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Graphs and Functions

Sequence of Functions and Mean Value Theorem

Let a<b. Let f_n: [a,b] -> R be a sequence of functions such that, for each n in N ( N set of natural numbers),f_n is differentiable on (a,b). Suppose that for all n in N, Sup on [a,b] of | f'_n(x) | < or = to M, where M is in R. ( Sup is supremum = least upper bound) Prove that for all n in N and all x, y in [a,b], one has

Uniformly Cauchy Sequence of Continuous Functions

Let f_n : [0,1] -> R be a sequence of continuous functions such that for each n in N (natural numbers), f_n is differentiable on (0,1). Suppose that f_n(0) converges to some number, denoted f(0), and also suppose that the sequence (f'_n) converges uniformly on (0,1) to some function g: (0,1) -> R. Prove that the sequence (f_n) c

Diffusion Equation : Energy Decreasing as a Function of Time

3. Suppose that u(x. t) satisfies the diffusion equation ut = kuxx for 0 < x < L and t > 0, and the Robin boundary conditions ux(0, t) ? aou(0, t) = 0 and ux(L, t) + aLu(L, t) = 0 where k, L, a0 and aL are all positive constants. Show that ... is a decreasing function of t. Please see the attached file for the fully for

Sketching a graph of a function

Sketch the graph of the function y= -x^3+3x^2-4. Be sure to include and label: 1.) x and y intercepts 2.) asymptotes 3.) 1st and 2nd derivatives 4.) increasing and decreasing intervals 5.) intervals of concavity 6.) inflection point(s) 7.) relative extrema (max and min)

Diminishing returns (point of inflexion)

Identify the point of diminishing returns for the input-output functions. R=1/50000(600x^2-x^3), 0<x<400 (those are < or = to signs) R=-4/9(x^3-9x^2-27), 0<x<5 (those are < or = to signs)

Distance

A car travels along a straight road, heading West for 3 hours, and then travels NE on another road for 2 hours. If the car has maintained a constant speed of 55 mi/hr, how far is it from its starting point?

Analytic Functions

Show that if f in analytic in {z: |z| < 1} and if Im f(1/k)=0 for all k=2,3... then Im f(x)=0 for -1<x<1. Please see attached for Hint.

Fields

Let F be a field and le p(x) E F[x]. If p(x) is not irreducible, then F[x]/{p(x)} is: a) always a field b) sometimes a field c) never a field. Give reasons for your assertion. Please see attached.

Graph of Parabola

Sketch the graph of the function y = 16 - x^2. What are the domain and range of the function? What are the x-intercepts?

Function Classification

Can the graphs (attached) be classified as functions? Explain. (A graph, using smooth lines that connect data in the graph)

Polar Coordinates

A sprinkler distributes water in a circular pattern, supplying water to a depth of {see attachment} feet per hour at a distance of {see attachment} feet from the sprinkler. A) What is the total amount of water supplied per hour inside of a circle of radius 17? B) What is the total amount of water that goes throught the

Sketching the graph of a swimming fish's energy

Any help is greatly appreciated; I found this problem pretty frustrating. I replaced the "less than" symbol with the words "less than" because the computer seemed to have a hard time recognizing the symbol. "For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v^3.

Singular Point : Pole and Residue

2. Show that the singular point of each of the following functions is a pole. Determine the order m of that pole and the corresponding residue B. {please see attachment for functions} Please specify the terms that you use if necessary and clearly explain each step of your solution.

Functions : Maximizing Profit, Diminishing Returns and Maximizing Volume

1. The demand for a product in dollars is given by p(x) = 53/(x)^1/2 Fixed cost are $608 and the cost to produce each item is $0.53. Find the production level of x that maximizes profit within the range of 0<(or equal to)x< (or equal to)7530. 2. An efficiency study of the afternoon shift (12:00-4p.m.) at a factory shows th

Parametric Equations : Canonical Form, Values of t

Consider the line with parametric equations, x = 2t + 3 and y = -4t + 1 a) Find the equation of the line in non-parametric form. b) Find the values of the parameter t which correspond to the points A (3, 1) and B (7, -7) on the line. c) Write down the range of values of t which, together with the given parametr

Intercepts, Vertex, Line of Symmetry and Image Set

This question concerns the parabola which is the graph of the function: f(x) = [1/4(x-2)^2] -1 a) Explain how the graph of the parabola can be obtained from the graph of y =x[squared] by using appropriate translation and scalings. b) Using your answer to part (a), or otherwise, write down the coordinates of the vertex of th

Length of Curve / Length of Arc of Curve

2. Find the length of the arc of the curve y=f(x) on the intervals given: {see attachment} 3 and 4. Find the length of the curve defined by: {see attachment}, between the points: {see attachment} 5. Find the surface area generated when the graph of each function on the interval is revolved about the x-axis. (Give answer to

Polynomial functions, inverses, half-life, investments

Please find the attached. 1) Fit a polynomial Function f(x) to the graph. The scale on the x-axis is 1 and the scale on the y-axis is 5. The point (1,12) is on the graph, Assume that if the graph appears to cross the x-axis at a mark, it really does. (1,12) 3 2 1 4 2) Noise level in decib

Fixed Point : Mean Value Theorem

A number (a) is called a fixed point of a function (f) if f(a)=a. Prove that, if f'(x) does NOT equal 1 for all real numbers (x), then f has at most one fixed point.

Inequalities and Line Equations (5 Problems)

1) A line L1 has a slope of -7/10. Determine whether the line through (5,3) and (-2,-7) is parallel or perpendicular to L1 2) Graph: x+y=4 3) What is the slope of the line 6x+2y=48 4) Graph: Y &#8804; x-1 5) Graph: 3x &#8804; 4y

Graphing Inequities

I don't need to be shown how to graph the below, But I do need help knowing what to graph. 1. Graph: Y &#8804; 3x-6 2. Graph: -4x &#8805; 5y 3. Graph: -8x &#8804; 2y

Graphing Questions

I have a lot of the attached problems to do. I know how to put points on the graph (for example (4,3)) but I am not sure how to get the information I need to graph on the attached. So I don't need to see these problems actually graphed out, just how to get to the stage before graphing. One: Graph: X &#8805; 1 Two:

Parallel Lines

Find the pair of parallel lines: 1: -y=-x+2 2: -2y-2x=2 3:-2x+2y=2 Not sure how to do the above problem.