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# Graphs and Functions

### Graphing using a Lagrange Interpolating Polynomial and Piecewise Linear Function

1. For medical purposes the level of sugar was measured in blood (in mg/dl). The samples were taken with 1 2hr increments, as the following table shows: initial sample 96 mg/dl after 30 min. 133 mg/dl after 60 min. 142 mg/dl after 90 min. 81 mg/dl after 120 min. 87 mg/dl Graph in MATLAB sugar curves corresponding to thes

### Pointwise Operations and Characteristic Functions

Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) =

### Operations of Functions: Injective, Surjective and Bijective

Let f: A ?> B and g: B ?> C be functions. a) Suppose that f and g are injective. Show that g o f is injective. b) Suppose that f and g are surjective. Show that g o f is surjective. c) Suppose that f and g are bijective. Then g o f is bijective by parts a) and b). Show that (gof)^-1 = f^-1 o g^-1. Please do number 3 in at

### Pointwise Operations and Characteristic Functions

Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) =

### What is the expected yield on notes from year 1 to 2, assuming the PEH holds? If inflation for the next year is expected to be 4.5%, what would the expected real rate of interest be on the 3-month T-bill? Plot the yield curve for these Treasury securities. Using the three theories we have discussed, explain how each one helps explain the shape of this yield curve.

1. If U.S. Treasury yields are as follows: 3 month 6.0% 6 month 6.3% 1 year 6.5% 2 year 6.6% 5 year 6.4% 10 year 7.5% 30 year 8.0% a. What is the expected yield on notes from year 1 to 2, assuming the PEH holds? b. What is the expected yield on notes from year 2 to 5, assuming th

### Algebra Review

(See attached file for full problem description with equations and diagrams) --- 1. Simplification of linear algebraic expressions and expressions with fractional coefficients and solve x; 2. Solving simple linear equations with fractional coefficient: 3. Solving inequalities with fractional coefficient: 4.

### Parabola and tangent lines

The parabola y= x^2+4 has two tangents which pass through the point (0,-2). One is tangent to the parabola at (a, a^2+4) and the other at (-a, a^2+4) where a is a certain positive number. The question is a=?

### Solve Finite Difference Equation

See attached file for full problem description with equation. --- Find analytically the solution of this difference equation with the given initial values: Without computing the solution recursively, predict whether such a computation would be stable. (Note: A numerical process is unstable if small errors made at one

### Fashion Plotting

A new fashion in clothes is introduced. It spreads slowly through the population at first but then speeds up as more people become aware of it. Eventually those willing to try the new fashions begin to dry up and while the number of people adopting the fashion continues to increase. It does so at a decreasing rate. Later the fas

### Relations Versus Functions : Real-Life Application and Celsius-Fahrenheit Temperature Conversion

1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? 2. There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahren

### Evaluate graphs of derivative functions

(a) Suppose the graph in Figure 4.1.78 is that of a function g(x). Sketch the graph of the derivative g; (b) On the other hand, suppose the graph above is that of the derivative of a function f. For the interval ..., tell where the function f is (i) increasing; (ii) decreasing. (iii) Tell whether f has any extrema, and if so

### Finding the derivative of a function given graphically

For the function of f, given below in graph (a) Sketch (b) Where does change its sign (c) Where does have local minima and maxima Using the graph of write a brief description of complete sentences to describe the relationship between the following features of the function of: (a) the local maxima and minima o

### Heart Disease and Cancer : Plotting Graphs Using Excel, Trends and Making Predictions

You have been invited to present statistical information at a conference. To prepare, you must perform the following tasks: The following data was retrieved from www.cdc.gov. It represents the number of deaths in the United States due to heart Disease and cancer in each of the years; 1985, 1990, 1995, and 2002. The f

### Find Domain, Graph, Height, Minimum Surface Area of a Box

Consider an open-top box with a square base and a volume of 108 cubic inches. Let x be the length of a side of the base. a) Calculate the height h as a function of x. Is this function even, odd, or neither? b) What is the domain of the function above? (Note that there may be physical and/or mathematical restrictions.)

