### 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

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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

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?

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

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

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) A) 0+3+12+...+3n2. B) 4x3x2x1+5x4x3x2+...+n(n-1)(n-2)(n-3)

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

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

Please see the attached file for the fully formatted problems.

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).

(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

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

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

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

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, then sketch the graph (including the axis of symmetry). y = -x2 + 3x - 3

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.)

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

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

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? ---

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.

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

Show that the function f(x) = √x is uniformly continuous on [0,∞). 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) ---

Please see the attached file for the fully formatted problem.

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.

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

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

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,.

(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.

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