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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

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

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}.

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.