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Functional Analysis

A Discussion On Rolle's Theorem

Please solve the following with as much explanation of each step as possible. Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, suppose that f(a) = f(b) and that c is a real number in the interval such that f'(c) = 0. Find an interval for the function g over which Rolle's

Poles & Singularities

Classify the behavior at infinity (analytic, pole, zero, or essential singularity; if a zero or pole, give its order) of the following functions: f1(z) = (z^3 + i)/z f2(z) = e^(tan(1/z))

Polar coordinates

X and y represents rectangular coordinates. What is the given equation using polar coordinates (r, theta). x^2 = 4y


Assume x>0 and find all x for which the given series converges... Please see attached for full questions.

Area, volume, and expantion

A.) let A be the area of a circle of radius r that is changing w/ respect to time. if dr/dt is a constant, is dA/dt a constant, explain. b.) let V be the volume of a sphere of radius r that is changing w/ respect to time. If dr/dt is constant, is dV/dt constant, explain. c.) All edges of a cube are expanding at a rate of

Cross Section, Bounded Region, Solid of Revolution - Washers or Disks

6. Sketch the region and then find the volume of the solid whose base is the given region and which has the property that each cross section to the x-axis is an equilateral triangle a) the region bounded by the curves {see attachment} 7. Same problem statement as #6 above, except cross-section to the x-axis is a semi-circle:

Functional Analysis : Continuity, Graphing

Problem: Let f be that function defined by setting (Please see the attached file for the fully formatted problem.) a. Describe graphically f(x). b. At what points is f continuous?

Bounded Numbers

Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all numbers -x where x E A. Prove that inf A = -sup(-A) (Please see the attached file for the fully formatted problem.) Included in the attachment is a copy of the solution, but please explain in your own words how the proof works; don't j


Part A: A soap bubble is given an initial velocity downward from a height of H directly above a vent. Warm air from the vent gently applies a small constant upward force on the soap bubble. As a result, the soap bubble descends only to within a few centimeters of the vent before turning around. For the entire motion o fthe s

Polar coordinates

Polar coordinates o a particular point are r=4, 0=pi/3. I need to ind the rectangular coordinates of the point. (x,y)=?

Mapping (Quarter-Plane; Half-Line)

Find the image of the quarter-plane {see attachment} under the mapping {see attachment}. Show graphs (shaded regions) in the w-plane and identify the images of the half-lines {see attachment}.


Please see the attached file for full problem description. - Prove that f is continuous at 0 - Determine whether f is differentiable at 0, give a careful proof


Sketch the region consisting of all points (x, y) such that x2 + y2. You need to include all steps in calculating the resulting region. (See attachment for full question)

Consider the shaded doman D

Problem 14 ONLY 14) Consider the shaded doman D in the attached figure bounded by the simple closed... (see attachment)

Finding Extrema: Lagrange Multipliers

Use the method of Lagrange multipliers to find the indicated extremum. Let f(x,y) = 8x^2 - 24xy + y^2. Find the maximum and minimum values of the function f(x,y) subject to the constraint 8x^2 + y^2 = 1.

Minimizing F(x)

Let Q=(0,7) and R=(10,11) be given points in the plane. We want to find the point P=(x,0) on the x-axis such that the sum of distances PQ+PR is as small as possible. To solve this problem, we need to minimize the following function of x. F(x)=?? over the closed interval [a,b] where a=?? and b=??

Min/Max problem

Find the length of the shortest line from the origin to the line y=1-6x.

Convergence Problem

Please tell me if the following (see attached) series converge. This is a problem from my text book and unfortunately there are no solutions for this problem. I need justification, not just a yes it converges or no it doesn't answer.

Help with setting up problem

In the problem we are asked to solve for x and y. I can solve for the triangles but not able to use them to find the answer. can someone guide me.

Help with setting up problem

I would like some assistance in setting up this problem to solve. In the diagram Theta is 50 degrees and x is the unknown. Thank You