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# Graphs and Functions

### 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) ---

### Fixed Point : Fixed-Point Iteration and Error Estimate

Please see the attached file for the fully formatted problem.

### 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

### 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

### 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.

### 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

### 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.

### Real-Life Applications of Functions and Graphs : Heart Disease / Cancer and Fahrenheit / Celsius Temperature Conversions

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

### Graphing Functions Differentiation

Differentiation of graphs for Celcius and Farenheit functions

### 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)

### Rectangular form

How do I write z=6(cospi/3+i sinpi/3) in rectangular form?

### Find the First Four Terms of the Sequence

What are the first four terms of the sequence whose general term is An=4(n-1)!/n!?

### Cylindrical Polar Coordinates

Find V in cylindrical polar coordinates. Is my working correct? (See attached file for full problem description).

### Cylindrical Polar Coordinates

Express the potential function U in cylindrical polar coordinates.. U(x,y,z) = -z(x^2+y^2)^2

### Even and odd functions

If g(x) is even and h(x) is odd, what is g(h(x))? what about h(g(x))?

### Spherical Polar Coordinates Confirmation

I have an answer for this problem, so it is just a check and confirmation I require. --- (See attached file for full problem description)

### 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.

### Sketch a system

0<=x<=300 0<=y<=12000 Sketch

### 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

### Conformal Mapping : Semi-Infinite Strip

Please see the attached file for the fully formatted problems.

### 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

### 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.