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Functional Analysis

Please explain in detail the following:

Please explain step by step how to complete the following: In the following determine graphically whether the equation could possibly be an identity. If it could, prove that it is. 5. sin^4 t - cos^4 t = 2 sin² t -1 ans. - sin^4t - cos^4 t = (sin²2t - cos²t) (sin²t + cos²t) = (sin²t-(1-sin²t)) (1) = 2 sin²t -

Please explain in detail

24. Graph the following parabola clearly labeling exact points for the vertx, x-intercepts(s), and y-intercept(s): y=2xsquare+4x-3

Uniform Convergence

Please see the attached file for the fully formatted problems. We are using the book Methods of Real Analysis by Richard R. Goldberg.

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

Mean value theorem

Find the points guaranteed by the mean value theorom for F(x) = x^.5 - 2x on the closed interval [ 0 , 4 ]

Find number of degrees...

Use the equation s=rO for the following: 1. Find the number of degrees in the central angle of a circle with radius of 20 inches if the angle subtends an arc of 13 inches. 2. Find the length of a degree on the equator considering the diameter of the equator to be 7912 miles. 3. A bicycle has a 28 inch wheel. How man

Outer Measure

Let A = union ( i from 1 to infinity) of M_i, Mi's are disjoint, show that m*(A) = sum (i from 1 to infinity) of |M_i| m*(A) is the outer measure of A, that is, m*(A) = inf sum (i from 1 to infinity) of M_i. PLEASE NOTICE THE = SIGN, A = the union, not a subset of the union.

Intermediate value theorem

Determine if the following conforms to the intermediate value theorem. Clearly give the conditions and, if appropriate, find an appropriate value for c. f(x)=10/((x^2)+1) [0,1] k=8

Volumes of revolution

Find the volume of the object rotated by the given points. 1. y=1/x, y = 0, x = 1, x = 3; about y = -1 2. y = x, y = 0, x = 2, x = 4; about x = 1

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))

Find a horizontal line that divides an area into two equal parts.

Find a horizontal line that divides an area between y=x^2 and y=9 into two equal parts. I need help in approaching this problem. I know how to calculate the area. I divided the area by two and through trial and error I found the horizontal line to be close to y=5.6. I don't know how to get an exact answer.

Polar coordinates

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

Polar coordinates

The polar coordinates of a point (-1, -pi/3), find the rectangular coordinates for that point.

Convergence

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

Graphing

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}.

Continuty

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