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# Graphs and Functions

### Functions

Suppose you throw a baseball straight up at a velocity of 64 feet per second. A function can be created by expressing distance above the ground, s, as a function of time, t. This function is s = -16t2 + v0t + s0 16 represents 1/2g, the gravitational pull due to gravity (measured in feet per second2). v0 is the initial veloci

### Function

For the function y = x2 - 4x - 5 How do I put the functionin the from of y = x2 - 4x - 5? and what would the line of symmetry be?

### Expressing a Word Problem as an Equation

If it is 300 miles from Chicago to St. Louis in a car traveling a constant speed of 60 mph, how do I write a linear function that expresses the distance to be traveled to reach St. Louis, s, as a function of time, t?

### Value of Y in a function

For the function y= 1/x-1 Give the y values for x=-2,-1,0,1,2,3?

### Revenue and Cost Functions and Break-Even Point

A revenue function is R(x) = 22x and a cost function is C(x) = -9x + 341. The break even point is: A. (17, 374) B. (242, 9) C. (11, 242) D. (31.5, 661.5)

### Find the equation of a line given two points : (8,6) and (2,-4)

Problem: The equation of the line through (8,6) and (2,-4) is A. 5x-3y = 22 B. y=3/5x = 8/5 C. 3x=4y = 48 D. -4x = 2y = -16

### Function : f(x) = (x+2)(2x-3). f(-2) =

Problem: f(x) = (x+2)(2x-3). f(-2) = A. 0 B. 4 C. -6 D. -14

### Prove Composition of Functions are Associative

Prove, from the definition of function (using ordered pairs), that composition of functions is associative. (i.e. prove that f * (g*h) = (f* g) *h) for suitable functions f, g, h I would like to know how to use ordered pairs to proof the associative of composited functions.

### Differentiability, Bounded Above and Supremums

1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a belongs to A and b belongs to B}. (a) Show that if A is open, then A+B is open. (b) If A and B are both closed, is A+B closed? Justify your answer. 2. Let f be differentiable for x > a and A as x --> infinity. Prove that there is a sequence x_n --> infi

### Properties of additive functions; Bounded; Continuous; Measurable

Let f : R --> R be an additive function i.e. f(x+y) = f(x) + f(y) for all x,y in R. 1. If f is bounded at a point, then f is continuous at that point. 2. If f is measurable, then f is linear i.e. f(x) = cx for some c in R. I have already proved that f is continuous if and only if f is linear, and I have proven that if f i

### Length of Polygonal Line Segments and Length of a Curve and Distance Formula

Find the total length of the polygonal line segments joining the points (xi, f(xi),i=0, 1,...,n, zwhere a= x0,x1,... xn=b is a regular partition of (a,b). use the indicated values for n (1) f(x) = sqrt x, a=0,b=4 (a) n=2, (b) n=4 (2) f(x) = sin^2 x, a=0, b= 2pi (a) n=2 (b) n=4 (c) n=8 (3) Use a y integration to find

### Solve the following linear program using the graphical solution

(See attached file for full problem description with proper equations and diagrams) --- Graphical solution procedure Please help solve this linear problem in the attachment using the graphical solution procedure & graph the feasible region: Solve the following linear program using the graphical solution procedure: M

### Directed Graphs, Vertices and Distinct Paths

6. Recall that R^3={(x,y,z):x,y,z(subset of R)}. Let G(V,E) be a directed graph, in which V= {(x,y,z)-(subset of R^3) :x,y,z(subset of R),-10<=x,y,z<=10}. Suppose that for any vertex, v=(x,y,z)--[subset of V], the only edges originating at v are the ones joining v to (x+1,y,z),(x,y+1,z),(x,y,z+1) . i.e. any path that originate

### What are all the intercepts of the graph of...? (6 Problems)

20. At what points does the graph of y = x^2 - 3x -10 cross the x-axis 21. What are all of the intercepts of the graph of y = 15x^2 + 89x - 6? 22. What are all the intercepts of the graph of y = 2x^2 - 11x + 5? 23. What are all the intercepts of the graph of y = 6x^2 + 13x + 6? 24. What are all the

### Maximum Modulus Theorem Problem

Let f be analytic in the disk B(0;R) and for 0 =< r < R define A(r) = max { Re f(z) : |z| = r}. Show that unless f is a constant, A(r) is a strictly increasing function of r. Please justify every step and claim and show how you used all what is given. Also refer to theorems or lemmas used in the proof. The section where I

