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Graphs and Functions

Graphs : Coloring Maps

Let G be a cubic plane graph. Prove that the map M(G) is 3-colourable iff each country has an even number of sides. note: if we omit the vertices and edges of a plane graph G from the plane, the remainder falls into connected components, called "faces." Clearly each plane graph has exactly one unbounded face. The boundary

Graphs : Coloring Maps

Show that a planar map M = M(G) can be 2 colored iff every vertex of G has even degree. [The map M(G) of a graph G is the collection of its faces, which are to be colored so that no adjacent faces have the same color.] [hint: if every vertex of G has even degree then G is a union of edge-disjoint cycles. for another solut

Cubic Graphs : Connectivity and Edge-connectivity

Connectivity and edge-connectivity are equal for cubic graphs. Construct examples showing that every possible connectivity number occurs (for graphs regular of degree 3):(Kappa) = 0, 1, 2, 3. Kappa = connectivity

Graphs, Vertices and Cycle Length

A nontrivial graph G is called prime if G = G_1 x G_2 implies that G_1 or G_2 is trivial. Show that if a connected graph G has a vertex which is not in a cycle of length four, then G is prime. where "x" is the cartesian product

Graph the Parabola

Graph the parabola 1/3 x^2 + 10/3 x + 28/3 = 0 Plot the vertex and four additional points, two on each side of the vertex.

Intepreting Data and Piecewise Functions

A factory begins emitting particulate matter into the atmosphere at 8 am each workday, with the emissions continuing until 4 pm. The level of pollutants, P(t), measured by a monitoring station 1/2 mile away is approximated as follows, where t represents the number of hours since 8 am: p(t)= 75t + 100 if 0 is les

Differentiable Functions and Inverses

1. Let be a differentiable function such that (0) = 0 and for all . a) Show that is strictly monotone and therefore its inverse exists. b) Assume that exists. Compute Please see the attached file for the fully formatted problems.

Equations of Tangent Lines and Areas Between Functions

Please see the attached file for the fully formatted problems. Determine the area between f(x) = 2x and on the domain determined by the points where the graphs of the functions cross. a. 0.1111 b. 1.0125 c. 1.6667 d. 1.3333 Find the function whose tangent line has the slope for each value of x, and whose grap

Graphs : Chromatic Number

Show that the chromatic number of G_1 + G_2 isX(G_1) + X(G_2) for any two graphs G_1 and G_2. Where X(G)is the Chromatic Number. defn: X(G) a proper colouring or simply a colouring of the vertices of G is an assignment of colours to the vertices in such a way that adjacent vertices have distinct colours; X(G) is then the mi

Graphs, Nodes and Partitions

Suppose there are 90 small towns in a country. From each town there is a direct bus route to at least 50 towns. Is it possible to go from one town to any other town by bus possibly changing from one bus and then taking another bus to another town?

Evans Price Adjustment Model : Supply and Demand Functions

Qn1)The Evans Price Adjustment model is a dynamic model in which price p denote the price of a particular commodity, S(p) and D(p) denote the supply and demand functions of that commodity respectively. These 3 parameters are regarded as function of time t. The time rate of change of price is assumed to be proportional to the sho

Maximum Value of a Function

Suppose that a projectile is fired at an angle of 45 degrees from the horizontal.... Please see attachment and show work.

Characteristic Function of Metric Space

Let S ⊂ M. (a) Define the characteristic function Xs : M --> R. (b) If M is a metric space, show that Xs(x) is discontinuous at x if and only if x is a boundary point of S. [Please see attached PDF file for full problem]. for part (a), I think something similar to

Lines Parallel To y=-x+6

Find an equation of the line that is parallel to the line y = -x + 6 and that passes through the point (5,9).

Basic 25

Graph function and state domain and range 2 y=x -2x+1


Here I go again with slope. Find the slope of (1,3) and (7,-6)

Vertical and Horizontal

Write and equation for the vertical and horizontal lines passing through the point (4,-4) in (x,y) coordinates. Please show the steps to help me understand how you accomplished this.

Line Equations from a Word Problems

Sue owens her brother $8000 and agrees to pay $250 a month until it is paid off. Let A represent the amount owed(in dollars) and let N represent the number of payments made. Write an equation relating A to N and then graph your equation using the the Y axis as A with a scale of 1000 to 8000 and the x axis represents N with a s

Graph from a Word Problem

Joe is saving money. He saves $50 in his piggy bank to start with and adds an additional $10 each month. Let A represent the total amount of money (in dollars) in the piggy bank and let N represent the number of months that Joe has been saving. Write an equation relating A to N and then graph the equation using the axis Y for

Graphing Word Problems

Kira owes $6125 and agrees to pay $175 a month until paid. Let A represent the amount owed (in dollars) and let N represent the number of payments made. Write an equation relating A to N then graph the equation. State the points being graphed.

Writing Equations and Graphing

Lisa is a software saleswoman. Lisa's monthly salary is $1700 plus an additional $90 for every copy of Math is Fun she sells. Let S represent Lisa's monthly salary (in dollars), and let N represent the number of copies of Math is Fun that she sold. Write an equation relating S to N, and then graph your equation using the axes

Point of Inflection

Please see the attached file for the fully formatted problems. Let . Which value of t corresponds to an inflection point for f(t)? A. B. C. D. There is no inflection point.