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Graphs and Functions

Parametric Equations

Find the parametric equations for the line through the point (1,0,-1) and parallel to the line (1/3)(x-4) = (1/2)y = z+2

Limits and Monotone Functions

First, I am looking for an example of a monotone function with (a,b)-->R that is unbounded and then I need to verify that the function has lim_x-->c^+ less than or equal to Lim_x-->d^- whenever a < c < d < b keywords: monotonic

Differentiable Functions : Lipschitz and Absolute Values

I have a function that is differentiable on [a,b] and I am trying to figure out which scenario is more restrictive: a) the function is a Lipschitz function with a Lipschitz constant L in (0,1) or b) the absolute value of f'(x) is less than one for all x in [a,b]

Proofs : Positive and Negative Functions

For a given function f, let f^+(x)=max{f(x), 0} and f^-(x)=max{-f(x), 0} Prove that for any function f, f=f^+ - f^- and |f|=f^+ + f^- Please see the attached file for the fully formatted problems.

Mean Value and Graphing

A) Sketch the Graph of y = cosx in the range -Pi < x < Pi. Hence Sketch the graph of y= |cosx| in the same range. (b) Find the mean Value of y= |cosx| in the range -Pi < x < Pi P.S its the |cosx| thats confusing me - I have never seen this format before.

Vertex, focus, directrix and latus rectum of Parabola and ellipse

1. Find the equation of a parabola whose vertex is (0,0) and directrix is the line y=3. 2. Find the vertex, focus, and directrix of (x-2)^2=12(y+1). Find the latus rectum and graph the parabola, making sure that all points and axis are labeled. 3. Find the equation of the ellipse whose center is the origin and has a

Bivariate Functions : Domain, Range and Intercepts

F(x,y) = x^2 + y^2 Are the xyz-intercepts all 0? The domain looks like it is all real #s. Is the range also all real #'s? I also see the graph is continuous on all points. Really I am just confused about the range.

Domain, range, intercepts

F (x,y) = 4 What are the domain, range and intercepts of this one? I am totally confused about this graph. Are there even any x and y intercepts? I don't think so, since I am assuming z = 4 and that would be the z-intercept, right?

Range, domain, intercepts

F(x,y) = cos (y) What would the domain of this be? (-infinity, +infinity)??? Is the range between -1 and 1? Are the z and x-intercepts 1 and does a y-intercept exist?

Domain, range and intercepts

I have graphed the following function: f(x,y) = 4-y^2 I am a little confused as to what the domain, range, and xyz-intercepts are. Can you help with that?

Mobius Functions, Euler Functions and Carmicheal's Conjecture

1) Prove that in is a positive integer. ( : is the Mobius function) Hint: one of the four argument is divisible by 4. 2) If is a prime and . Show that ( : is the Euler function) 3) a. Prove that is an integer if n is a prime and that it is not an integer b. Prove that is not an integer if n is divisi

Entire function

Show that if an entire function f maps the real axis into itself and the imaginary axis into itself, then f is an odd function, i.e., f(&#8722;z) = &#8722;f(z) for any z. Give two proofs, which are really different.

Analytic function

Prove that if f(z) : H -> H is an analytic function from the upper-half plane to itself, then:|f(z) &#8722; f(z_0)|/|f(z) &#8722; (f(z_0))bar|<=|z &#8722; z_0|/|z &#8722; (z_0)bar| where z,z_0 are in H and |f'(z)|/Im(f(z))<=1/Im(z) where z is in H. When does equality hold?

Analytic function

3. Let D = {z : |z| < 1}. Suppose that f : D -> D is analytic, f(1/3) = 0 and f'(1/3) = 0. Show that |f(0)| <= 1/9.

Increasing/decreasing holomorphic functions

1. Let f(z) be a holomorphic function in the disc |z| < R1 and set M(r) = sup|f(z)|(|z|=r), A(r) = supR(f(z)) (|z|<r) where 0<=r<R_1 (a) Show that M(r) is monotonic and, in fact, strictly increasing, unless f is a constant. (b) Show that A(r) is monotonic and, in fact, strictly increasing, unless f is constant.

Parameter problem on parabolas

For a parabola y^2 = 4ax: 1. Find the equation of the tangent at P ( at^2, 2at ) on the parabola. 2. Find the point Q on the parabola so that PQ passes through the focus F ( a, 0 ) of the parabola 3. Show that the tangents at P and Q intersect on the directrix of the parabola


Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-4,2) and perpendicular to the line whose equation is y = 1/3X + 7

Congruences, Equivalence Relations and Inverses

1. Show that a = b mod m is an equivalence relation on Z. I used = to mean "equal by definition to" and Z as integers. 2. Find the inverse of each of the following integers. r 1 2 3 4 5 6 ----------------------------------- r^-1 mod 7 3. Sh

Mappings, Injective and Surjective Functions and Cycles

1. Let f : X -> Y and g : Y -> Z be mappings. (1) Show that if f and g are both injective, then so is g o f : X -> Z (2) Show that if f and g are both surjective, then so is g o f : X -> Z. 2. Let alpha = 1 2 3 4 5 and Beta = 1 2 3 4 5 3 5 1 2 4 3 2 4 5 1 .

F(X) Positive and Negative

Consider the following graph of the function f(x) f(X) Positive and Negative. See attached file for full problem description.

How do you use linear equation in business?

In most businesses, increasing prices of their product can have a negative effect on the number of customers of the business. A bus company in a small town has an average number of riders of 1,000 per day. The bus company charges $2.00 for a ride. They conducted a survey of their customers and found that they will lose approxima

Petersen Graph Nonplanar

Show that the Petersen graph is nonplanar by a) showing that it has k3,3 as a subcontraction, and b) using the problem 1 show above part a) You don't have to solve problem 1. Can you explain about contraction. problem 1.-Let k>=3 be an integer , and let G be a plane graph of order n(>=k) and size m. a) If the length


Require histograms of: 1 )Arrival Rate Number of Arrivals vs. Number of Occurrences This should be one consolidated histogram representing all of the observations from each data sheet. In plotting this data, the horizontal axis should be the number of arrivals during the 5-minute interval and the vertical axis should be th