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Graphs and Functions

Instantaneous Rate of Change

An object is moving along the straight line as follows. It starts at x = 0 and then it moves to the right to x = 2. Then the object moves to the left to x = - 3, and finally to the right to stop at x = 1. Sketch a possible graph of the position function s(t). Sketch a possible graph of the velocity v(t) (the instantaneous rate o

Graphing, Trends and Forecasting

Please see the attached file for the complete questions. Complete the ordered pairs so that each is a solution for the given equation. 24. (0, ), ( , ), ( ,0), ( , ) 46. Science and medicine. Celsius temperature readings can be converted to Fahrenheit readings using the formula . What is the Fahrenheit tempera

Graphing and Linear Equations

Please see the attached file for the complete questions. MTH 212 Unit 2 - Individual Project A 1. The following table shows the number of hours five car salespeople worked and the number of cars they sold. Using Excel, plot each point on the same graph where the first coordinate is the number of hours and the second coord

Over what intervals are the following functions continuous?

Please see attached file for full problem description. Over what intervals are the following functions continuous? Justify your answer using the definition of continuity. a. b. Let f and g be twice differentiable functions such that for all x in the domain of f. If and What (if anything) can you say about f,

Modelling Data with Polynomial and Rational Functions

Many different kinds of data can be modeled using polynomial functions. An example of a polynomial function would be gas mileage for an automobile. If we compare gas mileage at two different speeds, V1 and V2, the gas required varies as (V1/V2), raised to the third power, (V1/V2)3. Rational functions are also useful. For examp

Closed Loop Transfer Function

Please see the attached file for the fully formatted problems. Closed-Loop Transfer Function Find the state equation and output equation solution. Basically transfer from Transfer Function to State Space. Hint: Output Part of the answer has

Graphing an Eliipse and Interval Notation

Complete the square and sketch a graph of the following ellipse. Give the coordinates of the centre and the four vertices. What is the length of the major axis? 4x^2 + y^2- 8x + 4y -8 = 0 Solve and graph your solutions on the number line and express your answers in interval notation. 1. |3x - 7| >= 23 2. 1/2 |5-2x|

Applications of Graphs of Linear Equations

Name: __________________________ Date: _____________ 1. Which of the ordered pairs (6, 1), (8, 0), (4, -2), (-4, 6) are solutions for the equation x + 2y = 8? 2. A small company did a poll of how their employees commuted to work. The data is shown in the bar graph below. (a) How many people commute to work v

Trees, Graphs and Multigraphs

By contracting an edge e = uv, we mean removing e and identifying the vertices u and v as a single new vertex. Let num_T(G) denote the number of spanning trees of the graph G. a. Show that the following recursive formula holds: num_T(G) = num_T(G - e) + num_T (G * e) where G * e means the multigraph obtained from G by contrac

Graceful Trees and Paths

A (p,q) graph G is called graceful if it is possible to label the vertices of G with distinct elements from the set {0,1,...,q} in such a way that the induced edge labeling, which assigns the integer |i - j| to the edge ij, assigns the labels 1,2,...,q to the q edges of G. The graceful tree conjecture states that every tree i

Left-Side and Right-Side Continuous Functions

We define the floor function [[x]] to be the greatest integer not exceeding x. For example[[4]]=4 [[2.37]]=2 [[-1]]=1[[-1.2]]-2 Sketch by hand the graph y=[[x]] by first tabulating the values pf [[1]] for several numbers x. Then compare your graph with the plot from a graphing calculator. What are the discontinuities of f(x

Randomly Eulerian Graphs

Let G be a graph that is randomly eulerian from a vertex v. Show that if deg u = Delta(G)"max degree in G", then G is randomly eulerian from u.

Forests and Eulerian Graphs

Let F be a forest. Add a vertex x to F and join x to each vertex of odd degree in F. Prove that the graph obtained in this way is randomly Eulerian from x, and every graph randomly Eulerian from x can be obtained in this way.

Randomly Eulerian Graphs

Recall that a graph G is randomly Eulerian from a vertex x if and maximal trail starting at x in an Euler circuit. (If T = xx_1 ... x_l, then T is a maximal trail starting at x iff x_l is an isolated vertex in G - E(T).) Prove that a nonempty graph G is randomly Eulerian from x iff G has an Euler circuit and x is contained in ev

Eulerian Graphs

Give examples of eulerian grpahs that are randomly eulerian from exactly none, one, two or all of their vertices.

Eulerian Graph

Let G be an eulerian graph of order n >= 3. Prove that G is randomly eulerian from exactly none, one, two or all of its vertices.

Area of ellipse and slopes of f & f(inverse)

1) find the area of the enclosed region between the curve and the coordinate axes x^2/8 + y^2/6 =9 the graph is just the coordinate system with an ellipse draw over it, no numbers or letter are present 2) if f(x) =x^2 (x>=0) and f(inverse) = x^1/2, show that the slopes of the graphs of f(x) and f(inverse) are recipr

Vertex of a Parabola

I need to understand how to find the vertex of a parabola two ways. Show examples

Graphing Second-Order and Third-Order Rational Functions

I do not understand how to conduct these equations I Excel so that I can show a graph of each. Choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function. Provide a graph for the second-order rational function (e.g., x2), choosing x values in the range from -10 through +10.

Graphing and Tangents

A) Graph b) Draw tangent to graph @ points where x coord. are -2, 0, and 1 c) f(x) by determining Limit h>0 f^'(x+h) - f(x) / h d) f^' (-2), f^' (0), and f^' (1) (should meet slopes in part b) f(x) = 1/2 x^2

Graphs and Linear Equations

1. The following table shows the height of a tree as it ages. In Excel, plot each point on the same graph where the first coordinate is the age of the tree and the second coordinate is the height of the tree (age, height). After plotting each point, explain if there is a linear relationship between the age and height of the tr

Graphs and Limits

Graph each function then find specified limits. Exist? f(x)=x^2 ; Find lim x--> -1 f(x) and lim x--->0 f(x)