability beta proportion X 10% 0.50 1.2 0.333 Y 8% 0.25 1.6 0.333 Z

Quantitative Method

I need help on this question- A quantitative method may be sensitive to changes in the parameters involved in comparing alternatives. A careful analyst conducts a sensitivity analysis to see if the situation's outcome is susceptible to small changes in such parameters (say 5%). Is Decision Analysis susceptible to such a s

Examine several newspapers and magazines

Examine several newspapers and magazines and describe at least three examples of functions that appear. What is the domain and range of each function? EXAMPLE OF ASSIGNMENT: The amount of money you pay for a gym membership is a function, because you pay a fee each month in addition to the initial fees for joining. In thi

Matlab help

Two related questions. First - I need a good way to display error bars on a scatter plot. I have a set of x and y data, that I have attached as a text file along with y+err and y-err data (denoted +/-). The err values vary from point to point, and +err and -err are not necessarily the same. Second - I am using the data to

Functions and equations and measurement

This Discussion Question will concentrate on functions and graphs. Understanding the definitions of words is the essence of mathematics. When we understand the meaning of words, finding a solution is much easier because we know what task the problem is asking us to complete. a. Part 1 b. In your own words, define the wo

Population Growth Problems

In a galaxy far, far away (my professor writes his own practice problems), on the planet Xylor, a herd of 100 Tybars was introduced for breeding. After 5 years, the herd had increased to 500. If the rate of herd growth is assumed to be directly proportional to the number of Tybars present on Xylor at any time t: a. How many T


Please see the attached file for the fully formatted problems. Problem 1 Consider the following two functions (see the attached file). Write a program called <name>1.sce generates plots of these functions over the range 0&#8804;x&#8804;8 on a single figure. Format the figure so that it is very readable and visually appealing

Modeling a Box as a Particle

See attached file for full problem description. Need help on 2b, 3. 2. A box of mass m is placed on a plane, which is inclined at an angle a to the horizontal, where tan a The plane is rough and the coefficient of friction between the box and the plane is . The box is kept in equilibrium on the plane by applying a horizontal

Banach Spaces

Show that (X, ||*||) is a Banach space if and only if {x in X: ||x||=1} is complete. Know that in the first direction, we must show that {x in X: ||x||=1} is closed subset of X. For the reverse direction, I know I have to take a cauchy sequence and translate it to the unit circle and then show that if it is convergent ther

Functions and Sequences of Iterates

Let f be a function defined on [a,b] and suppose that z is a point in (a,b) such that f(z) = z. Further suppose that there is a number alpha < 1 such that f ' (x) < alpha for all x contained in (a,b) and that 0 < f ' (x) for all x contained in (a, b). a.) Prove that if z<x, then z<f(x) and that f(x) - z < alpha(x-z).

Absolute and Relative Errors : Three-Digit Chopping and Rounding Arithmetic

Please help with the following problems. Provide step by step calculations. Compute the absolute error and the relative error in approximations of p by p*. a) p = pi , p* = 22/7 Perform the the following computations i) exactly, ii) using three-digit chopping arithmetic, iii) using three digit rounding arithmetic. iv) Com

Predicates, Logical Connectives and Quantifiers

Let P(x), Q(x) and R(x) be statements "x is a clear explanation", "x is satisfactory" and "x is an excuse" respectively. Suppose that the universe of discourse for x consists of all English test. Express the following statements using quantifiers, logical connectives and P(x), Q(x) and R(x) a) All clear explanations are satis

Is the packing process capable? Is an adjustment needed?

Canine Gourmet Super Breath dog treats are sold in boxes labeled with a net weight of 12 ounces (340grams) per box. Each box contains eight individual 1.5 ounce packets. To reduce the chances of shorting the customer, product design specifications call for the packet-filling process average to be set at 43.5 grams. Tolerances ar

Increasing Functions

Which of the functions below is increasing for all x-values? f(x) = x - 7 f(x) = x2 - 7 f(x) = x3 - 7 f(x) = x4 - 7

Metric Spaces

(See attached file for full problem description) 1. Show that the functions d defined below satisfy the properties of a metric. a. Let X be any nonempty set and let d be defined by The d is the call the discrete metric. b. If X is the set of all m-tuples of real numbers and, if for and , then (X,d) is a metric spac

Manufacturing Costs and Cost Functions

Manufacturing Cost The weekly production cost C (in dollars) of manufacturing x hand calculators is given by the formula C = 6000 + 8x - x 2 / 1000. What is the cost of producing 1000 hand calculators?

Bessel Functions and Sturm-Liouville Problem

(See attached file for full problem description) --- Use the following table to solve 3 and 4. J0(x) J1(x) Y0(x) Y1(x) 2.4048 0.0000 0.8936 2.1971 5.5201 3.8317 3.9577 5.4297 8.6537 7.0156 7.0861 8.5960 11.7915 10.1735 10.2223 11.7492 14.9309 13.3237 13.3611 14.8974 3. Find the first four &#945; i

Bessel Function, proofs

USING THE BESSEL FUNCTION OF ORDER ZERO: Verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative.