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Graphs and Functions

Continuity and Differentiability : Finding Values that Make a Function Differenentiable

Please explain how to find a and b such that f(x) = {a(x^3), x is less than or equal to 2 {x^2 + b, x is greater than 2 is differentiable everywhere

Tangent Line, Extrema, Intervals of Decrease and Concavity

Y=3x^4+4x^3-6x^2-12x What are the points where tangent line is horizontal? What is the relative maxima and minima? Intervals of decrease? Points of inflection? and interval(s) of concavity?

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

Value order to make a function continuous at a given point

If x<-4, f(x) = {[-2(x^3) - 6(x^2) +14x+24] / (x+4)} if x>=-4, f(x) = 5(x^2) +5x+a What value must be chosen for a in order to make this function continuous at -4? Please note the value of a does not equal to 66.

Evaluate the sum using generating functions.

Evaluate the sum (using generating functions) A) 0+3+12+...+3n2. B) 4x3x2x1+5x4x3x2+...+n(n-1)(n-2)(n-3)

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

For what values of a and b is the line 5x+y=b tangent to the curve y=ax2 at x=2? Find Horizontal asymptote of the function....

Please see the attached file for the fully formatted problems.

Borel Measurable and Borel Functions

1).Let f(X) : R -> R be the following: f(x) = { 1 if x is in Q (rationals) , 0 if x is not in Q ( irrational)} Prove that f(x) is Borel measurable ( Borel functions).

Evaluate graphs of derivative functions

(a) Suppose the graph in Figure 4.1.78 is that of a function g(x). Sketch the graph of the derivative g; (b) On the other hand, suppose the graph above is that of the derivative of a function f. For the interval ..., tell where the function f is (i) increasing; (ii) decreasing. (iii) Tell whether f has any extrema, and if so

Finding the derivative of a function given graphically

For the function of f, given below in graph (a) Sketch (b) Where does change its sign (c) Where does have local minima and maxima Using the graph of write a brief description of complete sentences to describe the relationship between the following features of the function of: (a) the local maxima and minima o

Heart Disease and Cancer : Plotting Graphs Using Excel, Trends and Making Predictions

You have been invited to present statistical information at a conference. To prepare, you must perform the following tasks: The following data was retrieved from www.cdc.gov. It represents the number of deaths in the United States due to heart Disease and cancer in each of the years; 1985, 1990, 1995, and 2002. The f

Heart Disease and Cancer : Plotting Graphs, Trends and Making Predictions

You have been invited to present statistical information at a conference. To prepare, you must perform the following tasks: 1. The following data was retrieved from www.cdc.gov. It represents the number of deaths in the United States due to heart Disease and cancer in each of the years; 1985, 1990, 1995, and 2002. Year Di

Axis of Symmetry Identifications

Identify the axis of symmetry, create a suitable table of values, then sketch the graph (including the axis of symmetry). y = -x2 + 3x - 3 Here, a = -1, b = 3, c = -3 The axis of symmetry is x = -b/2a = -3 (-2) = 1.5 See attached file for full problem description, equations, charts and diagrams.

Identify the axis of symmetry, create a suitable table of values

Identify the axis of symmetry, create a suitable table of values, then sketch the graph (including the axis of symmetry). y = -x2 + 3x - 3

Find Domain, Graph, Height, Minimum Surface Area of a Box

Consider an open-top box with a square base and a volume of 108 cubic inches. Let x be the length of a side of the base. a) Calculate the height h as a function of x. Is this function even, odd, or neither? b) What is the domain of the function above? (Note that there may be physical and/or mathematical restrictions.)

How to find vertex and intercepts for quadratic equations

Problem: Find the vertex and intercepts for y = x^2 + x - 6

Uniform Convergence of Sequnece

Prove : Let f1,f2.... be a sequence of continuous functions convergent uniformly on a bounded closed interval [a,b] and let c E[a,b] . For n = 1,2,...., define ..... Then the sequence g1,g2.... converges uniformly on [a,b]. Is the same true if [a,b] is replaced by ? Please see the attached file for the fully formatte

Uniformly Convergent Sequence of Functions and Weierstrass Test

Find an example of a sequence of continuous functions on fn on [0,1] such that the series...converges uniformly on [0,1] but the series ... diverges. Is it a counterexample for the Weierstrass test? ---

Uniform Convergence of a Sequence of Functions

Prove the following theorem. Let f1,f2,f3.... be continuous functions on a closed bounded interval [a,b] . Then fn--->f uniformly on [a,b] if and only if fn(x)-->f(x) for every xn-->x such that xn,x E[a,b] . Please see the attached file for the fully formatted problems.

Functions, Roots, Convergence, Fixed Point Method

Consider the function f(x) = 2sinx + e^-x - 1 on the interval r E [?2,2]. If you plot the function, you will see that it has two roots on this interval (a) Write down a first order fixed point method for finding one of the two roots. (b) Will this fixed point method converge for both of the roots (Justify)? If it does not co

Proof of Uniform Continuity

Show that the function f(x) = &#8730;x is uniformly continuous on [0,&#8734;). Note: This is from a beginning analysis class. We can only use the definition of uniform continuity. (In other words, cannot use compactness, etc to prove) ---

Fixed Point : Fixed-Point Iteration and Error Estimate

Please see the attached file for the fully formatted problem.

Distance and graphing in 3-D space

1. Find the distance from the origin to the line passing through the point P(3,1,5) and having the direction vector v=2i-j+k. 2. Graph z=x^2 in space.

Graphing binomials from trinomials.

Using graphing to check your answers is helpful. When you factor a trinomial into two binomials, each binomial represents a linear relationship. If you plot the two binomials (which are just lines) on a graph, what do they have in common with a plot of the trinomial itself? More important than that, how can this information be u

The relation between u,v and w where u,v,w are not independent

Independence and relations Real Analysis Jacobians (II) If u = (x + y)/z, v = (y + z)/x, w = y(x + y + z)/xz Show that u,v,w are not independent. Also find the

Measurable Functions

Suppose u(x) : X--> R v(x) : X --> R Both u(x) and v(x) are measurable Let f(x) : x --> R^2 f(x) = (u(x), v(x) ) Then f (x) is measurable Now prove a generalization of the above. That is, prove: if u_1(x) : X--> R u_2(x): X--> R . . . . u_n(x) : X--> R u_1,.

Use the Mean-Value Theorem- continuously differentiable function

(See attached file for full problem description) Let a sequence xn be defined inductively by . Suppose that as and . Show that . (Note that " " refers to "little oh") HINT: Use the Mean-Value Theorem and assume that F is a continuously differentiable function.

Find the linear equation that expresses temperature in degrees Fahrenheit as a function of temperature in degrees Celsius.

In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? 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 0 Boiling