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# Graphs and Functions

### Riemann Integrable Function : Upper and Lower Sums

Please see attachment. Q. Show directly that the function is integrable on R = [0,1] x [0,1] and find (Hint: Partition R into by squares and let N , limUp = limLp = integrable Up = upper Riemann sum of f respect to partition &#61664; U(f,p) = Lp = Lower Riemann sum of f respect to partition &#61664; L(f,p) =

### Vector Components : Force

A FORCE OF 85 N ACTS TO THE LEFT AND DOWNWARD AT AN ANGLE OF 45 DEGREES WITH THE HORIZONTAL. WHAT IS THE EFFECTIVE DOWNWARD FORCE AND WHAT COMPONENT IS ACTING TO THE LEFT?

### Examples of Functions : Exponentials, Graphs and Usefulness

1. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph of both the functions and post to the discussion forum. Discuss these functions and their graphs with your classmates. 2. Given the values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, x, and y, form each of the follow

### Graph of polar equation

Inspect the four given graphs, which are plotted in polar coordinates (and shown in an attached .doc file), and choose the graph that corresponds to the polar equation r = 6 - (cos theta).

### Functions : Pointwise Operations

If function f(x) = x^2-3x-4, find and simplify f(x+h)-f(x)/h

### Chromatic Numbers

Use words to describe the solution process. No programming. 2. Let G = (V,E) be a graph where V {1,2,3,4,5,6,7,8,9,10,11,12} and E contains all edges connecting to vertices a and b such that ab=0 (mod 3). What is the chromatic number of G? Is G planar? See the attached file.

### Graph Coloring Problem: Describe the Solution Process

Please use words to describe the solution process: Let G be a graph with exactly one cycle. Prove that x(G) is less than or equal t0 3. *(Please see attachment for proper symbols).

### Complete the Graph Problem

Please use words to describe the solution process: Let G be a graph with n vertices that is not a complete graph. Prove that x (G) < n HINT: If G does not contain k3 as a subgraph, then every face must have degree at least 4. *(Please see attachment for proper symbols).

### Graphs and Figures

Please use words to describe the solution process. Let G and H be the graphs in the following figure (see attachment): Please find x(G) and x(H). See the attachments.

### Evaluate the Binomial Coefficient

Evaluate the combination or binomial function in the completion 48 factorial over 37 factorial. The answer should equal to 22595200368.

### Lipschitz functions

A function f:A->R is called Lipschitz if there exists a bound M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x, y belong to A. Geometrically speaking a function f is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of f. a- Show that if f:A->R i

### Writing Equations Distributive Property

I must write an equation that has the solution x = -2 and graph it. My equation must include combining like terms on the left side and distributive property on the right side. I missed two weeks of class due to illness, and am having a hard time working this out. I just need a sample answer or starting point so that I underst

### Graphical Representation of Linear Equation

Solve the following questions on graphical representation of linear equations. Your response should include the graphs for each of these questions. 1. Plot the graph of the equations 3x - 2y = -6 and 4x + 2y = -11 and interpret the result. 2. Plot the graph of the equations 3x - 5y = 14 and -6x + 10y = 9 and interpret

### Graphing Trigonometric Functions

Graph the attached function (which of the attached graphs is correct?) See the attached file.

### Divergence Criterion - Corollary

Corollary:(Divergence criterion for function limits).let f be a function defined on A, and let c be a limit piont of A. If there exist two sequences (x_n) and (y_n) in A with x_n not =c and y_n not =c and lim x_n=lim y_n=c but lim f(x_n) not = lim f(y_n), then we conclude that the functional limit lim f(x) as x->c does not exist

### Graphs : Connectedness and Cycles

13. Let G be a connected graph with (please see the attachment). Prove that G contains exactly one cycle.

### Find the Critical Point Set for the Given System

Find the critical point set for the given system. *Please see attachment for system.

### Congruences : Multiplicative Inverse

In Z135, the element [4] is a unit. Why? Find the multiplicative inverse [4]^-1 (Please see attachment for proper format.)

### Differential Equations - Cross-Eyed Heart

27. y'' + 25y = sin(4t), y(0) = 0, y'(0) = 0. Plot the component curves and the orbit, the latter for the rectangle |y|< 0.25 and |y'|< 1. Any surprises? 28. Hearts and Eyes: Find a solution formula for y'' + 25y = sin&(wt), where w is not equal 5. Plot the solution curve of the IVP with y(0) = y'(0), where w=4. Plot t

### Eulerian and Non-Eulerian Graphs

Let G be a connected graph that is not Eulerian. Prove that it is possible to add a single vertex to G together with some edges from this new vertex to some old vertices so that the new graph is Eulerian. Please see attachment for background and hints.

### Graphs : Eulerian Trails

1. We noticed that a graph with more than two vertices of odd degree cannot have an Eulerian trail... (please see the attached file).

### Graphs : Connectedness, Vertices and Edges

11. Let G be a graph with n>= 2 vertices. a) Prove that if G has at least (n-1) + 1 edges the G is connected. ( 2 ) b) Show that the result in (a) is best possible; that is, for each n>= 2, prove there is a graph with (n- 1)

### How to Find the Area of a Region

Find the total area of the region between the graph of F and the x-axis. (See attachment).

### Count the graphs that have vertex set V = {1, 2, 3, ..., n}.

The problem is to let V = {1, 2, 3, ..., n}, and to determine the number of different graphs that can be formed with V as vertex set. See attached file for full problem description.

### How to Graph Equations and Determine the Slopes of Lines

Attached photo in word doc, please use zoom for better viewing.

### Density function

A random variable X has the density function f(x)=...Find E(e^2x/3) Please see attached for full question.

### Finding curves

Find the curve that passes through the points (3, 2) and has the property that if the tangent line is drawn at any point P on the curve, then the part of the tangent line that lies in the first quadrant is bisected at P.

### Critical Point Functions

(-x^2/2(x+1)^3/2) + (2x/(x+1)^1/2)=0 , Critical points are any point in the domain or a function where the derivative is undefined or equals zero. The critical points are candidates for maximum and minimum values. To find the critical points set: =0