### Problem Set

Solve this inequality state the solution set using interval notation and graph the solution # 26. 3 2 ---- > ------- x + 2 x - 1 Page 576 Find the vertex and intercepts for each quadratic function and sketch its graph. # 49. y = x^2 - 4x - 12 # 50. y = x^2 + 2x - 24

### Quadratic Function : Graphing and Maximizing Profit

The total profit, p(x) in dollars for a company to manufacture and sell x items per week is given by the function p(x)=-x^2+50x - Graph the function and label the x and y-intercepts on a scale where y-axis ranges from -100 to 900 and x-axis ranges from 0 to 55 - What is the maximum profit earned by the company in a week?

### How do you prove on the basis of the definition, the function

My definition goes: If the function has its derivatives at point (a,b) then the function is differentible at (a,b) How do you prove on the basis of the definition, the function f defined by f(x,y)=xy(x+y) is differentiable at every point of its domain?

### Properties of Continuous Functions : Intermediate Value Theorem

Suppose that functions f,g : [a,b] -> R are continuous, satisfy f(a) <= g(a) and f(b) >= g(b). Then there exists a real number c in [a,b] such that f(c) = g(c). Label the statement as true or false. If it is true, prove it. If not, give an example of why it is false and if possible, correct it to make it true.

### Graphing using a Lagrange Interpolating Polynomial and Piecewise Linear Function

1. For medical purposes the level of sugar was measured in blood (in mg/dl). The samples were taken with 1 2hr increments, as the following table shows: initial sample 96 mg/dl after 30 min. 133 mg/dl after 60 min. 142 mg/dl after 90 min. 81 mg/dl after 120 min. 87 mg/dl Graph in MATLAB sugar curves corresponding to thes

### Pointwise Operations and Characteristic Functions

Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) =

### Operations of Functions: Injective, Surjective and Bijective

Let f: A ?> B and g: B ?> C be functions. a) Suppose that f and g are injective. Show that g o f is injective. b) Suppose that f and g are surjective. Show that g o f is surjective. c) Suppose that f and g are bijective. Then g o f is bijective by parts a) and b). Show that (gof)^-1 = f^-1 o g^-1. Please do number 3 in at

### Pointwise Operations and Characteristic Functions

Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) =

### What is the expected yield on notes from year 1 to 2, assuming the PEH holds? If inflation for the next year is expected to be 4.5%, what would the expected real rate of interest be on the 3-month T-bill? Plot the yield curve for these Treasury securities. Using the three theories we have discussed, explain how each one helps explain the shape of this yield curve.

1. If U.S. Treasury yields are as follows: 3 month 6.0% 6 month 6.3% 1 year 6.5% 2 year 6.6% 5 year 6.4% 10 year 7.5% 30 year 8.0% a. What is the expected yield on notes from year 1 to 2, assuming the PEH holds? b. What is the expected yield on notes from year 2 to 5, assuming th

### Algebra Review

(See attached file for full problem description with equations and diagrams) --- 1. Simplification of linear algebraic expressions and expressions with fractional coefficients and solve x; 2. Solving simple linear equations with fractional coefficient: 3. Solving inequalities with fractional coefficient: 4.

### Parabola and tangent lines

The parabola y= x^2+4 has two tangents which pass through the point (0,-2). One is tangent to the parabola at (a, a^2+4) and the other at (-a, a^2+4) where a is a certain positive number. The question is a=?

### Relations Vs Functions and Celsius & Fahrenheit Temperature Conversion

1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? 2. There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahrenheit Celsius Freezing point of water 32

### Solve Finite Difference Equation

See attached file for full problem description with equation. --- Find analytically the solution of this difference equation with the given initial values: Without computing the solution recursively, predict whether such a computation would be stable. (Note: A numerical process is unstable if small errors made at one

### Fashion Plotting

A new fashion in clothes is introduced. It spreads slowly through the population at first but then speeds up as more people become aware of it. Eventually those willing to try the new fashions begin to dry up and while the number of people adopting the fashion continues to increase. It does so at a decreasing rate. Later the fas

### Relations Versus Functions : Real-Life Application and Celsius-Fahrenheit Temperature Conversion

1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? 2. There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahren