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Graphs and Functions

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) = √x is uniformly continuous on [0,∞). 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) ---

Distance and graphing in 3-D space

1. Find the distance from the origin to the line passing through the point P(3,1,5) and having the direction vector v=2i-j+k. 2. Graph z=x^2 in space.

Graphing binomials from trinomials.

Using graphing to check your answers is helpful. When you factor a trinomial into two binomials, each binomial represents a linear relationship. If you plot the two binomials (which are just lines) on a graph, what do they have in common with a plot of the trinomial itself? More important than that, how can this information be u

Measurable Functions

Suppose u(x) : X--> R v(x) : X --> R Both u(x) and v(x) are measurable Let f(x) : x --> R^2 f(x) = (u(x), v(x) ) Then f (x) is measurable Now prove a generalization of the above. That is, prove: if u_1(x) : X--> R u_2(x): X--> R . . . . u_n(x) : X--> R u_1,.

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.

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

Graph, Solve for X and Inverses

1. Graph for the function: f(x)=2-4^x 2. Solve for x: ln(7x-1)=6 3. Solve for x: lnx=3+ln(x-1) 4. Are the following funtions inverses of each other? 1. f(x)=x-1/3 g(x)=3x-1.

Functions: Lebesgue Spaces

Consider the following function: f(x) = 1/x for x in [1, infinity) = 1 for x in (-1,1) = -1/x for x in (-infinity, -1] Please explain why f(x) is in L^2(R)L^1(R)

Carmichael Number

Which one of the following is true A Carmichael number is: a) a 2 pseudo-prime b) a 3 pseudo-prime c) a 5 pseudo-prime d) All of the above e) None of the above f) Just (a) and (b)

Graph the linear equation ranges

Graph the linear equation for the indicated values of the independent variable.Show this on a Graph as well as the formula V=50n + 30, 0.1<= n <= 0.9

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.

Prove Functions and Sets

(See attached file for full problem description) 1) a. Let f be defined on [a, b] by f(x) prove directly that f is measurable b. let E be measurable subset of R, and let f be measurable function on E Define the function f and f on E as follows: f (x) = max { f(x), 0}, and f (x) = max{ -f(x), 0}, 1b. prove directly t

Graph, Factor and Solve Equations

Graph: 4x + y >= 4 Solve: (x - 10)(x + 9) Factor: 10r3s2 + 25r2s2 - 15r2s3 Factor : 3x2 + 7x + 2 Solve - factoring: x2 - 2x - 3 = 0