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Graphs and Functions

Quadratic Equation

A square picture is mounted in a frame 1 cm wide. The area of the picture is 2/3 of the total area. Find the length of a side of the picture.

Function solutions

2. Given the polynomial f(x) = 2x 3 -5x2-4x+3, find the solutions if the function is completed as a) f(x) =0 b) f(x+2)=0 d) f(2x) = 0

Prove A Variation of Fermat's Theorem

There always exists a real number n such that a^n = b^n + c^n , where a, b and c are any integers. The problem is not Fermat's Last Theorem, but a variation of it with real exponents.

Functions: Onto and One-to-one, Bijections and Functions

Please help with the following problems on graphs and functions. Provide step by step calculations. 1. Assuming A,B not equal to no solution, define m1:AxB->A and m2:AxB-> as follows: m1(x,y)=x and m2(x,y)=y. If f: A->B, show that a) f onto=>m2 |f is onto b)f one-to-one=>m2 f is one-to-one 2. Assuming f: A->B and g:

Functions and Graphs: Trends and Real World Implications

Plot your data for each disease as points in a rectangular coordinate system. Year...................1985..........1990..........1995......2002 Heart Disease 771,169 720,058 684,462 162,672 Cancer 461,563 505,322 554,643 557,271 AIDS * 8,000 25,188 39,979 14,095 - Use individu

Plotting a Surface Function

Consider the function u(x,t) = sin(4 pi x) e^(-pi t). Plot using a graphical tool and explain what you observe. Please see the attached file for the fully formatted problems.

Statistics

What is the "causal relationship" between independent and dependent variable?

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.