### Uniformly Convergent Sequence of Functions and Weierstrass Test

Find an example of a sequence of continuous functions on fn on [0,1] such that the series...converges uniformly on [0,1] but the series ... diverges. Is it a counterexample for the Weierstrass test? ---

### Uniform Convergence of a Sequence of Functions

Prove the following theorem. Let f1,f2,f3.... be continuous functions on a closed bounded interval [a,b] . Then fn--->f uniformly on [a,b] if and only if fn(x)-->f(x) for every xn-->x such that xn,x E[a,b] . Please see the attached file for the fully formatted problems.

### Functions, Roots, Convergence, Fixed Point Method

Consider the function f(x) = 2sinx + e^-x - 1 on the interval r E [?2,2]. If you plot the function, you will see that it has two roots on this interval (a) Write down a first order fixed point method for finding one of the two roots. (b) Will this fixed point method converge for both of the roots (Justify)? If it does not co

### Proof of Uniform Continuity

Show that the function f(x) = &#8730;x is uniformly continuous on [0,&#8734;). Note: This is from a beginning analysis class. We can only use the definition of uniform continuity. (In other words, cannot use compactness, etc to prove) ---

### The relation between u,v and w where u,v,w are not independent

Independence and relations Real Analysis Jacobians (II) If u = (x + y)/z, v = (y + z)/x, w = y(x + y + z)/xz Show that u,v,w are not independent. Also find the

### Use the Mean-Value Theorem- continuously differentiable function

(See attached file for full problem description) Let a sequence xn be defined inductively by . Suppose that as and . Show that . (Note that " " refers to "little oh") HINT: Use the Mean-Value Theorem and assume that F is a continuously differentiable function.

### Find the linear equation that expresses temperature in degrees Fahrenheit as a function of temperature in degrees Celsius.

In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahrenheit Celsius Freezing point of water 32 0 Boiling

### Real-World Applications of Graphs and Functions : Heart Disease and Cancer

You have been invited to present statistical information at a conference. To prepare, you must perform the following tasks: 1. The following data was retrieved from www.cdc.gov. It represents the number of deaths in the United States due to heart Disease and cancer in each of the years; 1985, 1990, 1995, and 2002. Year Di

### Minimum spanning tree

Hi. Is the statement below TRUE or FALSE. Why? Question : I have a connected weighted undirected graph G with a minimum spanning tree T. If I increase the weight of one edge, the new minimum spanning tree T' of the new graph G' differs from T in at most one edge.

### Vertical and Horizontal Asymptotes

Here is a question on finding the vertical asymptotes g(x)=x+3/x(x-3) Can you show me a horizontal asymptote as well? f(x)=12x/3x^2+1.

### Business - monthly revenue achieved by selling x boxes of candy...

The monthly revenue achieved by selling x boxes of candy is figured to be x(5 - 0.05x) dollars. The wholesale cost of each box of candy is \$1.50. a) How many boxes must be sold each month to achieve a profit of at least \$60? b) Using a graph in utility, graph the revenue function. c) What is the maximum re

### Constructing an Open Box : Writing Functions and Calculating and Minimizing Area

A open box with a square base is required to have a volume of 10 cubic feet. a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base. b) How much material is required for a base 1 foot by 1 foot? c) How much material is required for a base 2 feet by 2 feet? d) Gr

### Heart Disease, Cancer and AIDS : Plotting Graphs and Defining Functions

I have found the number of deaths in the United States due to each medical condition in each of the following years. 1985 1990 1995 2002 Heart Disease 390,000 720,000 850,000 960,000 Cancer 200,00 505,000 547,000 555,000

### Functions and Graphs

Trying to find the number of deaths in the United States due to each of the following medical conditions in each of these years; 1985, 1990, 1995, and 2002. heart disease, cancer and aids. How do I plot the data for each disease as points in a rectangular coordinate system? I must connect the data points. Using a curve how

### Graphing and Solving Quadratic Inequalities (3 Problems)

State the solution set using interval notation and graph the solution set. Please check these for me and graph them. I do not know how to use a graphing tool. If you also have any advice on how & what tool I can use to graph, it would be helpful. Please do not handwrite the Graphs because I can not view scanned photos

### Find the vertex and intercepts for each quadratic function, and sketch its graph.

Find the vertex and intercepts for each quadratic function, and sketch its graph. 49. 50. 53. 54. Please see the attached file for the fully formatted problems.