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Graphs and Functions

Pairwise Disjoint Graphs and Clique Numbers

Let be pairwise disjoint graphs, and let . Prove that is called clique number of the graph G, is the maximum order among the complete subgraphs of G . Please see the attached file for the fully formatted problems.

Asymptotes, Tangents and Intercepts

(4) Explain why the function f (x) = =has a vertical asymptote but no vertical tangent. (5) Sketch the graph of the curve....for [0,2a). Show all special features such as vertical asymptotes, horizontal asymptotes, cusps, vertical tangents, and intercepts. See attached file for full problem description.

Horizontal and Vertical Asymptotes, Vertical Tangents and Cusps

(1) Find constants a and b that guarantee that the graph of the function defined by f(x)= will have a vertical asymptote at x = 5 and a horizontal asymptote at y = -3. (2) Find all vertical tangents and vertical cusps for each of the following functions. Justify your work. See attached file for full problem description.

Trees and Graphs : Outerplanar Graphs

What bound is given for X(G) by the theorem "for every graph G, X(G)<=1+max &(G') ,where the maximum is taken over all induced subgraphs G' of G" in the case that G is a) a tree? b) an outerplanar graph. Note: -&(G') this sign represent the minimum degree of G'. Yes it is that minimum -A graph G is outerplana

Vertex Chromatic Numbers and Betti Numbers

Prove for every graph G of order n, that n/B(G)<=X(G)<=n+1-B(G). X(G) is the minimum integer k for which a graph G is k-colorable is called the vertex chromatic number In the page 82 B(G) is defined like independent sets like you say but in the page 187 it other kind of B and it is define like Betti number and it is defi

Slopes and Intercepts

1. The x-coordinate of (-5,3) 2. The slope through the points (-2,5),(6,-3) 3. The y- intercept of 3x +7y =21 4. The slope of -10x-5y=0 5. The y-coordinate of (7,-6) 6. The x- intercept of -6x+6y=-6 7. Evaluate the function f(x)=3x-7forx=4 8. The slope through (8,6),(-4,0) 9. The y-intercept of y = 1/2x 10. The slop

Calculating Work and Slope of a Tangent

1. The natural length of a spring is 10 cm. A force of 25 N stretches it to a length of 20cm. How much work, in units of N-cm, is done in stretching it from a length of 10cm to a length of 15cm? Hooke's law for a spring is given by f=kx, where f is the force, x is the distance the spring is stretched, and k is a constant. 2.

Power Series and Holomorphic Functions

Let f(z) be holomorphic in the region |z|<=R with power series expansion f(z)=sum(n=0 to infinity) a_nz^n. Let the partial sum of the series be defined as s_N(z)=sum(n=0 to N) a_nz^n Show that for |z|less than R we have s_n(z)= 1/i2pi(integral over |w|=R of f(w)[(w^N+1 - z^N+1)/(w-z)]dw/w^N+1)

Holomorphic Functions

If f(z) is holomorphic on |z|<1, f(0)=1, and for all |z|<=1 we have R(f(z))>=0, then show that -2<=R(f'(0))<=2 keywords: holomorphisms

Finding domain & range

Finding domain & range. See attached file for full problem description. (a) f(x) = (x -2)/ (3x + 4) (b) g(x) = -11/(4 +x) (c) g(x) = 4x^3 + 5x^2 -2x

Evaluating Functions and Applications of Functions

7.1 Determine whether the correspondence is a function. 8. Domain Range Colorado State University University of Colorado ____________ > Colorado All three colleges points to college University of Denver Gonzaga University University of Washin

Graphing and Solving Linear Equations

Graph and, if possible, determine the slope. Graph using the slope and the y -intercept. Determine whether the graphs of the given pair of lines are parallel. Determine whether the graphs of the given pair of lines are perpendicular. See attached file for full problem description. 7.4 Graph and, if possible, determi

Chromatic Numbers and Graph Coloring

Let G1 be a graph such that every two odd cycles intersect. Prove that X(G)=<5. (The minimum integer for which a graph is k-colorable is called the vertex chromatic number, or simply the chromatic number of , and is denote by , this problem is about graph coloring).


Please see the attached file for the fully formatted problems.

Graph the line

Graph the line with equation. See attached file for full problem description.

How to graph a simple equation

Please help me graph the line with equation: y=-5x-4 Also, show all of the steps so that I can learn how to do it myself.

Sample Question: Graph

1. Plot the graph of the equations 2x - 3y = 6 and 2x + y = -10 and interpret the result. 2. Plot the graph of the equations 2x + 4y = 10 and 3x + 6y = 12 and interpret the result. 3. Determine graphically the vertices of the triangle, the equation of whose sides are given as y = x; y = 0; 2x + 3y = 10. Interpret the res

Graphing Linear Functions and Finding the Point of Intersection

A. The solutions of line m are (3,3),(5,5),(15,15),(34,34),(678,678), and (1234,1234). b. The solutions of line n are (3,-3),5,-5),(15,-15),(34,-34),(678,-678), and (1234,-1234). c. Form the equations of both the lines d. What are the co ordinates of the point of intersection of lines m and n? e. Write the co-ordinat

Inverse Functions and Set Operations

Let f be a function from A to B. Let S and T be subsets of B. Show that: a) -1 -1 -1 f (S U T) = f (s) U f (T) b) -1 -1 -1 f (S n T) = f (S) n f ( T)

Hamiltonian Graphs and 2-Connected Graphs

Explain this problem with a graph to understand and explain it step by step. a) Show that if G is a 2-connected graph containing a vertex that is adjacent to at least three vertices of degree 2, then G is not hamiltonian. b) The subdivision graph S(G) of a graph G is that graph obtained from G by replacing each edge uv of